VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.3% → 99.6%

Time: 23.6s

Alternatives: 31

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -110000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 17500000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x \cdot \cos B}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \cos B \cdot \frac{x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -110000000.0)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 17500000.0)
     (-
      (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5))
      (/ (* x (cos B)) (sin B)))
     (- (/ 1.0 (sin B)) (* (cos B) (/ x (sin B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -110000000.0) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 17500000.0) {
		tmp = ((F / sin(B)) * pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - ((x * cos(B)) / sin(B));
	} else {
		tmp = (1.0 / sin(B)) - (cos(B) * (x / 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 (f <= (-110000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 17500000.0d0) then
        tmp = ((f / sin(b)) * ((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0))) - ((x * cos(b)) / sin(b))
    else
        tmp = (1.0d0 / sin(b)) - (cos(b) * (x / sin(b)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -110000000.0) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 17500000.0) {
		tmp = ((F / Math.sin(B)) * Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - ((x * Math.cos(B)) / Math.sin(B));
	} else {
		tmp = (1.0 / Math.sin(B)) - (Math.cos(B) * (x / Math.sin(B)));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -110000000.0:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 17500000.0:
		tmp = ((F / math.sin(B)) * math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - ((x * math.cos(B)) / math.sin(B))
	else:
		tmp = (1.0 / math.sin(B)) - (math.cos(B) * (x / math.sin(B)))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -110000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 17500000.0)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5)) - Float64(Float64(x * cos(B)) / sin(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(cos(B) * Float64(x / sin(B))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -110000000.0)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 17500000.0)
		tmp = ((F / sin(B)) * ((((F * F) + 2.0) + (x * 2.0)) ^ -0.5)) - ((x * cos(B)) / sin(B));
	else
		tmp = (1.0 / sin(B)) - (cos(B) * (x / sin(B)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -110000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 17500000.0], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[B], $MachinePrecision] * N[(x / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.1e8

   1. Initial program 58.6%

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

   3. Simplified 74.3%

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

   5. Taylor expanded in x around 0 74.4%

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

      1. associate-*l/ 74.3%

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

      2. *-lft-identity 74.3%

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

      3. +-commutative 74.3%

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

      4. unpow2 74.3%

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

      5. fma-udef 74.3%

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

   7. Simplified 74.3%

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

   8. Taylor expanded in F around -inf 99.8%

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

   if -1.1e8 < F < 1.75e7

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 99.7%

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

   if 1.75e7 < F

   1. Initial program 58.2%

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

   3. Simplified 74.7%

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

   5. Taylor expanded in F around inf 99.6%

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

      1. associate-/r* 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in F around 0 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*r/ 99.9%

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

   10. Simplified 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -110000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 17500000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x \cdot \cos B}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \cos B \cdot \frac{x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos B \cdot \frac{x}{\sin B}\\ \mathbf{if}\;F \leq -118000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 35000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (cos B) (/ x (sin B)))))
   (if (<= F -118000000.0)
     (- (/ -1.0 (sin B)) (/ x (tan B)))
     (if (<= F 35000000.0)
       (- (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5)) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = cos(B) * (x / sin(B));
	double tmp;
	if (F <= -118000000.0) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 35000000.0) {
		tmp = ((F / sin(B)) * pow((((F * F) + 2.0) + (x * 2.0)), -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 = cos(b) * (x / sin(b))
    if (f <= (-118000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 35000000.0d0) then
        tmp = ((f / sin(b)) * ((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-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 = Math.cos(B) * (x / Math.sin(B));
	double tmp;
	if (F <= -118000000.0) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 35000000.0) {
		tmp = ((F / Math.sin(B)) * Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = math.cos(B) * (x / math.sin(B))
	tmp = 0
	if F <= -118000000.0:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 35000000.0:
		tmp = ((F / math.sin(B)) * math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(cos(B) * Float64(x / sin(B)))
	tmp = 0.0
	if (F <= -118000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 35000000.0)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -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 = cos(B) * (x / sin(B));
	tmp = 0.0;
	if (F <= -118000000.0)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 35000000.0)
		tmp = ((F / sin(B)) * ((((F * F) + 2.0) + (x * 2.0)) ^ -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[(N[Cos[B], $MachinePrecision] * N[(x / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -118000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 35000000.0], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.18e8

   1. Initial program 58.6%

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

   3. Simplified 74.3%

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

   5. Taylor expanded in x around 0 74.4%

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

      1. associate-*l/ 74.3%

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

      2. *-lft-identity 74.3%

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

      3. +-commutative 74.3%

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

      4. unpow2 74.3%

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

      5. fma-udef 74.3%

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

   7. Simplified 74.3%

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

   8. Taylor expanded in F around -inf 99.8%

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

   if -1.18e8 < F < 3.5e7

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 99.7%

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

      1. associate-*l/ 99.6%

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

      2. *-commutative 99.6%

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

   5. Simplified 99.6%

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

   if 3.5e7 < F

   1. Initial program 58.2%

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

   3. Simplified 74.7%

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

   5. Taylor expanded in F around inf 99.6%

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

      1. associate-/r* 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in F around 0 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*r/ 99.9%

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

   10. Simplified 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -118000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 35000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \cos B \cdot \frac{x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \cos B \cdot \frac{x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -5 \cdot 10^{+37}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 47000000:\\ \;\;\;\;\frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \cos B \cdot \frac{x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -5e+37)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 47000000.0)
       (- (/ F (/ (sin B) (pow (fma F F 2.0) -0.5))) t_0)
       (- (/ 1.0 (sin B)) (* (cos B) (/ x (sin B))))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -5e+37) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 47000000.0) {
		tmp = (F / (sin(B) / pow(fma(F, F, 2.0), -0.5))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - (cos(B) * (x / sin(B)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -5e+37)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 47000000.0)
		tmp = Float64(Float64(F / Float64(sin(B) / (fma(F, F, 2.0) ^ -0.5))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(cos(B) * Float64(x / sin(B))));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -5e+37], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 47000000.0], N[(N[(F / N[(N[Sin[B], $MachinePrecision] / N[Power[N[(F * F + 2.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[B], $MachinePrecision] * N[(x / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.99999999999999989e37

   1. Initial program 52.7%

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

   3. Simplified 70.7%

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

   5. Taylor expanded in x around 0 70.8%

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

      1. associate-*l/ 70.7%

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

      2. *-lft-identity 70.7%

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

      3. +-commutative 70.7%

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

      4. unpow2 70.7%

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

      5. fma-udef 70.7%

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

   7. Simplified 70.7%

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

   8. Taylor expanded in F around -inf 99.8%

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

   if -4.99999999999999989e37 < F < 4.7e7

   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 x around 0 99.4%

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

      1. associate-*l/ 99.5%

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

      2. *-lft-identity 99.5%

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

      3. +-commutative 99.5%

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

      4. unpow2 99.5%

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

      5. fma-udef 99.5%

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

   7. Simplified 99.5%

      \[\leadsto F \cdot \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}} - \frac{x}{\tan B} \]
   8. Step-by-step derivation

      1. clear-num 99.5%

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

      2. un-div-inv 99.5%

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

      3. inv-pow 99.5%

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

      4. sqrt-pow1 99.5%

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

      5. metadata-eval 99.5%

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

   9. Applied egg-rr 99.5%

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

   if 4.7e7 < F

   1. Initial program 58.2%

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

   3. Simplified 74.7%

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

   5. Taylor expanded in F around inf 99.6%

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

      1. associate-/r* 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in F around 0 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*r/ 99.9%

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

   10. Simplified 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{+37}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 47000000:\\ \;\;\;\;\frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \cos B \cdot \frac{x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.4 \cdot 10^{+48}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 47000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} + x \cdot \frac{-1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \cos B \cdot \frac{x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.4e+48)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 47000000.0)
     (+
      (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5))
      (* x (/ -1.0 (tan B))))
     (- (/ 1.0 (sin B)) (* (cos B) (/ x (sin B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.4e+48) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 47000000.0) {
		tmp = ((F / sin(B)) * pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) + (x * (-1.0 / tan(B)));
	} else {
		tmp = (1.0 / sin(B)) - (cos(B) * (x / 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 (f <= (-3.4d+48)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 47000000.0d0) then
        tmp = ((f / sin(b)) * ((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0))) + (x * ((-1.0d0) / tan(b)))
    else
        tmp = (1.0d0 / sin(b)) - (cos(b) * (x / sin(b)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.4e+48) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 47000000.0) {
		tmp = ((F / Math.sin(B)) * Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) + (x * (-1.0 / Math.tan(B)));
	} else {
		tmp = (1.0 / Math.sin(B)) - (Math.cos(B) * (x / Math.sin(B)));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.4e+48:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 47000000.0:
		tmp = ((F / math.sin(B)) * math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) + (x * (-1.0 / math.tan(B)))
	else:
		tmp = (1.0 / math.sin(B)) - (math.cos(B) * (x / math.sin(B)))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.4e+48)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 47000000.0)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5)) + Float64(x * Float64(-1.0 / tan(B))));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(cos(B) * Float64(x / sin(B))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -3.4e+48)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 47000000.0)
		tmp = ((F / sin(B)) * ((((F * F) + 2.0) + (x * 2.0)) ^ -0.5)) + (x * (-1.0 / tan(B)));
	else
		tmp = (1.0 / sin(B)) - (cos(B) * (x / sin(B)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.4e+48], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 47000000.0], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] + N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[B], $MachinePrecision] * N[(x / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.4000000000000003e48

   1. Initial program 51.1%

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

   3. Simplified 69.7%

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

   5. Taylor expanded in x around 0 69.8%

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

      1. associate-*l/ 69.8%

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

      2. *-lft-identity 69.8%

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

      3. +-commutative 69.8%

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

      4. unpow2 69.8%

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

      5. fma-udef 69.8%

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

   7. Simplified 69.8%

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

   8. Taylor expanded in F around -inf 99.8%

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

   if -3.4000000000000003e48 < F < 4.7e7

   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 4.7e7 < F

   1. Initial program 58.2%

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

   3. Simplified 74.7%

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

   5. Taylor expanded in F around inf 99.6%

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

      1. associate-/r* 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in F around 0 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*r/ 99.9%

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

   10. Simplified 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.4 \cdot 10^{+48}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 47000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} + x \cdot \frac{-1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \cos B \cdot \frac{x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.037:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 47000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} + \frac{-1}{\frac{\tan B}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \cos B \cdot \frac{x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.037)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 47000000.0)
     (+
      (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5))
      (/ -1.0 (/ (tan B) x)))
     (- (/ 1.0 (sin B)) (* (cos B) (/ x (sin B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.037) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 47000000.0) {
		tmp = ((F / sin(B)) * pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) + (-1.0 / (tan(B) / x));
	} else {
		tmp = (1.0 / sin(B)) - (cos(B) * (x / 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 (f <= (-0.037d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 47000000.0d0) then
        tmp = ((f / sin(b)) * ((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0))) + ((-1.0d0) / (tan(b) / x))
    else
        tmp = (1.0d0 / sin(b)) - (cos(b) * (x / sin(b)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.037) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 47000000.0) {
		tmp = ((F / Math.sin(B)) * Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) + (-1.0 / (Math.tan(B) / x));
	} else {
		tmp = (1.0 / Math.sin(B)) - (Math.cos(B) * (x / Math.sin(B)));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -0.037:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 47000000.0:
		tmp = ((F / math.sin(B)) * math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) + (-1.0 / (math.tan(B) / x))
	else:
		tmp = (1.0 / math.sin(B)) - (math.cos(B) * (x / math.sin(B)))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.037)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 47000000.0)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5)) + Float64(-1.0 / Float64(tan(B) / x)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(cos(B) * Float64(x / sin(B))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -0.037)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 47000000.0)
		tmp = ((F / sin(B)) * ((((F * F) + 2.0) + (x * 2.0)) ^ -0.5)) + (-1.0 / (tan(B) / x));
	else
		tmp = (1.0 / sin(B)) - (cos(B) * (x / sin(B)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.037], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 47000000.0], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[B], $MachinePrecision] * N[(x / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.0369999999999999982

   1. Initial program 59.1%

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

   3. Simplified 74.6%

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

   5. Taylor expanded in x around 0 74.8%

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

      1. associate-*l/ 74.7%

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

      2. *-lft-identity 74.7%

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

      3. +-commutative 74.7%

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

      4. unpow2 74.7%

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

      5. fma-udef 74.7%

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

   7. Simplified 74.7%

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

   8. Taylor expanded in F around -inf 99.8%

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

   if -0.0369999999999999982 < F < 4.7e7

   1. Initial program 99.5%

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

      1. div-inv 99.6%

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

      2. clear-num 99.5%

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

   4. Applied egg-rr 99.5%

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

   if 4.7e7 < F

   1. Initial program 58.2%

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

   3. Simplified 74.7%

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

   5. Taylor expanded in F around inf 99.6%

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

      1. associate-/r* 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in F around 0 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*r/ 99.9%

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

   10. Simplified 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.037:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 47000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} + \frac{-1}{\frac{\tan B}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \cos B \cdot \frac{x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ t_1 := \frac{1}{\sin B}\\ t_2 := {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \left(F \cdot t\_1\right) - \frac{x}{B}\\ \mathbf{if}\;F \leq -0.037:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq -2.3 \cdot 10^{-146}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;F \leq 3 \cdot 10^{-33}:\\ \;\;\;\;\frac{F}{\sin B \cdot \left(F + \frac{1}{F}\right)} - t\_0\\ \mathbf{elif}\;F \leq 7500000:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1 - \cos B \cdot \frac{x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B)))
        (t_1 (/ 1.0 (sin B)))
        (t_2
         (- (* (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5) (* F t_1)) (/ x B))))
   (if (<= F -0.037)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F -2.3e-146)
       t_2
       (if (<= F 3e-33)
         (- (/ F (* (sin B) (+ F (/ 1.0 F)))) t_0)
         (if (<= F 7500000.0) t_2 (- t_1 (* (cos B) (/ x (sin B))))))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double t_1 = 1.0 / sin(B);
	double t_2 = (pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F * t_1)) - (x / B);
	double tmp;
	if (F <= -0.037) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= -2.3e-146) {
		tmp = t_2;
	} else if (F <= 3e-33) {
		tmp = (F / (sin(B) * (F + (1.0 / F)))) - t_0;
	} else if (F <= 7500000.0) {
		tmp = t_2;
	} else {
		tmp = t_1 - (cos(B) * (x / sin(B)));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = x / tan(b)
    t_1 = 1.0d0 / sin(b)
    t_2 = (((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0)) * (f * t_1)) - (x / b)
    if (f <= (-0.037d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= (-2.3d-146)) then
        tmp = t_2
    else if (f <= 3d-33) then
        tmp = (f / (sin(b) * (f + (1.0d0 / f)))) - t_0
    else if (f <= 7500000.0d0) then
        tmp = t_2
    else
        tmp = t_1 - (cos(b) * (x / sin(b)))
    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 t_2 = (Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F * t_1)) - (x / B);
	double tmp;
	if (F <= -0.037) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= -2.3e-146) {
		tmp = t_2;
	} else if (F <= 3e-33) {
		tmp = (F / (Math.sin(B) * (F + (1.0 / F)))) - t_0;
	} else if (F <= 7500000.0) {
		tmp = t_2;
	} else {
		tmp = t_1 - (Math.cos(B) * (x / Math.sin(B)));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	t_1 = 1.0 / math.sin(B)
	t_2 = (math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F * t_1)) - (x / B)
	tmp = 0
	if F <= -0.037:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= -2.3e-146:
		tmp = t_2
	elif F <= 3e-33:
		tmp = (F / (math.sin(B) * (F + (1.0 / F)))) - t_0
	elif F <= 7500000.0:
		tmp = t_2
	else:
		tmp = t_1 - (math.cos(B) * (x / math.sin(B)))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	t_1 = Float64(1.0 / sin(B))
	t_2 = Float64(Float64((Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5) * Float64(F * t_1)) - Float64(x / B))
	tmp = 0.0
	if (F <= -0.037)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= -2.3e-146)
		tmp = t_2;
	elseif (F <= 3e-33)
		tmp = Float64(Float64(F / Float64(sin(B) * Float64(F + Float64(1.0 / F)))) - t_0);
	elseif (F <= 7500000.0)
		tmp = t_2;
	else
		tmp = Float64(t_1 - Float64(cos(B) * Float64(x / sin(B))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	t_1 = 1.0 / sin(B);
	t_2 = (((((F * F) + 2.0) + (x * 2.0)) ^ -0.5) * (F * t_1)) - (x / B);
	tmp = 0.0;
	if (F <= -0.037)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= -2.3e-146)
		tmp = t_2;
	elseif (F <= 3e-33)
		tmp = (F / (sin(B) * (F + (1.0 / F)))) - t_0;
	elseif (F <= 7500000.0)
		tmp = t_2;
	else
		tmp = t_1 - (cos(B) * (x / sin(B)));
	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]}, Block[{t$95$2 = N[(N[(N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.037], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -2.3e-146], t$95$2, If[LessEqual[F, 3e-33], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * N[(F + N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 7500000.0], t$95$2, N[(t$95$1 - N[(N[Cos[B], $MachinePrecision] * N[(x / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.0369999999999999982

   1. Initial program 59.1%

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

   3. Simplified 74.6%

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

   5. Taylor expanded in x around 0 74.8%

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

      1. associate-*l/ 74.7%

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

      2. *-lft-identity 74.7%

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

      3. +-commutative 74.7%

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

      4. unpow2 74.7%

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

      5. fma-udef 74.7%

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

   7. Simplified 74.7%

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

   8. Taylor expanded in F around -inf 99.8%

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

   if -0.0369999999999999982 < F < -2.3000000000000001e-146 or 3.0000000000000002e-33 < F < 7.5e6

   1. Initial program 99.5%

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

   3. Taylor expanded in B around 0 87.0%

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

      1. div-inv 87.2%

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

   5. Applied egg-rr 87.2%

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

   if -2.3000000000000001e-146 < F < 3.0000000000000002e-33

   1. Initial program 99.4%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 99.6%

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

      1. associate-*l/ 99.6%

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

      2. *-lft-identity 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. fma-udef 99.6%

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

   7. Simplified 99.6%

      \[\leadsto F \cdot \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}} - \frac{x}{\tan B} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 90.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(F \cdot \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}\right)\right)} - \frac{x}{\tan B} \]

      2. expm1-udef 83.4%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(F \cdot \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}\right)} - 1\right)} - \frac{x}{\tan B} \]

      3. associate-*r/ 83.4%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{F \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}}\right)} - 1\right) - \frac{x}{\tan B} \]

      4. sqrt-div 83.4%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}}{\sin B}\right)} - 1\right) - \frac{x}{\tan B} \]

      5. metadata-eval 83.4%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F \cdot \frac{\color{blue}{1}}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}\right)} - 1\right) - \frac{x}{\tan B} \]

      6. un-div-inv 83.4%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}}{\sin B}\right)} - 1\right) - \frac{x}{\tan B} \]

   9. Applied egg-rr 83.4%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}\right)} - 1\right)} - \frac{x}{\tan B} \]
   10. Step-by-step derivation

       1. expm1-def 90.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}\right)\right)} - \frac{x}{\tan B} \]

       2. expm1-log1p 99.6%

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

       3. associate-/l/ 99.7%

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

   11. Simplified 99.7%

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

   12. Taylor expanded in F around inf 91.3%

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

   if 7.5e6 < F

   1. Initial program 58.2%

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

   3. Simplified 74.7%

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

   5. Taylor expanded in F around inf 99.6%

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

      1. associate-/r* 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in F around 0 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*r/ 99.9%

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

   10. Simplified 99.9%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.037:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -2.3 \cdot 10^{-146}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \left(F \cdot \frac{1}{\sin B}\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3 \cdot 10^{-33}:\\ \;\;\;\;\frac{F}{\sin B \cdot \left(F + \frac{1}{F}\right)} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 7500000:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \left(F \cdot \frac{1}{\sin B}\right) - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \cos B \cdot \frac{x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.75:\\ \;\;\;\;\frac{-1}{\sin B} - t\_1\\ \mathbf{elif}\;F \leq 1.75:\\ \;\;\;\;F \cdot \left(t\_0 \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\right) - t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 - \cos B \cdot \frac{x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))) (t_1 (/ x (tan B))))
   (if (<= F -1.75)
     (- (/ -1.0 (sin B)) t_1)
     (if (<= F 1.75)
       (- (* F (* t_0 (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))))) t_1)
       (- t_0 (* (cos B) (/ x (sin B))))))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / sin(B);
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -1.75) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= 1.75) {
		tmp = (F * (t_0 * sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_1;
	} else {
		tmp = t_0 - (cos(B) * (x / sin(B)));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 / sin(b)
    t_1 = x / tan(b)
    if (f <= (-1.75d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_1
    else if (f <= 1.75d0) then
        tmp = (f * (t_0 * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))))) - t_1
    else
        tmp = t_0 - (cos(b) * (x / sin(b)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = 1.0 / Math.sin(B);
	double t_1 = x / Math.tan(B);
	double tmp;
	if (F <= -1.75) {
		tmp = (-1.0 / Math.sin(B)) - t_1;
	} else if (F <= 1.75) {
		tmp = (F * (t_0 * Math.sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_1;
	} else {
		tmp = t_0 - (Math.cos(B) * (x / Math.sin(B)));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = 1.0 / math.sin(B)
	t_1 = x / math.tan(B)
	tmp = 0
	if F <= -1.75:
		tmp = (-1.0 / math.sin(B)) - t_1
	elif F <= 1.75:
		tmp = (F * (t_0 * math.sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_1
	else:
		tmp = t_0 - (math.cos(B) * (x / math.sin(B)))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / sin(B))
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.75)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= 1.75)
		tmp = Float64(Float64(F * Float64(t_0 * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0)))))) - t_1);
	else
		tmp = Float64(t_0 - Float64(cos(B) * Float64(x / sin(B))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = 1.0 / sin(B);
	t_1 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.75)
		tmp = (-1.0 / sin(B)) - t_1;
	elseif (F <= 1.75)
		tmp = (F * (t_0 * sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_1;
	else
		tmp = t_0 - (cos(B) * (x / sin(B)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.75], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, 1.75], N[(N[(F * N[(t$95$0 * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(t$95$0 - N[(N[Cos[B], $MachinePrecision] * N[(x / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.75

   1. Initial program 58.6%

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

   3. Simplified 74.3%

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

   5. Taylor expanded in x around 0 74.4%

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

      1. associate-*l/ 74.3%

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

      2. *-lft-identity 74.3%

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

      3. +-commutative 74.3%

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

      4. unpow2 74.3%

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

      5. fma-udef 74.3%

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

   7. Simplified 74.3%

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

   8. Taylor expanded in F around -inf 99.8%

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

   if -1.75 < F < 1.75

   1. Initial program 99.4%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in F around 0 98.7%

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

   if 1.75 < F

   1. Initial program 58.7%

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

   3. Simplified 75.0%

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

   5. Taylor expanded in F around inf 98.7%

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

      1. associate-/r* 98.7%

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

   7. Simplified 98.7%

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

   8. Taylor expanded in F around 0 98.9%

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

      1. *-commutative 98.9%

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

      2. associate-*r/ 99.0%

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

   10. Simplified 99.0%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.75:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.75:\\ \;\;\;\;F \cdot \left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\right) - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \cos B \cdot \frac{x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1.45:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{2}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \cos B \cdot \frac{x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.4)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.45)
       (- (/ F (* (sin B) (sqrt 2.0))) t_0)
       (- (/ 1.0 (sin B)) (* (cos B) (/ x (sin B))))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.4) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.45) {
		tmp = (F / (sin(B) * sqrt(2.0))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - (cos(B) * (x / sin(B)));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.4d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.45d0) then
        tmp = (f / (sin(b) * sqrt(2.0d0))) - t_0
    else
        tmp = (1.0d0 / sin(b)) - (cos(b) * (x / sin(b)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.4) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.45) {
		tmp = (F / (Math.sin(B) * Math.sqrt(2.0))) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - (Math.cos(B) * (x / Math.sin(B)));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.4:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.45:
		tmp = (F / (math.sin(B) * math.sqrt(2.0))) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - (math.cos(B) * (x / math.sin(B)))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.4)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.45)
		tmp = Float64(Float64(F / Float64(sin(B) * sqrt(2.0))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(cos(B) * Float64(x / sin(B))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.4)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.45)
		tmp = (F / (sin(B) * sqrt(2.0))) - t_0;
	else
		tmp = (1.0 / sin(B)) - (cos(B) * (x / sin(B)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.4], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.45], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[B], $MachinePrecision] * N[(x / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.3999999999999999

   1. Initial program 58.6%

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

   3. Simplified 74.3%

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

   5. Taylor expanded in x around 0 74.4%

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

      1. associate-*l/ 74.3%

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

      2. *-lft-identity 74.3%

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

      3. +-commutative 74.3%

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

      4. unpow2 74.3%

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

      5. fma-udef 74.3%

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

   7. Simplified 74.3%

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

   8. Taylor expanded in F around -inf 99.8%

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

   if -1.3999999999999999 < F < 1.44999999999999996

   1. Initial program 99.4%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 99.4%

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

      1. associate-*l/ 99.5%

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

      2. *-lft-identity 99.5%

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

      3. +-commutative 99.5%

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

      4. unpow2 99.5%

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

      5. fma-udef 99.5%

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

   7. Simplified 99.5%

      \[\leadsto F \cdot \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}} - \frac{x}{\tan B} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 80.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(F \cdot \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}\right)\right)} - \frac{x}{\tan B} \]

      2. expm1-udef 67.2%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(F \cdot \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}\right)} - 1\right)} - \frac{x}{\tan B} \]

      3. associate-*r/ 67.2%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{F \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}}\right)} - 1\right) - \frac{x}{\tan B} \]

      4. sqrt-div 67.2%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}}{\sin B}\right)} - 1\right) - \frac{x}{\tan B} \]

      5. metadata-eval 67.2%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F \cdot \frac{\color{blue}{1}}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}\right)} - 1\right) - \frac{x}{\tan B} \]

      6. un-div-inv 67.2%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}}{\sin B}\right)} - 1\right) - \frac{x}{\tan B} \]

   9. Applied egg-rr 67.2%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}\right)} - 1\right)} - \frac{x}{\tan B} \]
   10. Step-by-step derivation

       1. expm1-def 80.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}\right)\right)} - \frac{x}{\tan B} \]

       2. expm1-log1p 99.5%

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

       3. associate-/l/ 99.5%

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

   11. Simplified 99.5%

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

   12. Taylor expanded in F around 0 98.5%

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

       1. *-commutative 98.5%

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

   14. Simplified 98.5%

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

   if 1.44999999999999996 < F

   1. Initial program 58.7%

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

   3. Simplified 75.0%

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

   5. Taylor expanded in F around inf 98.7%

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

      1. associate-/r* 98.7%

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

   7. Simplified 98.7%

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

   8. Taylor expanded in F around 0 98.9%

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

      1. *-commutative 98.9%

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

      2. associate-*r/ 99.0%

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

   10. Simplified 99.0%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.45:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{2}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \cos B \cdot \frac{x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1.45:\\ \;\;\;\;\frac{F}{\frac{\sin B}{\sqrt{0.5}}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \cos B \cdot \frac{x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.4)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.45)
       (- (/ F (/ (sin B) (sqrt 0.5))) t_0)
       (- (/ 1.0 (sin B)) (* (cos B) (/ x (sin B))))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.4) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.45) {
		tmp = (F / (sin(B) / sqrt(0.5))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - (cos(B) * (x / sin(B)));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.4d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.45d0) then
        tmp = (f / (sin(b) / sqrt(0.5d0))) - t_0
    else
        tmp = (1.0d0 / sin(b)) - (cos(b) * (x / sin(b)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.4) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.45) {
		tmp = (F / (Math.sin(B) / Math.sqrt(0.5))) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - (Math.cos(B) * (x / Math.sin(B)));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.4:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.45:
		tmp = (F / (math.sin(B) / math.sqrt(0.5))) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - (math.cos(B) * (x / math.sin(B)))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.4)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.45)
		tmp = Float64(Float64(F / Float64(sin(B) / sqrt(0.5))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(cos(B) * Float64(x / sin(B))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.4)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.45)
		tmp = (F / (sin(B) / sqrt(0.5))) - t_0;
	else
		tmp = (1.0 / sin(B)) - (cos(B) * (x / sin(B)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.4], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.45], 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] - N[(N[Cos[B], $MachinePrecision] * N[(x / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.3999999999999999

   1. Initial program 58.6%

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

   3. Simplified 74.3%

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

   5. Taylor expanded in x around 0 74.4%

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

      1. associate-*l/ 74.3%

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

      2. *-lft-identity 74.3%

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

      3. +-commutative 74.3%

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

      4. unpow2 74.3%

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

      5. fma-udef 74.3%

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

   7. Simplified 74.3%

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

   8. Taylor expanded in F around -inf 99.8%

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

   if -1.3999999999999999 < F < 1.44999999999999996

   1. Initial program 99.4%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 99.4%

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

      1. associate-*l/ 99.5%

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

      2. *-lft-identity 99.5%

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

      3. +-commutative 99.5%

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

      4. unpow2 99.5%

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

      5. fma-udef 99.5%

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

   7. Simplified 99.5%

      \[\leadsto F \cdot \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}} - \frac{x}{\tan B} \]
   8. Step-by-step derivation

      1. clear-num 99.5%

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

      2. un-div-inv 99.5%

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

      3. inv-pow 99.5%

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

      4. sqrt-pow1 99.5%

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

      5. metadata-eval 99.5%

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

   9. Applied egg-rr 99.5%

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

   10. Taylor expanded in F around 0 98.6%

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

   if 1.44999999999999996 < F

   1. Initial program 58.7%

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

   3. Simplified 75.0%

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

   5. Taylor expanded in F around inf 98.7%

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

      1. associate-/r* 98.7%

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

   7. Simplified 98.7%

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

   8. Taylor expanded in F around 0 98.9%

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

      1. *-commutative 98.9%

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

      2. associate-*r/ 99.0%

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

   10. Simplified 99.0%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.45:\\ \;\;\;\;\frac{F}{\frac{\sin B}{\sqrt{0.5}}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \cos B \cdot \frac{x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1.45:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \cos B \cdot \frac{x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.4)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.45)
       (- (/ (* F (sqrt 0.5)) (sin B)) t_0)
       (- (/ 1.0 (sin B)) (* (cos B) (/ x (sin B))))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.4) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.45) {
		tmp = ((F * sqrt(0.5)) / sin(B)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - (cos(B) * (x / sin(B)));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.4d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.45d0) then
        tmp = ((f * sqrt(0.5d0)) / sin(b)) - t_0
    else
        tmp = (1.0d0 / sin(b)) - (cos(b) * (x / sin(b)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.4) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.45) {
		tmp = ((F * Math.sqrt(0.5)) / Math.sin(B)) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - (Math.cos(B) * (x / Math.sin(B)));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.4:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.45:
		tmp = ((F * math.sqrt(0.5)) / math.sin(B)) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - (math.cos(B) * (x / math.sin(B)))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.4)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.45)
		tmp = Float64(Float64(Float64(F * sqrt(0.5)) / sin(B)) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(cos(B) * Float64(x / sin(B))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.4)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.45)
		tmp = ((F * sqrt(0.5)) / sin(B)) - t_0;
	else
		tmp = (1.0 / sin(B)) - (cos(B) * (x / sin(B)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.4], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.45], N[(N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[B], $MachinePrecision] * N[(x / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.3999999999999999

   1. Initial program 58.6%

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

   3. Simplified 74.3%

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

   5. Taylor expanded in x around 0 74.4%

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

      1. associate-*l/ 74.3%

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

      2. *-lft-identity 74.3%

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

      3. +-commutative 74.3%

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

      4. unpow2 74.3%

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

      5. fma-udef 74.3%

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

   7. Simplified 74.3%

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

   8. Taylor expanded in F around -inf 99.8%

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

   if -1.3999999999999999 < F < 1.44999999999999996

   1. Initial program 99.4%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 99.4%

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

      1. associate-*l/ 99.5%

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

      2. *-lft-identity 99.5%

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

      3. +-commutative 99.5%

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

      4. unpow2 99.5%

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

      5. fma-udef 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in F around 0 98.6%

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

      1. *-commutative 98.6%

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

   10. Simplified 98.6%

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

   if 1.44999999999999996 < F

   1. Initial program 58.7%

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

   3. Simplified 75.0%

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

   5. Taylor expanded in F around inf 98.7%

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

      1. associate-/r* 98.7%

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

   7. Simplified 98.7%

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

   8. Taylor expanded in F around 0 98.9%

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

      1. *-commutative 98.9%

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

      2. associate-*r/ 99.0%

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

   10. Simplified 99.0%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.45:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \cos B \cdot \frac{x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 89.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ t_1 := \frac{1}{\sin B}\\ t_2 := {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \left(F \cdot t\_1\right) - \frac{x}{B}\\ \mathbf{if}\;F \leq -0.037:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq -5.6 \cdot 10^{-144}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;F \leq 3.1 \cdot 10^{-33}:\\ \;\;\;\;\frac{F}{\sin B \cdot \left(F + \frac{1}{F}\right)} - t\_0\\ \mathbf{elif}\;F \leq 27000:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1 - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B)))
        (t_1 (/ 1.0 (sin B)))
        (t_2
         (- (* (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5) (* F t_1)) (/ x B))))
   (if (<= F -0.037)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F -5.6e-144)
       t_2
       (if (<= F 3.1e-33)
         (- (/ F (* (sin B) (+ F (/ 1.0 F)))) t_0)
         (if (<= F 27000.0) t_2 (- 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 t_2 = (pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F * t_1)) - (x / B);
	double tmp;
	if (F <= -0.037) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= -5.6e-144) {
		tmp = t_2;
	} else if (F <= 3.1e-33) {
		tmp = (F / (sin(B) * (F + (1.0 / F)))) - t_0;
	} else if (F <= 27000.0) {
		tmp = t_2;
	} 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) :: t_2
    real(8) :: tmp
    t_0 = x / tan(b)
    t_1 = 1.0d0 / sin(b)
    t_2 = (((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0)) * (f * t_1)) - (x / b)
    if (f <= (-0.037d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= (-5.6d-144)) then
        tmp = t_2
    else if (f <= 3.1d-33) then
        tmp = (f / (sin(b) * (f + (1.0d0 / f)))) - t_0
    else if (f <= 27000.0d0) then
        tmp = t_2
    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 t_2 = (Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F * t_1)) - (x / B);
	double tmp;
	if (F <= -0.037) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= -5.6e-144) {
		tmp = t_2;
	} else if (F <= 3.1e-33) {
		tmp = (F / (Math.sin(B) * (F + (1.0 / F)))) - t_0;
	} else if (F <= 27000.0) {
		tmp = t_2;
	} 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)
	t_2 = (math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F * t_1)) - (x / B)
	tmp = 0
	if F <= -0.037:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= -5.6e-144:
		tmp = t_2
	elif F <= 3.1e-33:
		tmp = (F / (math.sin(B) * (F + (1.0 / F)))) - t_0
	elif F <= 27000.0:
		tmp = t_2
	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))
	t_2 = Float64(Float64((Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5) * Float64(F * t_1)) - Float64(x / B))
	tmp = 0.0
	if (F <= -0.037)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= -5.6e-144)
		tmp = t_2;
	elseif (F <= 3.1e-33)
		tmp = Float64(Float64(F / Float64(sin(B) * Float64(F + Float64(1.0 / F)))) - t_0);
	elseif (F <= 27000.0)
		tmp = t_2;
	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);
	t_2 = (((((F * F) + 2.0) + (x * 2.0)) ^ -0.5) * (F * t_1)) - (x / B);
	tmp = 0.0;
	if (F <= -0.037)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= -5.6e-144)
		tmp = t_2;
	elseif (F <= 3.1e-33)
		tmp = (F / (sin(B) * (F + (1.0 / F)))) - t_0;
	elseif (F <= 27000.0)
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(N[(N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.037], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -5.6e-144], t$95$2, If[LessEqual[F, 3.1e-33], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * N[(F + N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 27000.0], t$95$2, N[(t$95$1 - t$95$0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.0369999999999999982

   1. Initial program 59.1%

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

   3. Simplified 74.6%

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

   5. Taylor expanded in x around 0 74.8%

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

      1. associate-*l/ 74.7%

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

      2. *-lft-identity 74.7%

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

      3. +-commutative 74.7%

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

      4. unpow2 74.7%

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

      5. fma-udef 74.7%

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

   7. Simplified 74.7%

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

   8. Taylor expanded in F around -inf 99.8%

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

   if -0.0369999999999999982 < F < -5.59999999999999995e-144 or 3.09999999999999997e-33 < F < 27000

   1. Initial program 99.5%

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

   3. Taylor expanded in B around 0 87.0%

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

      1. div-inv 87.2%

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

   5. Applied egg-rr 87.2%

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

   if -5.59999999999999995e-144 < F < 3.09999999999999997e-33

   1. Initial program 99.4%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 99.6%

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

      1. associate-*l/ 99.6%

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

      2. *-lft-identity 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. fma-udef 99.6%

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

   7. Simplified 99.6%

      \[\leadsto F \cdot \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}} - \frac{x}{\tan B} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 90.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(F \cdot \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}\right)\right)} - \frac{x}{\tan B} \]

      2. expm1-udef 83.4%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(F \cdot \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}\right)} - 1\right)} - \frac{x}{\tan B} \]

      3. associate-*r/ 83.4%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{F \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}}\right)} - 1\right) - \frac{x}{\tan B} \]

      4. sqrt-div 83.4%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}}{\sin B}\right)} - 1\right) - \frac{x}{\tan B} \]

      5. metadata-eval 83.4%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F \cdot \frac{\color{blue}{1}}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}\right)} - 1\right) - \frac{x}{\tan B} \]

      6. un-div-inv 83.4%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}}{\sin B}\right)} - 1\right) - \frac{x}{\tan B} \]

   9. Applied egg-rr 83.4%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}\right)} - 1\right)} - \frac{x}{\tan B} \]
   10. Step-by-step derivation

       1. expm1-def 90.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}\right)\right)} - \frac{x}{\tan B} \]

       2. expm1-log1p 99.6%

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

       3. associate-/l/ 99.7%

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

   11. Simplified 99.7%

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

   12. Taylor expanded in F around inf 91.3%

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

   if 27000 < F

   1. Initial program 58.2%

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

   3. Simplified 74.7%

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

   5. Taylor expanded in x around 0 74.7%

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

      1. associate-*l/ 74.7%

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

      2. *-lft-identity 74.7%

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

      3. +-commutative 74.7%

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

      4. unpow2 74.7%

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

      5. fma-udef 74.7%

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

   7. Simplified 74.7%

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

   8. 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 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.037:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -5.6 \cdot 10^{-144}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \left(F \cdot \frac{1}{\sin B}\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.1 \cdot 10^{-33}:\\ \;\;\;\;\frac{F}{\sin B \cdot \left(F + \frac{1}{F}\right)} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 27000:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \left(F \cdot \frac{1}{\sin B}\right) - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 89.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.037:\\ \;\;\;\;\frac{-1}{\sin B} - t\_1\\ \mathbf{elif}\;F \leq -4.4 \cdot 10^{-147}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 3 \cdot 10^{-33}:\\ \;\;\;\;\frac{F}{\sin B \cdot \left(F + \frac{1}{F}\right)} - t\_1\\ \mathbf{elif}\;F \leq 2600000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0
         (-
          (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5))
          (/ x B)))
        (t_1 (/ x (tan B))))
   (if (<= F -0.037)
     (- (/ -1.0 (sin B)) t_1)
     (if (<= F -4.4e-147)
       t_0
       (if (<= F 3e-33)
         (- (/ F (* (sin B) (+ F (/ 1.0 F)))) t_1)
         (if (<= F 2600000.0) t_0 (- (/ 1.0 (sin B)) t_1)))))))

C

double code(double F, double B, double x) {
	double t_0 = ((F / sin(B)) * pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - (x / B);
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -0.037) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= -4.4e-147) {
		tmp = t_0;
	} else if (F <= 3e-33) {
		tmp = (F / (sin(B) * (F + (1.0 / F)))) - t_1;
	} else if (F <= 2600000.0) {
		tmp = t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_1;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((f / sin(b)) * ((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0))) - (x / b)
    t_1 = x / tan(b)
    if (f <= (-0.037d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_1
    else if (f <= (-4.4d-147)) then
        tmp = t_0
    else if (f <= 3d-33) then
        tmp = (f / (sin(b) * (f + (1.0d0 / f)))) - t_1
    else if (f <= 2600000.0d0) then
        tmp = t_0
    else
        tmp = (1.0d0 / sin(b)) - t_1
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = ((F / Math.sin(B)) * Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - (x / B);
	double t_1 = x / Math.tan(B);
	double tmp;
	if (F <= -0.037) {
		tmp = (-1.0 / Math.sin(B)) - t_1;
	} else if (F <= -4.4e-147) {
		tmp = t_0;
	} else if (F <= 3e-33) {
		tmp = (F / (Math.sin(B) * (F + (1.0 / F)))) - t_1;
	} else if (F <= 2600000.0) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = ((F / math.sin(B)) * math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - (x / B)
	t_1 = x / math.tan(B)
	tmp = 0
	if F <= -0.037:
		tmp = (-1.0 / math.sin(B)) - t_1
	elif F <= -4.4e-147:
		tmp = t_0
	elif F <= 3e-33:
		tmp = (F / (math.sin(B) * (F + (1.0 / F)))) - t_1
	elif F <= 2600000.0:
		tmp = t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5)) - Float64(x / B))
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -0.037)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= -4.4e-147)
		tmp = t_0;
	elseif (F <= 3e-33)
		tmp = Float64(Float64(F / Float64(sin(B) * Float64(F + Float64(1.0 / F)))) - t_1);
	elseif (F <= 2600000.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = ((F / sin(B)) * ((((F * F) + 2.0) + (x * 2.0)) ^ -0.5)) - (x / B);
	t_1 = x / tan(B);
	tmp = 0.0;
	if (F <= -0.037)
		tmp = (-1.0 / sin(B)) - t_1;
	elseif (F <= -4.4e-147)
		tmp = t_0;
	elseif (F <= 3e-33)
		tmp = (F / (sin(B) * (F + (1.0 / F)))) - t_1;
	elseif (F <= 2600000.0)
		tmp = t_0;
	else
		tmp = (1.0 / sin(B)) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.037], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, -4.4e-147], t$95$0, If[LessEqual[F, 3e-33], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * N[(F + N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, 2600000.0], t$95$0, N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.0369999999999999982

   1. Initial program 59.1%

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

   3. Simplified 74.6%

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

   5. Taylor expanded in x around 0 74.8%

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

      1. associate-*l/ 74.7%

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

      2. *-lft-identity 74.7%

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

      3. +-commutative 74.7%

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

      4. unpow2 74.7%

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

      5. fma-udef 74.7%

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

   7. Simplified 74.7%

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

   8. Taylor expanded in F around -inf 99.8%

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

   if -0.0369999999999999982 < F < -4.4000000000000002e-147 or 3.0000000000000002e-33 < F < 2.6e6

   1. Initial program 99.5%

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

   3. Taylor expanded in B around 0 87.0%

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

   if -4.4000000000000002e-147 < F < 3.0000000000000002e-33

   1. Initial program 99.4%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 99.6%

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

      1. associate-*l/ 99.6%

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

      2. *-lft-identity 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. fma-udef 99.6%

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

   7. Simplified 99.6%

      \[\leadsto F \cdot \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}} - \frac{x}{\tan B} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 90.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(F \cdot \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}\right)\right)} - \frac{x}{\tan B} \]

      2. expm1-udef 83.4%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(F \cdot \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}\right)} - 1\right)} - \frac{x}{\tan B} \]

      3. associate-*r/ 83.4%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{F \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}}\right)} - 1\right) - \frac{x}{\tan B} \]

      4. sqrt-div 83.4%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}}{\sin B}\right)} - 1\right) - \frac{x}{\tan B} \]

      5. metadata-eval 83.4%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F \cdot \frac{\color{blue}{1}}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}\right)} - 1\right) - \frac{x}{\tan B} \]

      6. un-div-inv 83.4%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}}{\sin B}\right)} - 1\right) - \frac{x}{\tan B} \]

   9. Applied egg-rr 83.4%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}\right)} - 1\right)} - \frac{x}{\tan B} \]
   10. Step-by-step derivation

       1. expm1-def 90.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}\right)\right)} - \frac{x}{\tan B} \]

       2. expm1-log1p 99.6%

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

       3. associate-/l/ 99.7%

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

   11. Simplified 99.7%

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

   12. Taylor expanded in F around inf 91.3%

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

   if 2.6e6 < F

   1. Initial program 58.2%

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

   3. Simplified 74.7%

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

   5. Taylor expanded in x around 0 74.7%

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

      1. associate-*l/ 74.7%

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

      2. *-lft-identity 74.7%

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

      3. +-commutative 74.7%

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

      4. unpow2 74.7%

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

      5. fma-udef 74.7%

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

   7. Simplified 74.7%

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

   8. 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 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.037:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -4.4 \cdot 10^{-147}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3 \cdot 10^{-33}:\\ \;\;\;\;\frac{F}{\sin B \cdot \left(F + \frac{1}{F}\right)} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2600000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 87.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.0037:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq -1.2 \cdot 10^{-145}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{\sin B \cdot \left(F + \frac{1}{F}\right)} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -0.0037)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F -1.2e-145)
       (- (* (/ F (sin B)) (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))) (/ x B))
       (- (/ F (* (sin B) (+ F (/ 1.0 F)))) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -0.0037) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= -1.2e-145) {
		tmp = ((F / sin(B)) * sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B);
	} else {
		tmp = (F / (sin(B) * (F + (1.0 / F)))) - 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 <= (-0.0037d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= (-1.2d-145)) then
        tmp = ((f / sin(b)) * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))) - (x / b)
    else
        tmp = (f / (sin(b) * (f + (1.0d0 / f)))) - 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 <= -0.0037) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= -1.2e-145) {
		tmp = ((F / Math.sin(B)) * Math.sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B);
	} else {
		tmp = (F / (Math.sin(B) * (F + (1.0 / F)))) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -0.0037:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= -1.2e-145:
		tmp = ((F / math.sin(B)) * math.sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B)
	else:
		tmp = (F / (math.sin(B) * (F + (1.0 / F)))) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -0.0037)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= -1.2e-145)
		tmp = Float64(Float64(Float64(F / sin(B)) * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))) - Float64(x / B));
	else
		tmp = Float64(Float64(F / Float64(sin(B) * Float64(F + Float64(1.0 / F)))) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -0.0037)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= -1.2e-145)
		tmp = ((F / sin(B)) * sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B);
	else
		tmp = (F / (sin(B) * (F + (1.0 / F)))) - 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, -0.0037], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -1.2e-145], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * N[(F + N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.0037000000000000002

   1. Initial program 59.1%

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

   3. Simplified 74.6%

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

   5. Taylor expanded in x around 0 74.8%

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

      1. associate-*l/ 74.7%

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

      2. *-lft-identity 74.7%

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

      3. +-commutative 74.7%

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

      4. unpow2 74.7%

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

      5. fma-udef 74.7%

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

   7. Simplified 74.7%

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

   8. Taylor expanded in F around -inf 99.8%

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

   if -0.0037000000000000002 < F < -1.20000000000000008e-145

   1. Initial program 99.6%

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

   3. Taylor expanded in B around 0 83.7%

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

   4. Taylor expanded in F around 0 82.3%

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

   if -1.20000000000000008e-145 < F

   1. Initial program 78.4%

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

   3. Simplified 86.9%

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

   5. Taylor expanded in x around 0 86.9%

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

      1. associate-*l/ 86.9%

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

      2. *-lft-identity 86.9%

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

      3. +-commutative 86.9%

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

      4. unpow2 86.9%

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

      5. fma-udef 86.9%

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

   7. Simplified 86.9%

      \[\leadsto F \cdot \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}} - \frac{x}{\tan B} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 68.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(F \cdot \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}\right)\right)} - \frac{x}{\tan B} \]

      2. expm1-udef 64.9%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(F \cdot \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}\right)} - 1\right)} - \frac{x}{\tan B} \]

      3. associate-*r/ 64.9%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{F \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}}\right)} - 1\right) - \frac{x}{\tan B} \]

      4. sqrt-div 64.9%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}}{\sin B}\right)} - 1\right) - \frac{x}{\tan B} \]

      5. metadata-eval 64.9%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F \cdot \frac{\color{blue}{1}}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}\right)} - 1\right) - \frac{x}{\tan B} \]

      6. un-div-inv 64.9%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}}{\sin B}\right)} - 1\right) - \frac{x}{\tan B} \]

   9. Applied egg-rr 64.9%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}\right)} - 1\right)} - \frac{x}{\tan B} \]
   10. Step-by-step derivation

       1. expm1-def 68.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}\right)\right)} - \frac{x}{\tan B} \]

       2. expm1-log1p 87.0%

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

       3. associate-/l/ 87.0%

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

   11. Simplified 87.0%

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

   12. Taylor expanded in F around inf 92.8%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.0037:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.2 \cdot 10^{-145}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{\sin B \cdot \left(F + \frac{1}{F}\right)} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 85.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.017:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq -1.92 \cdot 10^{-90}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} + x \cdot \left(\frac{-1}{B} - B \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{\sin B \cdot \left(F + \frac{1}{F}\right)} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -0.017)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F -1.92e-90)
       (+
        (* (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5) (/ F B))
        (* x (- (/ -1.0 B) (* B -0.3333333333333333))))
       (- (/ F (* (sin B) (+ F (/ 1.0 F)))) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -0.017) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= -1.92e-90) {
		tmp = (pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B)) + (x * ((-1.0 / B) - (B * -0.3333333333333333)));
	} else {
		tmp = (F / (sin(B) * (F + (1.0 / F)))) - 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 <= (-0.017d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= (-1.92d-90)) then
        tmp = (((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0)) * (f / b)) + (x * (((-1.0d0) / b) - (b * (-0.3333333333333333d0))))
    else
        tmp = (f / (sin(b) * (f + (1.0d0 / f)))) - 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 <= -0.017) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= -1.92e-90) {
		tmp = (Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B)) + (x * ((-1.0 / B) - (B * -0.3333333333333333)));
	} else {
		tmp = (F / (Math.sin(B) * (F + (1.0 / F)))) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -0.017:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= -1.92e-90:
		tmp = (math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B)) + (x * ((-1.0 / B) - (B * -0.3333333333333333)))
	else:
		tmp = (F / (math.sin(B) * (F + (1.0 / F)))) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -0.017)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= -1.92e-90)
		tmp = Float64(Float64((Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5) * Float64(F / B)) + Float64(x * Float64(Float64(-1.0 / B) - Float64(B * -0.3333333333333333))));
	else
		tmp = Float64(Float64(F / Float64(sin(B) * Float64(F + Float64(1.0 / F)))) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -0.017)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= -1.92e-90)
		tmp = (((((F * F) + 2.0) + (x * 2.0)) ^ -0.5) * (F / B)) + (x * ((-1.0 / B) - (B * -0.3333333333333333)));
	else
		tmp = (F / (sin(B) * (F + (1.0 / F)))) - 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, -0.017], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -1.92e-90], N[(N[(N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[(-1.0 / B), $MachinePrecision] - N[(B * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * N[(F + N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.017000000000000001

   1. Initial program 59.1%

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

   3. Simplified 74.6%

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

   5. Taylor expanded in x around 0 74.8%

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

      1. associate-*l/ 74.7%

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

      2. *-lft-identity 74.7%

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

      3. +-commutative 74.7%

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

      4. unpow2 74.7%

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

      5. fma-udef 74.7%

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

   7. Simplified 74.7%

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

   8. Taylor expanded in F around -inf 99.8%

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

   if -0.017000000000000001 < F < -1.92000000000000009e-90

   1. Initial program 99.7%

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

   3. Taylor expanded in B around 0 81.6%

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

   4. Taylor expanded in B around 0 77.4%

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

   if -1.92000000000000009e-90 < F

   1. Initial program 79.9%

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

   3. Simplified 87.9%

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

   5. Taylor expanded in x around 0 87.8%

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

      1. associate-*l/ 87.9%

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

      2. *-lft-identity 87.9%

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

      3. +-commutative 87.9%

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

      4. unpow2 87.9%

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

      5. fma-udef 87.9%

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

   7. Simplified 87.9%

      \[\leadsto F \cdot \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}} - \frac{x}{\tan B} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 69.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(F \cdot \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}\right)\right)} - \frac{x}{\tan B} \]

      2. expm1-udef 63.8%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(F \cdot \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}\right)} - 1\right)} - \frac{x}{\tan B} \]

      3. associate-*r/ 63.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{F \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}}\right)} - 1\right) - \frac{x}{\tan B} \]

      4. sqrt-div 63.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}}{\sin B}\right)} - 1\right) - \frac{x}{\tan B} \]

      5. metadata-eval 63.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F \cdot \frac{\color{blue}{1}}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}\right)} - 1\right) - \frac{x}{\tan B} \]

      6. un-div-inv 63.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}}{\sin B}\right)} - 1\right) - \frac{x}{\tan B} \]

   9. Applied egg-rr 63.8%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}\right)} - 1\right)} - \frac{x}{\tan B} \]
   10. Step-by-step derivation

       1. expm1-def 69.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}\right)\right)} - \frac{x}{\tan B} \]

       2. expm1-log1p 87.9%

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

       3. associate-/l/ 87.9%

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

   11. Simplified 87.9%

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

   12. Taylor expanded in F around inf 89.2%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.017:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.92 \cdot 10^{-90}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} + x \cdot \left(\frac{-1}{B} - B \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{\sin B \cdot \left(F + \frac{1}{F}\right)} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 85.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.018:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq -1.26 \cdot 10^{-85}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} + x \cdot \left(\frac{-1}{B} - B \cdot -0.3333333333333333\right)\\ \mathbf{elif}\;F \leq 2.65 \cdot 10^{-76}:\\ \;\;\;\;\cos B \cdot \frac{-x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -0.018)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F -1.26e-85)
       (+
        (* (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5) (/ F B))
        (* x (- (/ -1.0 B) (* B -0.3333333333333333))))
       (if (<= F 2.65e-76)
         (* (cos B) (/ (- x) (sin B)))
         (- (/ 1.0 (sin B)) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -0.018) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= -1.26e-85) {
		tmp = (pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B)) + (x * ((-1.0 / B) - (B * -0.3333333333333333)));
	} else if (F <= 2.65e-76) {
		tmp = cos(B) * (-x / sin(B));
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-0.018d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= (-1.26d-85)) then
        tmp = (((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0)) * (f / b)) + (x * (((-1.0d0) / b) - (b * (-0.3333333333333333d0))))
    else if (f <= 2.65d-76) then
        tmp = cos(b) * (-x / sin(b))
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -0.018) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= -1.26e-85) {
		tmp = (Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B)) + (x * ((-1.0 / B) - (B * -0.3333333333333333)));
	} else if (F <= 2.65e-76) {
		tmp = Math.cos(B) * (-x / Math.sin(B));
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -0.018:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= -1.26e-85:
		tmp = (math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B)) + (x * ((-1.0 / B) - (B * -0.3333333333333333)))
	elif F <= 2.65e-76:
		tmp = math.cos(B) * (-x / math.sin(B))
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -0.018)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= -1.26e-85)
		tmp = Float64(Float64((Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5) * Float64(F / B)) + Float64(x * Float64(Float64(-1.0 / B) - Float64(B * -0.3333333333333333))));
	elseif (F <= 2.65e-76)
		tmp = Float64(cos(B) * Float64(Float64(-x) / sin(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -0.018)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= -1.26e-85)
		tmp = (((((F * F) + 2.0) + (x * 2.0)) ^ -0.5) * (F / B)) + (x * ((-1.0 / B) - (B * -0.3333333333333333)));
	elseif (F <= 2.65e-76)
		tmp = cos(B) * (-x / sin(B));
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.018], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -1.26e-85], N[(N[(N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[(-1.0 / B), $MachinePrecision] - N[(B * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.65e-76], N[(N[Cos[B], $MachinePrecision] * N[((-x) / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.0179999999999999986

   1. Initial program 59.1%

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

   3. Simplified 74.6%

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

   5. Taylor expanded in x around 0 74.8%

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

      1. associate-*l/ 74.7%

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

      2. *-lft-identity 74.7%

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

      3. +-commutative 74.7%

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

      4. unpow2 74.7%

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

      5. fma-udef 74.7%

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

   7. Simplified 74.7%

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

   8. Taylor expanded in F around -inf 99.8%

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

   if -0.0179999999999999986 < F < -1.26e-85

   1. Initial program 99.7%

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

   3. Taylor expanded in B around 0 81.6%

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

   4. Taylor expanded in B around 0 77.4%

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

   if -1.26e-85 < F < 2.65e-76

   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 99.7%

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

   5. Taylor expanded in x around 0 99.6%

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

      1. associate-*l/ 99.7%

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

      2. *-lft-identity 99.7%

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

      3. +-commutative 99.7%

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

      4. unpow2 99.7%

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

      5. fma-udef 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in F around -inf 34.6%

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

   9. Taylor expanded in x around inf 84.6%

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

       1. associate-*l/ 84.6%

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

       2. neg-mul-1 84.6%

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

       3. distribute-rgt-neg-in 84.6%

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

   11. Simplified 84.6%

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

   if 2.65e-76 < F

   1. Initial program 66.0%

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

   3. Simplified 79.4%

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

   5. Taylor expanded in x around 0 79.4%

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

      1. associate-*l/ 79.4%

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

      2. *-lft-identity 79.4%

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

      3. +-commutative 79.4%

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

      4. unpow2 79.4%

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

      5. fma-udef 79.4%

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

   7. Simplified 79.4%

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

   8. Taylor expanded in F around inf 92.5%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.018:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.26 \cdot 10^{-85}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} + x \cdot \left(\frac{-1}{B} - B \cdot -0.3333333333333333\right)\\ \mathbf{elif}\;F \leq 2.65 \cdot 10^{-76}:\\ \;\;\;\;\cos B \cdot \frac{-x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 72.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.037:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -8 \cdot 10^{-86}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} + x \cdot \left(\frac{-1}{B} - B \cdot -0.3333333333333333\right)\\ \mathbf{elif}\;F \leq 1.9 \cdot 10^{-20}:\\ \;\;\;\;\cos B \cdot \frac{-x}{\sin B}\\ \mathbf{elif}\;F \leq 8.5 \cdot 10^{+120}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.037)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F -8e-86)
     (+
      (* (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5) (/ F B))
      (* x (- (/ -1.0 B) (* B -0.3333333333333333))))
     (if (<= F 1.9e-20)
       (* (cos B) (/ (- x) (sin B)))
       (if (<= F 8.5e+120)
         (- (/ 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 <= -0.037) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -8e-86) {
		tmp = (pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B)) + (x * ((-1.0 / B) - (B * -0.3333333333333333)));
	} else if (F <= 1.9e-20) {
		tmp = cos(B) * (-x / sin(B));
	} else if (F <= 8.5e+120) {
		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 <= (-0.037d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-8d-86)) then
        tmp = (((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0)) * (f / b)) + (x * (((-1.0d0) / b) - (b * (-0.3333333333333333d0))))
    else if (f <= 1.9d-20) then
        tmp = cos(b) * (-x / sin(b))
    else if (f <= 8.5d+120) 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 <= -0.037) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -8e-86) {
		tmp = (Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B)) + (x * ((-1.0 / B) - (B * -0.3333333333333333)));
	} else if (F <= 1.9e-20) {
		tmp = Math.cos(B) * (-x / Math.sin(B));
	} else if (F <= 8.5e+120) {
		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 <= -0.037:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -8e-86:
		tmp = (math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B)) + (x * ((-1.0 / B) - (B * -0.3333333333333333)))
	elif F <= 1.9e-20:
		tmp = math.cos(B) * (-x / math.sin(B))
	elif F <= 8.5e+120:
		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 <= -0.037)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -8e-86)
		tmp = Float64(Float64((Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5) * Float64(F / B)) + Float64(x * Float64(Float64(-1.0 / B) - Float64(B * -0.3333333333333333))));
	elseif (F <= 1.9e-20)
		tmp = Float64(cos(B) * Float64(Float64(-x) / sin(B)));
	elseif (F <= 8.5e+120)
		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 <= -0.037)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -8e-86)
		tmp = (((((F * F) + 2.0) + (x * 2.0)) ^ -0.5) * (F / B)) + (x * ((-1.0 / B) - (B * -0.3333333333333333)));
	elseif (F <= 1.9e-20)
		tmp = cos(B) * (-x / sin(B));
	elseif (F <= 8.5e+120)
		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, -0.037], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -8e-86], N[(N[(N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[(-1.0 / B), $MachinePrecision] - N[(B * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.9e-20], N[(N[Cos[B], $MachinePrecision] * N[((-x) / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 8.5e+120], 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 5 regimes

2. if F < -0.0369999999999999982

   1. Initial program 59.1%

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

   3. Taylor expanded in B around 0 38.8%

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

   4. Taylor expanded in F around -inf 78.6%

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

      1. distribute-lft-in 78.6%

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

      2. mul-1-neg 78.6%

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

      3. unsub-neg 78.6%

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

      4. associate-*r/ 78.6%

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

      5. metadata-eval 78.6%

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

   6. Simplified 78.6%

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

   if -0.0369999999999999982 < F < -8.00000000000000068e-86

   1. Initial program 99.7%

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

   3. Taylor expanded in B around 0 81.6%

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

   4. Taylor expanded in B around 0 77.4%

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

   if -8.00000000000000068e-86 < F < 1.8999999999999999e-20

   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 99.7%

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

   5. Taylor expanded in x around 0 99.6%

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

      1. associate-*l/ 99.7%

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

      2. *-lft-identity 99.7%

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

      3. +-commutative 99.7%

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

      4. unpow2 99.7%

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

      5. fma-udef 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in F around -inf 39.7%

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

   9. Taylor expanded in x around inf 83.9%

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

       1. associate-*l/ 83.9%

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

       2. neg-mul-1 83.9%

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

       3. distribute-rgt-neg-in 83.9%

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

   11. Simplified 83.9%

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

   if 1.8999999999999999e-20 < F < 8.50000000000000026e120

   1. Initial program 94.2%

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

   3. Taylor expanded in B around 0 88.7%

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

   4. Taylor expanded in F around inf 79.7%

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

   if 8.50000000000000026e120 < F

   1. Initial program 40.7%

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

   3. Simplified 62.0%

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

   5. Taylor expanded in F around inf 99.6%

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

      1. associate-/r* 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in B around 0 82.9%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.037:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -8 \cdot 10^{-86}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} + x \cdot \left(\frac{-1}{B} - B \cdot -0.3333333333333333\right)\\ \mathbf{elif}\;F \leq 1.9 \cdot 10^{-20}:\\ \;\;\;\;\cos B \cdot \frac{-x}{\sin B}\\ \mathbf{elif}\;F \leq 8.5 \cdot 10^{+120}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 72.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.037:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -2.25 \cdot 10^{-88}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} + x \cdot \left(\frac{-1}{B} - B \cdot -0.3333333333333333\right)\\ \mathbf{elif}\;F \leq 1.9 \cdot 10^{-20}:\\ \;\;\;\;\frac{\cos B \cdot \left(-x\right)}{\sin B}\\ \mathbf{elif}\;F \leq 1.01 \cdot 10^{+121}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.037)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F -2.25e-88)
     (+
      (* (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5) (/ F B))
      (* x (- (/ -1.0 B) (* B -0.3333333333333333))))
     (if (<= F 1.9e-20)
       (/ (* (cos B) (- x)) (sin B))
       (if (<= F 1.01e+121)
         (- (/ 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 <= -0.037) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -2.25e-88) {
		tmp = (pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B)) + (x * ((-1.0 / B) - (B * -0.3333333333333333)));
	} else if (F <= 1.9e-20) {
		tmp = (cos(B) * -x) / sin(B);
	} else if (F <= 1.01e+121) {
		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 <= (-0.037d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-2.25d-88)) then
        tmp = (((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0)) * (f / b)) + (x * (((-1.0d0) / b) - (b * (-0.3333333333333333d0))))
    else if (f <= 1.9d-20) then
        tmp = (cos(b) * -x) / sin(b)
    else if (f <= 1.01d+121) 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 <= -0.037) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -2.25e-88) {
		tmp = (Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B)) + (x * ((-1.0 / B) - (B * -0.3333333333333333)));
	} else if (F <= 1.9e-20) {
		tmp = (Math.cos(B) * -x) / Math.sin(B);
	} else if (F <= 1.01e+121) {
		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 <= -0.037:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -2.25e-88:
		tmp = (math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B)) + (x * ((-1.0 / B) - (B * -0.3333333333333333)))
	elif F <= 1.9e-20:
		tmp = (math.cos(B) * -x) / math.sin(B)
	elif F <= 1.01e+121:
		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 <= -0.037)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -2.25e-88)
		tmp = Float64(Float64((Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5) * Float64(F / B)) + Float64(x * Float64(Float64(-1.0 / B) - Float64(B * -0.3333333333333333))));
	elseif (F <= 1.9e-20)
		tmp = Float64(Float64(cos(B) * Float64(-x)) / sin(B));
	elseif (F <= 1.01e+121)
		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 <= -0.037)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -2.25e-88)
		tmp = (((((F * F) + 2.0) + (x * 2.0)) ^ -0.5) * (F / B)) + (x * ((-1.0 / B) - (B * -0.3333333333333333)));
	elseif (F <= 1.9e-20)
		tmp = (cos(B) * -x) / sin(B);
	elseif (F <= 1.01e+121)
		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, -0.037], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -2.25e-88], N[(N[(N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[(-1.0 / B), $MachinePrecision] - N[(B * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.9e-20], N[(N[(N[Cos[B], $MachinePrecision] * (-x)), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.01e+121], 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 5 regimes

2. if F < -0.0369999999999999982

   1. Initial program 59.1%

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

   3. Taylor expanded in B around 0 38.8%

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

   4. Taylor expanded in F around -inf 78.6%

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

      1. distribute-lft-in 78.6%

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

      2. mul-1-neg 78.6%

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

      3. unsub-neg 78.6%

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

      4. associate-*r/ 78.6%

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

      5. metadata-eval 78.6%

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

   6. Simplified 78.6%

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

   if -0.0369999999999999982 < F < -2.24999999999999996e-88

   1. Initial program 99.7%

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

   3. Taylor expanded in B around 0 81.6%

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

   4. Taylor expanded in B around 0 77.4%

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

   if -2.24999999999999996e-88 < F < 1.8999999999999999e-20

   1. Initial program 99.4%

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

   3. Taylor expanded in F around -inf 39.6%

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

   4. Taylor expanded in x around inf 83.9%

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

      1. associate-*r/ 83.9%

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

      2. *-commutative 83.9%

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

      3. neg-mul-1 83.9%

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

      4. distribute-rgt-neg-in 83.9%

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

   6. Simplified 83.9%

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

   if 1.8999999999999999e-20 < F < 1.01e121

   1. Initial program 94.2%

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

   3. Taylor expanded in B around 0 88.7%

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

   4. Taylor expanded in F around inf 79.7%

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

   if 1.01e121 < F

   1. Initial program 40.7%

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

   3. Simplified 62.0%

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

   5. Taylor expanded in F around inf 99.6%

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

      1. associate-/r* 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in B around 0 82.9%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.037:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -2.25 \cdot 10^{-88}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} + x \cdot \left(\frac{-1}{B} - B \cdot -0.3333333333333333\right)\\ \mathbf{elif}\;F \leq 1.9 \cdot 10^{-20}:\\ \;\;\;\;\frac{\cos B \cdot \left(-x\right)}{\sin B}\\ \mathbf{elif}\;F \leq 1.01 \cdot 10^{+121}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 78.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.031:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq -4.4 \cdot 10^{-87}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} + x \cdot \left(\frac{-1}{B} - B \cdot -0.3333333333333333\right)\\ \mathbf{elif}\;F \leq 1.9 \cdot 10^{-20}:\\ \;\;\;\;\frac{\cos B \cdot \left(-x\right)}{\sin B}\\ \mathbf{elif}\;F \leq 7.4 \cdot 10^{+120}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -0.031)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F -4.4e-87)
       (+
        (* (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5) (/ F B))
        (* x (- (/ -1.0 B) (* B -0.3333333333333333))))
       (if (<= F 1.9e-20)
         (/ (* (cos B) (- x)) (sin B))
         (if (<= F 7.4e+120)
           (- (/ 1.0 (sin B)) (/ x B))
           (- (/ 1.0 B) t_0)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -0.031) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= -4.4e-87) {
		tmp = (pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B)) + (x * ((-1.0 / B) - (B * -0.3333333333333333)));
	} else if (F <= 1.9e-20) {
		tmp = (cos(B) * -x) / sin(B);
	} else if (F <= 7.4e+120) {
		tmp = (1.0 / sin(B)) - (x / B);
	} else {
		tmp = (1.0 / B) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-0.031d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= (-4.4d-87)) then
        tmp = (((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0)) * (f / b)) + (x * (((-1.0d0) / b) - (b * (-0.3333333333333333d0))))
    else if (f <= 1.9d-20) then
        tmp = (cos(b) * -x) / sin(b)
    else if (f <= 7.4d+120) then
        tmp = (1.0d0 / sin(b)) - (x / b)
    else
        tmp = (1.0d0 / b) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -0.031) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= -4.4e-87) {
		tmp = (Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B)) + (x * ((-1.0 / B) - (B * -0.3333333333333333)));
	} else if (F <= 1.9e-20) {
		tmp = (Math.cos(B) * -x) / Math.sin(B);
	} else if (F <= 7.4e+120) {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	} else {
		tmp = (1.0 / B) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -0.031:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= -4.4e-87:
		tmp = (math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B)) + (x * ((-1.0 / B) - (B * -0.3333333333333333)))
	elif F <= 1.9e-20:
		tmp = (math.cos(B) * -x) / math.sin(B)
	elif F <= 7.4e+120:
		tmp = (1.0 / math.sin(B)) - (x / B)
	else:
		tmp = (1.0 / B) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -0.031)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= -4.4e-87)
		tmp = Float64(Float64((Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5) * Float64(F / B)) + Float64(x * Float64(Float64(-1.0 / B) - Float64(B * -0.3333333333333333))));
	elseif (F <= 1.9e-20)
		tmp = Float64(Float64(cos(B) * Float64(-x)) / sin(B));
	elseif (F <= 7.4e+120)
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / B) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -0.031)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= -4.4e-87)
		tmp = (((((F * F) + 2.0) + (x * 2.0)) ^ -0.5) * (F / B)) + (x * ((-1.0 / B) - (B * -0.3333333333333333)));
	elseif (F <= 1.9e-20)
		tmp = (cos(B) * -x) / sin(B);
	elseif (F <= 7.4e+120)
		tmp = (1.0 / sin(B)) - (x / B);
	else
		tmp = (1.0 / B) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.031], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -4.4e-87], N[(N[(N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[(-1.0 / B), $MachinePrecision] - N[(B * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.9e-20], N[(N[(N[Cos[B], $MachinePrecision] * (-x)), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7.4e+120], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -0.031

   1. Initial program 59.1%

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

   3. Simplified 74.6%

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

   5. Taylor expanded in x around 0 74.8%

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

      1. associate-*l/ 74.7%

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

      2. *-lft-identity 74.7%

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

      3. +-commutative 74.7%

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

      4. unpow2 74.7%

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

      5. fma-udef 74.7%

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

   7. Simplified 74.7%

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

   8. Taylor expanded in F around -inf 99.8%

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

   if -0.031 < F < -4.39999999999999976e-87

   1. Initial program 99.7%

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

   3. Taylor expanded in B around 0 81.6%

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

   4. Taylor expanded in B around 0 77.4%

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

   if -4.39999999999999976e-87 < F < 1.8999999999999999e-20

   1. Initial program 99.4%

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

   3. Taylor expanded in F around -inf 39.6%

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

   4. Taylor expanded in x around inf 83.9%

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

      1. associate-*r/ 83.9%

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

      2. *-commutative 83.9%

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

      3. neg-mul-1 83.9%

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

      4. distribute-rgt-neg-in 83.9%

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

   6. Simplified 83.9%

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

   if 1.8999999999999999e-20 < F < 7.40000000000000048e120

   1. Initial program 94.2%

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

   3. Taylor expanded in B around 0 88.7%

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

   4. Taylor expanded in F around inf 79.7%

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

   if 7.40000000000000048e120 < F

   1. Initial program 40.7%

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

   3. Simplified 62.0%

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

   5. Taylor expanded in F around inf 99.6%

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

      1. associate-/r* 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in B around 0 82.9%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.031:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -4.4 \cdot 10^{-87}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} + x \cdot \left(\frac{-1}{B} - B \cdot -0.3333333333333333\right)\\ \mathbf{elif}\;F \leq 1.9 \cdot 10^{-20}:\\ \;\;\;\;\frac{\cos B \cdot \left(-x\right)}{\sin B}\\ \mathbf{elif}\;F \leq 7.4 \cdot 10^{+120}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 64.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.037:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 5.3 \cdot 10^{-170}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} + x \cdot \left(\frac{-1}{B} - B \cdot -0.3333333333333333\right)\\ \mathbf{elif}\;F \leq 1.85 \cdot 10^{-20} \lor \neg \left(F \leq 8.2 \cdot 10^{+120}\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.037)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 5.3e-170)
     (+
      (* (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5) (/ F B))
      (* x (- (/ -1.0 B) (* B -0.3333333333333333))))
     (if (or (<= F 1.85e-20) (not (<= F 8.2e+120)))
       (- (/ 1.0 B) (/ x (tan B)))
       (- (/ 1.0 (sin B)) (/ x B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.037) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 5.3e-170) {
		tmp = (pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B)) + (x * ((-1.0 / B) - (B * -0.3333333333333333)));
	} else if ((F <= 1.85e-20) || !(F <= 8.2e+120)) {
		tmp = (1.0 / B) - (x / tan(B));
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-0.037d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 5.3d-170) then
        tmp = (((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0)) * (f / b)) + (x * (((-1.0d0) / b) - (b * (-0.3333333333333333d0))))
    else if ((f <= 1.85d-20) .or. (.not. (f <= 8.2d+120))) then
        tmp = (1.0d0 / b) - (x / tan(b))
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.037) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 5.3e-170) {
		tmp = (Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B)) + (x * ((-1.0 / B) - (B * -0.3333333333333333)));
	} else if ((F <= 1.85e-20) || !(F <= 8.2e+120)) {
		tmp = (1.0 / B) - (x / Math.tan(B));
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -0.037:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 5.3e-170:
		tmp = (math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B)) + (x * ((-1.0 / B) - (B * -0.3333333333333333)))
	elif (F <= 1.85e-20) or not (F <= 8.2e+120):
		tmp = (1.0 / B) - (x / math.tan(B))
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.037)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 5.3e-170)
		tmp = Float64(Float64((Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5) * Float64(F / B)) + Float64(x * Float64(Float64(-1.0 / B) - Float64(B * -0.3333333333333333))));
	elseif ((F <= 1.85e-20) || !(F <= 8.2e+120))
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -0.037)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 5.3e-170)
		tmp = (((((F * F) + 2.0) + (x * 2.0)) ^ -0.5) * (F / B)) + (x * ((-1.0 / B) - (B * -0.3333333333333333)));
	elseif ((F <= 1.85e-20) || ~((F <= 8.2e+120)))
		tmp = (1.0 / B) - (x / tan(B));
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.037], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.3e-170], N[(N[(N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[(-1.0 / B), $MachinePrecision] - N[(B * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[F, 1.85e-20], N[Not[LessEqual[F, 8.2e+120]], $MachinePrecision]], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.0369999999999999982

   1. Initial program 59.1%

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

   3. Taylor expanded in B around 0 38.8%

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

   4. Taylor expanded in F around -inf 78.6%

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

      1. distribute-lft-in 78.6%

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

      2. mul-1-neg 78.6%

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

      3. unsub-neg 78.6%

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

      4. associate-*r/ 78.6%

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

      5. metadata-eval 78.6%

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

   6. Simplified 78.6%

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

   if -0.0369999999999999982 < F < 5.3e-170

   1. Initial program 99.5%

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

   3. Taylor expanded in B around 0 62.8%

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

   4. Taylor expanded in B around 0 61.7%

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

   if 5.3e-170 < F < 1.85e-20 or 8.2e120 < F

   1. Initial program 57.3%

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

   3. Simplified 72.6%

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

   5. Taylor expanded in F around inf 90.4%

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

      1. associate-/r* 90.4%

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

   7. Simplified 90.4%

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

   8. Taylor expanded in B around 0 83.3%

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

   if 1.85e-20 < F < 8.2e120

   1. Initial program 94.2%

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

   3. Taylor expanded in B around 0 88.7%

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

   4. Taylor expanded in F around inf 79.7%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.037:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 5.3 \cdot 10^{-170}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} + x \cdot \left(\frac{-1}{B} - B \cdot -0.3333333333333333\right)\\ \mathbf{elif}\;F \leq 1.85 \cdot 10^{-20} \lor \neg \left(F \leq 8.2 \cdot 10^{+120}\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 64.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -7500:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 7.8 \cdot 10^{-168}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \left(\frac{F}{B} + 0.16666666666666666 \cdot \left(F \cdot B\right)\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.9 \cdot 10^{-20} \lor \neg \left(F \leq 7.5 \cdot 10^{+120}\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -7500.0)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 7.8e-168)
     (-
      (*
       (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5)
       (+ (/ F B) (* 0.16666666666666666 (* F B))))
      (/ x B))
     (if (or (<= F 1.9e-20) (not (<= F 7.5e+120)))
       (- (/ 1.0 B) (/ x (tan B)))
       (- (/ 1.0 (sin B)) (/ x B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -7500.0) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 7.8e-168) {
		tmp = (pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * ((F / B) + (0.16666666666666666 * (F * B)))) - (x / B);
	} else if ((F <= 1.9e-20) || !(F <= 7.5e+120)) {
		tmp = (1.0 / B) - (x / tan(B));
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-7500.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 7.8d-168) then
        tmp = (((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0)) * ((f / b) + (0.16666666666666666d0 * (f * b)))) - (x / b)
    else if ((f <= 1.9d-20) .or. (.not. (f <= 7.5d+120))) then
        tmp = (1.0d0 / b) - (x / tan(b))
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -7500.0) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 7.8e-168) {
		tmp = (Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * ((F / B) + (0.16666666666666666 * (F * B)))) - (x / B);
	} else if ((F <= 1.9e-20) || !(F <= 7.5e+120)) {
		tmp = (1.0 / B) - (x / Math.tan(B));
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -7500.0:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 7.8e-168:
		tmp = (math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * ((F / B) + (0.16666666666666666 * (F * B)))) - (x / B)
	elif (F <= 1.9e-20) or not (F <= 7.5e+120):
		tmp = (1.0 / B) - (x / math.tan(B))
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -7500.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 7.8e-168)
		tmp = Float64(Float64((Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5) * Float64(Float64(F / B) + Float64(0.16666666666666666 * Float64(F * B)))) - Float64(x / B));
	elseif ((F <= 1.9e-20) || !(F <= 7.5e+120))
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -7500.0)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 7.8e-168)
		tmp = (((((F * F) + 2.0) + (x * 2.0)) ^ -0.5) * ((F / B) + (0.16666666666666666 * (F * B)))) - (x / B);
	elseif ((F <= 1.9e-20) || ~((F <= 7.5e+120)))
		tmp = (1.0 / B) - (x / tan(B));
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -7500.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7.8e-168], N[(N[(N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(N[(F / B), $MachinePrecision] + N[(0.16666666666666666 * N[(F * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[F, 1.9e-20], N[Not[LessEqual[F, 7.5e+120]], $MachinePrecision]], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -7500

   1. Initial program 58.6%

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

   3. Taylor expanded in B around 0 39.3%

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

   4. Taylor expanded in F around -inf 79.7%

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

      1. distribute-lft-in 79.7%

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

      2. mul-1-neg 79.7%

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

      3. unsub-neg 79.7%

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

      4. associate-*r/ 79.7%

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

      5. metadata-eval 79.7%

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

   6. Simplified 79.7%

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

   if -7500 < F < 7.80000000000000025e-168

   1. Initial program 99.4%

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

   3. Taylor expanded in B around 0 71.3%

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

   4. Taylor expanded in B around 0 61.0%

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

   if 7.80000000000000025e-168 < F < 1.8999999999999999e-20 or 7.5000000000000006e120 < F

   1. Initial program 57.3%

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

   3. Simplified 72.6%

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

   5. Taylor expanded in F around inf 90.4%

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

      1. associate-/r* 90.4%

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

   7. Simplified 90.4%

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

   8. Taylor expanded in B around 0 83.3%

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

   if 1.8999999999999999e-20 < F < 7.5000000000000006e120

   1. Initial program 94.2%

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

   3. Taylor expanded in B around 0 88.7%

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

   4. Taylor expanded in F around inf 79.7%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7500:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 7.8 \cdot 10^{-168}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \left(\frac{F}{B} + 0.16666666666666666 \cdot \left(F \cdot B\right)\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.9 \cdot 10^{-20} \lor \neg \left(F \leq 7.5 \cdot 10^{+120}\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 62.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -4500000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -8.6 \cdot 10^{-216}:\\ \;\;\;\;\frac{-1}{B} - t\_0\\ \mathbf{elif}\;F \leq 5.3 \cdot 10^{-295}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{-20} \lor \neg \left(F \leq 6 \cdot 10^{+120}\right):\\ \;\;\;\;\frac{1}{B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -4500000.0)
     (- (/ -1.0 (sin B)) (/ x B))
     (if (<= F -8.6e-216)
       (- (/ -1.0 B) t_0)
       (if (<= F 5.3e-295)
         (/ (- x) B)
         (if (or (<= F 1.8e-20) (not (<= F 6e+120)))
           (- (/ 1.0 B) t_0)
           (- (/ 1.0 (sin B)) (/ x B))))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -4500000.0) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -8.6e-216) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= 5.3e-295) {
		tmp = -x / B;
	} else if ((F <= 1.8e-20) || !(F <= 6e+120)) {
		tmp = (1.0 / B) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-4500000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-8.6d-216)) then
        tmp = ((-1.0d0) / b) - t_0
    else if (f <= 5.3d-295) then
        tmp = -x / b
    else if ((f <= 1.8d-20) .or. (.not. (f <= 6d+120))) then
        tmp = (1.0d0 / b) - t_0
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -4500000.0) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -8.6e-216) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= 5.3e-295) {
		tmp = -x / B;
	} else if ((F <= 1.8e-20) || !(F <= 6e+120)) {
		tmp = (1.0 / B) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -4500000.0:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -8.6e-216:
		tmp = (-1.0 / B) - t_0
	elif F <= 5.3e-295:
		tmp = -x / B
	elif (F <= 1.8e-20) or not (F <= 6e+120):
		tmp = (1.0 / B) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -4500000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -8.6e-216)
		tmp = Float64(Float64(-1.0 / B) - t_0);
	elseif (F <= 5.3e-295)
		tmp = Float64(Float64(-x) / B);
	elseif ((F <= 1.8e-20) || !(F <= 6e+120))
		tmp = Float64(Float64(1.0 / B) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -4500000.0)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -8.6e-216)
		tmp = (-1.0 / B) - t_0;
	elseif (F <= 5.3e-295)
		tmp = -x / B;
	elseif ((F <= 1.8e-20) || ~((F <= 6e+120)))
		tmp = (1.0 / B) - t_0;
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -4500000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -8.6e-216], N[(N[(-1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 5.3e-295], N[((-x) / B), $MachinePrecision], If[Or[LessEqual[F, 1.8e-20], N[Not[LessEqual[F, 6e+120]], $MachinePrecision]], N[(N[(1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -4.5e6

   1. Initial program 58.6%

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

   3. Taylor expanded in B around 0 39.3%

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

   4. Taylor expanded in F around -inf 79.7%

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

      1. distribute-lft-in 79.7%

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

      2. mul-1-neg 79.7%

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

      3. unsub-neg 79.7%

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

      4. associate-*r/ 79.7%

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

      5. metadata-eval 79.7%

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

   6. Simplified 79.7%

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

   if -4.5e6 < F < -8.5999999999999995e-216

   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 x around 0 99.2%

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

      1. associate-*l/ 99.3%

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

      2. *-lft-identity 99.3%

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

      3. +-commutative 99.3%

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

      4. unpow2 99.3%

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

      5. fma-udef 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in F around -inf 30.7%

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

   9. Taylor expanded in B around 0 42.2%

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

   if -8.5999999999999995e-216 < F < 5.3000000000000004e-295

   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 99.8%

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

   5. Taylor expanded in F around inf 11.7%

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

      1. associate-/r* 11.7%

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

   7. Simplified 11.7%

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

   8. Taylor expanded in B around 0 30.1%

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

   9. Taylor expanded in x around inf 81.0%

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

       1. neg-mul-1 81.0%

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

       2. distribute-neg-frac 81.0%

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

   11. Simplified 81.0%

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

   if 5.3000000000000004e-295 < F < 1.79999999999999987e-20 or 6e120 < F

   1. Initial program 65.4%

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

   3. Simplified 77.8%

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

   5. Taylor expanded in F around inf 78.4%

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

      1. associate-/r* 78.4%

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

   7. Simplified 78.4%

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

   8. Taylor expanded in B around 0 76.6%

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

   if 1.79999999999999987e-20 < F < 6e120

   1. Initial program 94.2%

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

   3. Taylor expanded in B around 0 88.7%

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

   4. Taylor expanded in F around inf 79.7%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4500000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -8.6 \cdot 10^{-216}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 5.3 \cdot 10^{-295}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{-20} \lor \neg \left(F \leq 6 \cdot 10^{+120}\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 64.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.037:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.85 \cdot 10^{-297}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4.6 \cdot 10^{-22} \lor \neg \left(F \leq 1.01 \cdot 10^{+121}\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.037)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 1.85e-297)
     (- (* (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5) (/ F B)) (/ x B))
     (if (or (<= F 4.6e-22) (not (<= F 1.01e+121)))
       (- (/ 1.0 B) (/ x (tan B)))
       (- (/ 1.0 (sin B)) (/ x B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.037) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 1.85e-297) {
		tmp = (pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else if ((F <= 4.6e-22) || !(F <= 1.01e+121)) {
		tmp = (1.0 / B) - (x / tan(B));
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-0.037d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 1.85d-297) then
        tmp = (((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0)) * (f / b)) - (x / b)
    else if ((f <= 4.6d-22) .or. (.not. (f <= 1.01d+121))) then
        tmp = (1.0d0 / b) - (x / tan(b))
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.037) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 1.85e-297) {
		tmp = (Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else if ((F <= 4.6e-22) || !(F <= 1.01e+121)) {
		tmp = (1.0 / B) - (x / Math.tan(B));
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -0.037:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 1.85e-297:
		tmp = (math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B)) - (x / B)
	elif (F <= 4.6e-22) or not (F <= 1.01e+121):
		tmp = (1.0 / B) - (x / math.tan(B))
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.037)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 1.85e-297)
		tmp = Float64(Float64((Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5) * Float64(F / B)) - Float64(x / B));
	elseif ((F <= 4.6e-22) || !(F <= 1.01e+121))
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -0.037)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 1.85e-297)
		tmp = (((((F * F) + 2.0) + (x * 2.0)) ^ -0.5) * (F / B)) - (x / B);
	elseif ((F <= 4.6e-22) || ~((F <= 1.01e+121)))
		tmp = (1.0 / B) - (x / tan(B));
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.037], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.85e-297], N[(N[(N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[F, 4.6e-22], N[Not[LessEqual[F, 1.01e+121]], $MachinePrecision]], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.0369999999999999982

   1. Initial program 59.1%

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

   3. Taylor expanded in B around 0 38.8%

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

   4. Taylor expanded in F around -inf 78.6%

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

      1. distribute-lft-in 78.6%

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

      2. mul-1-neg 78.6%

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

      3. unsub-neg 78.6%

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

      4. associate-*r/ 78.6%

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

      5. metadata-eval 78.6%

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

   6. Simplified 78.6%

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

   if -0.0369999999999999982 < F < 1.85e-297

   1. Initial program 99.5%

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

   3. Taylor expanded in B around 0 78.1%

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

   4. Taylor expanded in B around 0 65.1%

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

   if 1.85e-297 < F < 4.5999999999999996e-22 or 1.01e121 < F

   1. Initial program 65.4%

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

   3. Simplified 77.8%

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

   5. Taylor expanded in F around inf 78.4%

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

      1. associate-/r* 78.4%

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

   7. Simplified 78.4%

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

   8. Taylor expanded in B around 0 76.6%

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

   if 4.5999999999999996e-22 < F < 1.01e121

   1. Initial program 94.2%

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

   3. Taylor expanded in B around 0 88.7%

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

   4. Taylor expanded in F around inf 79.7%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.037:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.85 \cdot 10^{-297}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4.6 \cdot 10^{-22} \lor \neg \left(F \leq 1.01 \cdot 10^{+121}\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 61.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -105000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -5 \cdot 10^{-211}:\\ \;\;\;\;\frac{-1}{B} - t\_0\\ \mathbf{elif}\;F \leq 3.4 \cdot 10^{-294}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -105000.0)
     (- (/ -1.0 (sin B)) (/ x B))
     (if (<= F -5e-211)
       (- (/ -1.0 B) t_0)
       (if (<= F 3.4e-294) (/ (- x) B) (- (/ 1.0 B) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -105000.0) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -5e-211) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= 3.4e-294) {
		tmp = -x / B;
	} else {
		tmp = (1.0 / B) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-105000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-5d-211)) then
        tmp = ((-1.0d0) / b) - t_0
    else if (f <= 3.4d-294) then
        tmp = -x / b
    else
        tmp = (1.0d0 / b) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -105000.0) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -5e-211) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= 3.4e-294) {
		tmp = -x / B;
	} else {
		tmp = (1.0 / B) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -105000.0:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -5e-211:
		tmp = (-1.0 / B) - t_0
	elif F <= 3.4e-294:
		tmp = -x / B
	else:
		tmp = (1.0 / B) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -105000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -5e-211)
		tmp = Float64(Float64(-1.0 / B) - t_0);
	elseif (F <= 3.4e-294)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(1.0 / B) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -105000.0)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -5e-211)
		tmp = (-1.0 / B) - t_0;
	elseif (F <= 3.4e-294)
		tmp = -x / B;
	else
		tmp = (1.0 / B) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -105000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -5e-211], N[(N[(-1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 3.4e-294], N[((-x) / B), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -105000

   1. Initial program 58.6%

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

   3. Taylor expanded in B around 0 39.3%

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

   4. Taylor expanded in F around -inf 79.7%

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

      1. distribute-lft-in 79.7%

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

      2. mul-1-neg 79.7%

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

      3. unsub-neg 79.7%

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

      4. associate-*r/ 79.7%

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

      5. metadata-eval 79.7%

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

   6. Simplified 79.7%

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

   if -105000 < F < -5.0000000000000002e-211

   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 x around 0 99.2%

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

      1. associate-*l/ 99.3%

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

      2. *-lft-identity 99.3%

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

      3. +-commutative 99.3%

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

      4. unpow2 99.3%

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

      5. fma-udef 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in F around -inf 30.7%

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

   9. Taylor expanded in B around 0 42.2%

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

   if -5.0000000000000002e-211 < F < 3.39999999999999981e-294

   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 99.8%

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

   5. Taylor expanded in F around inf 11.7%

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

      1. associate-/r* 11.7%

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

   7. Simplified 11.7%

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

   8. Taylor expanded in B around 0 30.1%

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

   9. Taylor expanded in x around inf 81.0%

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

       1. neg-mul-1 81.0%

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

       2. distribute-neg-frac 81.0%

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

   11. Simplified 81.0%

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

   if 3.39999999999999981e-294 < F

   1. Initial program 73.6%

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

   3. Simplified 84.0%

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

   5. Taylor expanded in F around inf 80.3%

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

      1. associate-/r* 80.3%

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

   7. Simplified 80.3%

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

   8. Taylor expanded in B around 0 70.4%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -105000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -5 \cdot 10^{-211}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 3.4 \cdot 10^{-294}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 55.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-79} \lor \neg \left(x \leq 1.5 \cdot 10^{-35}\right):\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (or (<= x -8e-79) (not (<= x 1.5e-35)))
   (- (/ -1.0 B) (/ x (tan B)))
   (- (/ -1.0 (sin B)) (/ x B))))

C

double code(double F, double B, double x) {
	double tmp;
	if ((x <= -8e-79) || !(x <= 1.5e-35)) {
		tmp = (-1.0 / B) - (x / tan(B));
	} else {
		tmp = (-1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-8d-79)) .or. (.not. (x <= 1.5d-35))) then
        tmp = ((-1.0d0) / b) - (x / tan(b))
    else
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if ((x <= -8e-79) || !(x <= 1.5e-35)) {
		tmp = (-1.0 / B) - (x / Math.tan(B));
	} else {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if (x <= -8e-79) or not (x <= 1.5e-35):
		tmp = (-1.0 / B) - (x / math.tan(B))
	else:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if ((x <= -8e-79) || !(x <= 1.5e-35))
		tmp = Float64(Float64(-1.0 / B) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if ((x <= -8e-79) || ~((x <= 1.5e-35)))
		tmp = (-1.0 / B) - (x / tan(B));
	else
		tmp = (-1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[Or[LessEqual[x, -8e-79], N[Not[LessEqual[x, 1.5e-35]], $MachinePrecision]], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8e-79 or 1.49999999999999994e-35 < x

   1. Initial program 81.2%

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

   3. Simplified 94.1%

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

   5. Taylor expanded in x around 0 94.0%

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

      1. associate-*l/ 94.0%

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

      2. *-lft-identity 94.0%

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

      3. +-commutative 94.0%

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

      4. unpow2 94.0%

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

      5. fma-udef 94.0%

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

   7. Simplified 94.0%

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

   8. Taylor expanded in F around -inf 80.1%

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

   9. Taylor expanded in B around 0 82.8%

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

   if -8e-79 < x < 1.49999999999999994e-35

   1. Initial program 68.4%

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

   3. Taylor expanded in B around 0 62.4%

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

   4. Taylor expanded in F around -inf 31.6%

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

      1. distribute-lft-in 31.6%

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

      2. mul-1-neg 31.6%

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

      3. unsub-neg 31.6%

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

      4. associate-*r/ 31.6%

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

      5. metadata-eval 31.6%

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

   6. Simplified 31.6%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-79} \lor \neg \left(x \leq 1.5 \cdot 10^{-35}\right):\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 41.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 5.2 \cdot 10^{-216}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;B \leq 1.7 \cdot 10^{-53}:\\ \;\;\;\;\frac{1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 5.2e-216)
   (/ (- x) B)
   (if (<= B 1.7e-53) (/ (- 1.0 x) B) (- (/ -1.0 B) (/ x (tan B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 5.2e-216) {
		tmp = -x / B;
	} else if (B <= 1.7e-53) {
		tmp = (1.0 - 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 <= 5.2d-216) then
        tmp = -x / b
    else if (b <= 1.7d-53) then
        tmp = (1.0d0 - 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 <= 5.2e-216) {
		tmp = -x / B;
	} else if (B <= 1.7e-53) {
		tmp = (1.0 - x) / B;
	} else {
		tmp = (-1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if B <= 5.2e-216:
		tmp = -x / B
	elif B <= 1.7e-53:
		tmp = (1.0 - x) / B
	else:
		tmp = (-1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 5.2e-216)
		tmp = Float64(Float64(-x) / B);
	elseif (B <= 1.7e-53)
		tmp = Float64(Float64(1.0 - 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 <= 5.2e-216)
		tmp = -x / B;
	elseif (B <= 1.7e-53)
		tmp = (1.0 - x) / B;
	else
		tmp = (-1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 5.2e-216], N[((-x) / B), $MachinePrecision], If[LessEqual[B, 1.7e-53], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 5.1999999999999997e-216

   1. Initial program 72.8%

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

   3. Simplified 83.5%

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

   5. Taylor expanded in F around inf 52.7%

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

      1. associate-/r* 52.7%

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

   7. Simplified 52.7%

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

   8. Taylor expanded in B around 0 31.5%

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

   9. Taylor expanded in x around inf 35.4%

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

       1. neg-mul-1 35.4%

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

       2. distribute-neg-frac 35.4%

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

   11. Simplified 35.4%

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

   if 5.1999999999999997e-216 < B < 1.7e-53

   1. Initial program 65.9%

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

   3. Simplified 82.1%

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

   5. Taylor expanded in F around inf 68.1%

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

      1. associate-/r* 68.0%

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

   7. Simplified 68.0%

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

   8. Taylor expanded in B around 0 69.8%

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

   if 1.7e-53 < B

   1. Initial program 87.1%

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

   3. Simplified 90.1%

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

   5. Taylor expanded in x around 0 90.2%

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

      1. associate-*l/ 90.1%

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

      2. *-lft-identity 90.1%

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

      3. +-commutative 90.1%

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

      4. unpow2 90.1%

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

      5. fma-udef 90.1%

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

   7. Simplified 90.1%

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

   8. Taylor expanded in F around -inf 57.2%

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

   9. Taylor expanded in B around 0 54.4%

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

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 5.2 \cdot 10^{-216}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;B \leq 1.7 \cdot 10^{-53}:\\ \;\;\;\;\frac{1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 43.8% accurate, 17.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -6.3 \cdot 10^{-9}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 9.5 \cdot 10^{-15}:\\ \;\;\;\;x \cdot \left(\frac{-1}{B} - B \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -6.3e-9)
   (/ (- -1.0 x) B)
   (if (<= F 9.5e-15)
     (* x (- (/ -1.0 B) (* B -0.3333333333333333)))
     (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -6.3e-9) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 9.5e-15) {
		tmp = x * ((-1.0 / B) - (B * -0.3333333333333333));
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-6.3d-9)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 9.5d-15) then
        tmp = x * (((-1.0d0) / b) - (b * (-0.3333333333333333d0)))
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -6.3e-9) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 9.5e-15) {
		tmp = x * ((-1.0 / B) - (B * -0.3333333333333333));
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -6.3e-9:
		tmp = (-1.0 - x) / B
	elif F <= 9.5e-15:
		tmp = x * ((-1.0 / B) - (B * -0.3333333333333333))
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -6.3e-9)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 9.5e-15)
		tmp = Float64(x * Float64(Float64(-1.0 / B) - Float64(B * -0.3333333333333333)));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -6.3e-9)
		tmp = (-1.0 - x) / B;
	elseif (F <= 9.5e-15)
		tmp = x * ((-1.0 / B) - (B * -0.3333333333333333));
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -6.3e-9], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 9.5e-15], N[(x * N[(N[(-1.0 / B), $MachinePrecision] - N[(B * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -6.3000000000000002e-9

   1. Initial program 59.7%

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

   3. Taylor expanded in F around -inf 98.4%

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

   4. Taylor expanded in B around 0 46.3%

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

      1. associate-*r/ 46.3%

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

      2. distribute-lft-in 46.3%

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

      3. metadata-eval 46.3%

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

      4. neg-mul-1 46.3%

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

   6. Simplified 46.3%

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

   if -6.3000000000000002e-9 < F < 9.5000000000000005e-15

   1. Initial program 99.5%

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

   3. Taylor expanded in B around 0 56.0%

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

   4. Taylor expanded in F around inf 24.0%

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

   5. Taylor expanded in x around inf 43.7%

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

   if 9.5000000000000005e-15 < F

   1. Initial program 60.7%

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

   3. Simplified 76.2%

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

   5. Taylor expanded in F around inf 94.5%

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

      1. associate-/r* 94.4%

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

   7. Simplified 94.4%

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

   8. Taylor expanded in B around 0 51.3%

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

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6.3 \cdot 10^{-9}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 9.5 \cdot 10^{-15}:\\ \;\;\;\;x \cdot \left(\frac{-1}{B} - B \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 43.8% accurate, 17.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.85 \cdot 10^{-9}:\\ \;\;\;\;\frac{-1 - x}{B} + B \cdot \left(x \cdot 0.3333333333333333 - 0.16666666666666666\right)\\ \mathbf{elif}\;F \leq 1.15 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \left(\frac{-1}{B} - B \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.85e-9)
   (+ (/ (- -1.0 x) B) (* B (- (* x 0.3333333333333333) 0.16666666666666666)))
   (if (<= F 1.15e-14)
     (* x (- (/ -1.0 B) (* B -0.3333333333333333)))
     (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.85e-9) {
		tmp = ((-1.0 - x) / B) + (B * ((x * 0.3333333333333333) - 0.16666666666666666));
	} else if (F <= 1.15e-14) {
		tmp = x * ((-1.0 / B) - (B * -0.3333333333333333));
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.85d-9)) then
        tmp = (((-1.0d0) - x) / b) + (b * ((x * 0.3333333333333333d0) - 0.16666666666666666d0))
    else if (f <= 1.15d-14) then
        tmp = x * (((-1.0d0) / b) - (b * (-0.3333333333333333d0)))
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.85e-9) {
		tmp = ((-1.0 - x) / B) + (B * ((x * 0.3333333333333333) - 0.16666666666666666));
	} else if (F <= 1.15e-14) {
		tmp = x * ((-1.0 / B) - (B * -0.3333333333333333));
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.85e-9:
		tmp = ((-1.0 - x) / B) + (B * ((x * 0.3333333333333333) - 0.16666666666666666))
	elif F <= 1.15e-14:
		tmp = x * ((-1.0 / B) - (B * -0.3333333333333333))
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.85e-9)
		tmp = Float64(Float64(Float64(-1.0 - x) / B) + Float64(B * Float64(Float64(x * 0.3333333333333333) - 0.16666666666666666)));
	elseif (F <= 1.15e-14)
		tmp = Float64(x * Float64(Float64(-1.0 / B) - Float64(B * -0.3333333333333333)));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.85e-9)
		tmp = ((-1.0 - x) / B) + (B * ((x * 0.3333333333333333) - 0.16666666666666666));
	elseif (F <= 1.15e-14)
		tmp = x * ((-1.0 / B) - (B * -0.3333333333333333));
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.85e-9], N[(N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision] + N[(B * N[(N[(x * 0.3333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.15e-14], N[(x * N[(N[(-1.0 / B), $MachinePrecision] - N[(B * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.85e-9

   1. Initial program 59.7%

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

   3. Taylor expanded in F around -inf 98.4%

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

   4. Taylor expanded in B around 0 46.3%

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

   if -1.85e-9 < F < 1.14999999999999999e-14

   1. Initial program 99.5%

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

   3. Taylor expanded in B around 0 56.0%

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

   4. Taylor expanded in F around inf 24.0%

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

   5. Taylor expanded in x around inf 43.7%

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

   if 1.14999999999999999e-14 < F

   1. Initial program 60.7%

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

   3. Simplified 76.2%

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

   5. Taylor expanded in F around inf 94.5%

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

      1. associate-/r* 94.4%

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

   7. Simplified 94.4%

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

   8. Taylor expanded in B around 0 51.3%

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

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.85 \cdot 10^{-9}:\\ \;\;\;\;\frac{-1 - x}{B} + B \cdot \left(x \cdot 0.3333333333333333 - 0.16666666666666666\right)\\ \mathbf{elif}\;F \leq 1.15 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \left(\frac{-1}{B} - B \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 28: 43.7% accurate, 21.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.75 \cdot 10^{-9}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 9.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.75e-9)
   (/ (- -1.0 x) B)
   (if (<= F 9.2e-12) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.75e-9) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 9.2e-12) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-2.75d-9)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 9.2d-12) then
        tmp = -x / b
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.75e-9) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 9.2e-12) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.75e-9:
		tmp = (-1.0 - x) / B
	elif F <= 9.2e-12:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.75e-9)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 9.2e-12)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.75e-9)
		tmp = (-1.0 - x) / B;
	elseif (F <= 9.2e-12)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.75e-9], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 9.2e-12], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.7499999999999998e-9

   1. Initial program 59.7%

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

   3. Taylor expanded in F around -inf 98.4%

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

   4. Taylor expanded in B around 0 46.3%

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

      1. associate-*r/ 46.3%

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

      2. distribute-lft-in 46.3%

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

      3. metadata-eval 46.3%

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

      4. neg-mul-1 46.3%

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

   6. Simplified 46.3%

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

   if -2.7499999999999998e-9 < F < 9.19999999999999957e-12

   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 inf 33.2%

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

      1. associate-/r* 33.2%

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

   7. Simplified 33.2%

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

   8. Taylor expanded in B around 0 23.9%

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

   9. Taylor expanded in x around inf 43.1%

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

       1. neg-mul-1 43.1%

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

       2. distribute-neg-frac 43.1%

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

   11. Simplified 43.1%

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

   if 9.19999999999999957e-12 < F

   1. Initial program 60.2%

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

   3. Simplified 75.9%

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

   5. Taylor expanded in F around inf 95.5%

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

      1. associate-/r* 95.5%

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

   7. Simplified 95.5%

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

   8. Taylor expanded in B around 0 51.9%

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

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.75 \cdot 10^{-9}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 9.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 29: 36.3% accurate, 32.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 1.9 \cdot 10^{-12}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F 1.9e-12) (/ (- x) B) (/ (- 1.0 x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= 1.9e-12) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= 1.9d-12) then
        tmp = -x / b
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= 1.9e-12) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= 1.9e-12:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= 1.9e-12)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= 1.9e-12)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, 1.9e-12], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 1.89999999999999998e-12

   1. Initial program 82.9%

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

   3. Simplified 89.4%

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

   5. Taylor expanded in F around inf 36.2%

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

      1. associate-/r* 36.2%

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

   7. Simplified 36.2%

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

   8. Taylor expanded in B around 0 22.9%

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

   9. Taylor expanded in x around inf 34.3%

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

       1. neg-mul-1 34.3%

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

       2. distribute-neg-frac 34.3%

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

   11. Simplified 34.3%

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

   if 1.89999999999999998e-12 < F

   1. Initial program 60.2%

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

   3. Simplified 75.9%

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

   5. Taylor expanded in F around inf 95.5%

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

      1. associate-/r* 95.5%

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

   7. Simplified 95.5%

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

   8. Taylor expanded in B around 0 51.9%

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

4. Final simplification 39.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.9 \cdot 10^{-12}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 30: 29.0% accurate, 81.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.7%

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

3. Simplified 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 54.9%

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

   1. associate-/r* 54.9%

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

7. Simplified 54.9%

   \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

8. Taylor expanded in B around 0 32.0%

   \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

9. Taylor expanded in x around inf 32.4%

   \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
10. Step-by-step derivation

    1. neg-mul-1 32.4%

       \[\leadsto \color{blue}{-\frac{x}{B}} \]

    2. distribute-neg-frac 32.4%

       \[\leadsto \color{blue}{\frac{-x}{B}} \]

11. Simplified 32.4%

    \[\leadsto \color{blue}{\frac{-x}{B}} \]

12. Final simplification 32.4%

    \[\leadsto \frac{-x}{B} \]
13. Add Preprocessing

Alternative 31: 10.0% 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 75.7%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

3. Simplified 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 54.9%

   \[\leadsto F \cdot \color{blue}{\frac{1}{F \cdot \sin B}} - \frac{x}{\tan B} \]
6. Step-by-step derivation

   1. associate-/r* 54.9%

      \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

7. Simplified 54.9%

   \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

8. Taylor expanded in B around 0 32.0%

   \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

9. Taylor expanded in x around 0 9.5%

   \[\leadsto \color{blue}{\frac{1}{B}} \]

10. Final simplification 9.5%

    \[\leadsto \frac{1}{B} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024027 
(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))))))