VandenBroeck and Keller, Equation (23)

Percentage Accurate: 77.4% → 98.9%

Time: 18.3s

Alternatives: 22

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.4% 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: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -420000:\\ \;\;\;\;\frac{-1}{\sin B} - t_1\\ \mathbf{elif}\;F \leq 13:\\ \;\;\;\;F \cdot \left(t_0 \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \left(t_0 + \frac{-1 - x}{F \cdot \left(F \cdot \sin B\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))) (t_1 (/ x (tan B))))
   (if (<= F -420000.0)
     (- (/ -1.0 (sin B)) t_1)
     (if (<= F 13.0)
       (- (* F (* t_0 (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))))) t_1)
       (+
        (* x (/ -1.0 (tan B)))
        (+ t_0 (/ (- -1.0 x) (* F (* F (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 <= -420000.0) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= 13.0) {
		tmp = (F * (t_0 * sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_1;
	} else {
		tmp = (x * (-1.0 / tan(B))) + (t_0 + ((-1.0 - x) / (F * (F * 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 <= (-420000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_1
    else if (f <= 13.0d0) then
        tmp = (f * (t_0 * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))))) - t_1
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (t_0 + (((-1.0d0) - x) / (f * (f * 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 <= -420000.0) {
		tmp = (-1.0 / Math.sin(B)) - t_1;
	} else if (F <= 13.0) {
		tmp = (F * (t_0 * Math.sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_1;
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (t_0 + ((-1.0 - x) / (F * (F * 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 <= -420000.0:
		tmp = (-1.0 / math.sin(B)) - t_1
	elif F <= 13.0:
		tmp = (F * (t_0 * math.sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_1
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (t_0 + ((-1.0 - x) / (F * (F * 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 <= -420000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= 13.0)
		tmp = Float64(Float64(F * Float64(t_0 * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0)))))) - t_1);
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(t_0 + Float64(Float64(-1.0 - x) / Float64(F * Float64(F * 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 <= -420000.0)
		tmp = (-1.0 / sin(B)) - t_1;
	elseif (F <= 13.0)
		tmp = (F * (t_0 * sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_1;
	else
		tmp = (x * (-1.0 / tan(B))) + (t_0 + ((-1.0 - x) / (F * (F * 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, -420000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, 13.0], 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[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 + N[(N[(-1.0 - x), $MachinePrecision] / N[(F * N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.2e5

   1. Initial program 50.7%

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

   2. Taylor expanded in F around -inf 99.8%

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

      1. *-un-lft-identity 99.8%

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

      2. +-commutative 99.8%

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

      3. div-inv 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. *-lft-identity 99.8%

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

      2. unsub-neg 99.8%

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

   6. Simplified 99.8%

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

   if -4.2e5 < F < 13

   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. Step-by-step derivation

      1. +-commutative 99.5%

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

      2. unsub-neg 99.5%

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

      3. associate-*l/ 99.4%

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

      4. associate-*r/ 99.4%

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

      5. *-commutative 99.4%

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

   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. Taylor expanded in F around 0 99.6%

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

   if 13 < F

   1. Initial program 57.7%

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

   2. Taylor expanded in F around inf 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-*r/ 99.9%

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

      3. +-commutative 99.9%

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

      4. distribute-lft-in 99.9%

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

      5. metadata-eval 99.9%

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

      6. associate-*r* 99.9%

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

      7. metadata-eval 99.9%

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

      8. neg-mul-1 99.9%

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

      9. *-commutative 99.9%

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

      10. unpow2 99.9%

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

      11. associate-*l* 99.9%

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

   4. Simplified 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -420000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 13:\\ \;\;\;\;F \cdot \left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\right) - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \left(\frac{1}{\sin B} + \frac{-1 - x}{F \cdot \left(F \cdot \sin B\right)}\right)\\ \end{array} \]

Alternative 2: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -420000:\\ \;\;\;\;\frac{-1}{\sin B} - t_1\\ \mathbf{elif}\;F \leq 1.45:\\ \;\;\;\;F \cdot \left(t_0 \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 - t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))) (t_1 (/ x (tan B))))
   (if (<= F -420000.0)
     (- (/ -1.0 (sin B)) t_1)
     (if (<= F 1.45)
       (- (* F (* t_0 (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))))) t_1)
       (- t_0 t_1)))))

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 <= -420000.0) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= 1.45) {
		tmp = (F * (t_0 * sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_1;
	} else {
		tmp = t_0 - 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 = 1.0d0 / sin(b)
    t_1 = x / tan(b)
    if (f <= (-420000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_1
    else if (f <= 1.45d0) then
        tmp = (f * (t_0 * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))))) - t_1
    else
        tmp = t_0 - t_1
    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 <= -420000.0) {
		tmp = (-1.0 / Math.sin(B)) - t_1;
	} else if (F <= 1.45) {
		tmp = (F * (t_0 * Math.sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_1;
	} else {
		tmp = t_0 - t_1;
	}
	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 <= -420000.0:
		tmp = (-1.0 / math.sin(B)) - t_1
	elif F <= 1.45:
		tmp = (F * (t_0 * math.sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_1
	else:
		tmp = t_0 - t_1
	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 <= -420000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= 1.45)
		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 - t_1);
	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 <= -420000.0)
		tmp = (-1.0 / sin(B)) - t_1;
	elseif (F <= 1.45)
		tmp = (F * (t_0 * sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_1;
	else
		tmp = t_0 - t_1;
	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, -420000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, 1.45], 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 - t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.2e5

   1. Initial program 50.7%

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

   2. Taylor expanded in F around -inf 99.8%

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

      1. *-un-lft-identity 99.8%

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

      2. +-commutative 99.8%

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

      3. div-inv 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. *-lft-identity 99.8%

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

      2. unsub-neg 99.8%

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

   6. Simplified 99.8%

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

   if -4.2e5 < F < 1.44999999999999996

   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. Step-by-step derivation

      1. +-commutative 99.5%

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

      2. unsub-neg 99.5%

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

      3. associate-*l/ 99.4%

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

      4. associate-*r/ 99.4%

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

      5. *-commutative 99.4%

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

   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. Taylor expanded in F around 0 99.6%

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

   if 1.44999999999999996 < F

   1. Initial program 58.3%

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

      1. +-commutative 58.3%

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

      2. unsub-neg 58.3%

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

      3. associate-*l/ 72.8%

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

      4. associate-*r/ 72.7%

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

      5. *-commutative 72.7%

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

   3. Simplified 72.7%

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

   4. Taylor expanded in F around inf 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-/r* 99.2%

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

   6. Simplified 99.2%

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

      1. sub-neg 99.2%

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

      2. associate-/l/ 99.3%

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

   8. Applied egg-rr 99.3%

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

      1. sub-neg 99.3%

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

      2. associate-*r/ 99.3%

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

      3. *-commutative 99.3%

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

      4. times-frac 99.5%

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

      5. *-rgt-identity 99.5%

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

      6. associate-*r/ 99.4%

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

      7. rgt-mult-inverse 99.5%

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

      8. *-lft-identity 99.5%

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

   10. Simplified 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -420000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.45:\\ \;\;\;\;F \cdot \left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\right) - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 3: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -420000:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{2 + x \cdot 2}}}{\sin B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -420000.0)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.4)
       (- (* F (/ (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))) (sin B))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -420000.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.4) {
		tmp = (F * (sqrt((1.0 / (2.0 + (x * 2.0)))) / sin(B))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-420000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.4d0) then
        tmp = (f * (sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))) / sin(b))) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -420000.0) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.4) {
		tmp = (F * (Math.sqrt((1.0 / (2.0 + (x * 2.0)))) / Math.sin(B))) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -420000.0:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.4:
		tmp = (F * (math.sqrt((1.0 / (2.0 + (x * 2.0)))) / math.sin(B))) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -420000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.4)
		tmp = Float64(Float64(F * Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0)))) / sin(B))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -420000.0)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.4)
		tmp = (F * (sqrt((1.0 / (2.0 + (x * 2.0)))) / sin(B))) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -420000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.4], N[(N[(F * N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.2e5

   1. Initial program 50.7%

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

   2. Taylor expanded in F around -inf 99.8%

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

      1. *-un-lft-identity 99.8%

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

      2. +-commutative 99.8%

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

      3. div-inv 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. *-lft-identity 99.8%

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

      2. unsub-neg 99.8%

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

   6. Simplified 99.8%

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

   if -4.2e5 < F < 1.3999999999999999

   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. Step-by-step derivation

      1. +-commutative 99.5%

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

      2. unsub-neg 99.5%

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

      3. associate-*l/ 99.4%

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

      4. associate-*r/ 99.4%

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

      5. *-commutative 99.4%

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

   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. Taylor expanded in F around 0 99.6%

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

   if 1.3999999999999999 < F

   1. Initial program 58.3%

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

      1. +-commutative 58.3%

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

      2. unsub-neg 58.3%

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

      3. associate-*l/ 72.8%

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

      4. associate-*r/ 72.7%

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

      5. *-commutative 72.7%

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

   3. Simplified 72.7%

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

   4. Taylor expanded in F around inf 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-/r* 99.2%

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

   6. Simplified 99.2%

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

      1. sub-neg 99.2%

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

      2. associate-/l/ 99.3%

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

   8. Applied egg-rr 99.3%

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

      1. sub-neg 99.3%

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

      2. associate-*r/ 99.3%

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

      3. *-commutative 99.3%

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

      4. times-frac 99.5%

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

      5. *-rgt-identity 99.5%

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

      6. associate-*r/ 99.4%

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

      7. rgt-mult-inverse 99.5%

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

      8. *-lft-identity 99.5%

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

   10. Simplified 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -420000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{2 + x \cdot 2}}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 4: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -420000:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -420000.0)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.4)
       (- (* F (/ (sqrt 0.5) (sin B))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -420000.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.4) {
		tmp = (F * (sqrt(0.5) / sin(B))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-420000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.4d0) then
        tmp = (f * (sqrt(0.5d0) / sin(b))) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -420000.0) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.4) {
		tmp = (F * (Math.sqrt(0.5) / Math.sin(B))) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -420000.0:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.4:
		tmp = (F * (math.sqrt(0.5) / math.sin(B))) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -420000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.4)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / sin(B))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -420000.0)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.4)
		tmp = (F * (sqrt(0.5) / sin(B))) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -420000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.4], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.2e5

   1. Initial program 50.7%

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

   2. Taylor expanded in F around -inf 99.8%

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

      1. *-un-lft-identity 99.8%

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

      2. +-commutative 99.8%

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

      3. div-inv 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. *-lft-identity 99.8%

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

      2. unsub-neg 99.8%

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

   6. Simplified 99.8%

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

   if -4.2e5 < F < 1.3999999999999999

   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. Step-by-step derivation

      1. +-commutative 99.5%

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

      2. unsub-neg 99.5%

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

      3. associate-*l/ 99.4%

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

      4. associate-*r/ 99.4%

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

      5. *-commutative 99.4%

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

   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. Taylor expanded in F around 0 99.6%

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

   5. Taylor expanded in x around 0 99.4%

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

   if 1.3999999999999999 < F

   1. Initial program 58.3%

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

      1. +-commutative 58.3%

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

      2. unsub-neg 58.3%

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

      3. associate-*l/ 72.8%

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

      4. associate-*r/ 72.7%

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

      5. *-commutative 72.7%

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

   3. Simplified 72.7%

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

   4. Taylor expanded in F around inf 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-/r* 99.2%

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

   6. Simplified 99.2%

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

      1. sub-neg 99.2%

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

      2. associate-/l/ 99.3%

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

   8. Applied egg-rr 99.3%

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

      1. sub-neg 99.3%

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

      2. associate-*r/ 99.3%

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

      3. *-commutative 99.3%

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

      4. times-frac 99.5%

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

      5. *-rgt-identity 99.5%

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

      6. associate-*r/ 99.4%

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

      7. rgt-mult-inverse 99.5%

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

      8. *-lft-identity 99.5%

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

   10. Simplified 99.5%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -420000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 5: 91.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.1 \cdot 10^{-9}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 0.135:\\ \;\;\;\;F \cdot \left(\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{1}{B}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.1e-9)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 0.135)
       (- (* F (* (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))) (/ 1.0 B))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.1e-9) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 0.135) {
		tmp = (F * (sqrt((1.0 / (2.0 + (x * 2.0)))) * (1.0 / B))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.1d-9)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 0.135d0) then
        tmp = (f * (sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))) * (1.0d0 / b))) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.1e-9) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 0.135) {
		tmp = (F * (Math.sqrt((1.0 / (2.0 + (x * 2.0)))) * (1.0 / B))) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.1e-9:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 0.135:
		tmp = (F * (math.sqrt((1.0 / (2.0 + (x * 2.0)))) * (1.0 / B))) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.1e-9)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 0.135)
		tmp = Float64(Float64(F * Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0)))) * Float64(1.0 / B))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.1e-9)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 0.135)
		tmp = (F * (sqrt((1.0 / (2.0 + (x * 2.0)))) * (1.0 / B))) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.1e-9], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.135], N[(N[(F * N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.0999999999999999e-9

   1. Initial program 52.6%

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

   2. Taylor expanded in F around -inf 97.4%

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

      1. *-un-lft-identity 97.4%

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

      2. +-commutative 97.4%

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

      3. div-inv 97.4%

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

   4. Applied egg-rr 97.4%

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

      1. *-lft-identity 97.4%

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

      2. unsub-neg 97.4%

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

   6. Simplified 97.4%

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

   if -1.0999999999999999e-9 < F < 0.13500000000000001

   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. Step-by-step derivation

      1. +-commutative 99.5%

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

      2. unsub-neg 99.5%

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

      3. associate-*l/ 99.4%

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

      4. associate-*r/ 99.4%

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

      5. *-commutative 99.4%

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

   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. Taylor expanded in F around 0 99.6%

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

   5. Taylor expanded in B around 0 85.9%

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

   if 0.13500000000000001 < F

   1. Initial program 58.3%

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

      1. +-commutative 58.3%

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

      2. unsub-neg 58.3%

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

      3. associate-*l/ 72.8%

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

      4. associate-*r/ 72.7%

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

      5. *-commutative 72.7%

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

   3. Simplified 72.7%

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

   4. Taylor expanded in F around inf 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-/r* 99.2%

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

   6. Simplified 99.2%

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

      1. sub-neg 99.2%

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

      2. associate-/l/ 99.3%

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

   8. Applied egg-rr 99.3%

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

      1. sub-neg 99.3%

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

      2. associate-*r/ 99.3%

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

      3. *-commutative 99.3%

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

      4. times-frac 99.5%

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

      5. *-rgt-identity 99.5%

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

      6. associate-*r/ 99.4%

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

      7. rgt-mult-inverse 99.5%

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

      8. *-lft-identity 99.5%

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

   10. Simplified 99.5%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.1 \cdot 10^{-9}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.135:\\ \;\;\;\;F \cdot \left(\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{1}{B}\right) - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 6: 91.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.1 \cdot 10^{-9}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 0.41:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.1e-9)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 0.41)
       (- (* (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))) (/ F B)) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.1e-9) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 0.41) {
		tmp = (sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.1d-9)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 0.41d0) then
        tmp = (sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))) * (f / b)) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.1e-9) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 0.41) {
		tmp = (Math.sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.1e-9:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 0.41:
		tmp = (math.sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.1e-9)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 0.41)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0)))) * Float64(F / B)) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.1e-9)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 0.41)
		tmp = (sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.1e-9], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.41], N[(N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / B), $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 < -1.0999999999999999e-9

   1. Initial program 52.6%

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

   2. Taylor expanded in F around -inf 97.4%

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

      1. *-un-lft-identity 97.4%

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

      2. +-commutative 97.4%

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

      3. div-inv 97.4%

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

   4. Applied egg-rr 97.4%

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

      1. *-lft-identity 97.4%

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

      2. unsub-neg 97.4%

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

   6. Simplified 97.4%

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

   if -1.0999999999999999e-9 < F < 0.409999999999999976

   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. Step-by-step derivation

      1. +-commutative 99.5%

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

      2. unsub-neg 99.5%

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

      3. associate-*l/ 99.4%

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

      4. associate-*r/ 99.4%

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

      5. *-commutative 99.4%

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

   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. Taylor expanded in F around 0 99.6%

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

   5. Taylor expanded in B around 0 85.9%

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

   if 0.409999999999999976 < F

   1. Initial program 58.3%

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

      1. +-commutative 58.3%

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

      2. unsub-neg 58.3%

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

      3. associate-*l/ 72.8%

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

      4. associate-*r/ 72.7%

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

      5. *-commutative 72.7%

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

   3. Simplified 72.7%

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

   4. Taylor expanded in F around inf 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-/r* 99.2%

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

   6. Simplified 99.2%

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

      1. sub-neg 99.2%

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

      2. associate-/l/ 99.3%

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

   8. Applied egg-rr 99.3%

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

      1. sub-neg 99.3%

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

      2. associate-*r/ 99.3%

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

      3. *-commutative 99.3%

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

      4. times-frac 99.5%

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

      5. *-rgt-identity 99.5%

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

      6. associate-*r/ 99.4%

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

      7. rgt-mult-inverse 99.5%

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

      8. *-lft-identity 99.5%

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

   10. Simplified 99.5%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.1 \cdot 10^{-9}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.41:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 7: 70.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3.3 \cdot 10^{+209}:\\ \;\;\;\;\frac{-1}{B} - t_0\\ \mathbf{elif}\;F \leq -0.016:\\ \;\;\;\;\frac{-1 - x}{\sin B}\\ \mathbf{elif}\;F \leq -7.6 \cdot 10^{-90}:\\ \;\;\;\;\sqrt{0.5} \cdot \frac{F}{\sin B}\\ \mathbf{elif}\;F \leq 2.7 \cdot 10^{-95}:\\ \;\;\;\;-\frac{\cos B}{\frac{\sin B}{x}}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{F \cdot B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -3.3e+209)
     (- (/ -1.0 B) t_0)
     (if (<= F -0.016)
       (/ (- -1.0 x) (sin B))
       (if (<= F -7.6e-90)
         (* (sqrt 0.5) (/ F (sin B)))
         (if (<= F 2.7e-95)
           (- (/ (cos B) (/ (sin B) x)))
           (- (* F (/ 1.0 (* F B))) t_0)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -3.3e+209) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= -0.016) {
		tmp = (-1.0 - x) / sin(B);
	} else if (F <= -7.6e-90) {
		tmp = sqrt(0.5) * (F / sin(B));
	} else if (F <= 2.7e-95) {
		tmp = -(cos(B) / (sin(B) / x));
	} else {
		tmp = (F * (1.0 / (F * B))) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-3.3d+209)) then
        tmp = ((-1.0d0) / b) - t_0
    else if (f <= (-0.016d0)) then
        tmp = ((-1.0d0) - x) / sin(b)
    else if (f <= (-7.6d-90)) then
        tmp = sqrt(0.5d0) * (f / sin(b))
    else if (f <= 2.7d-95) then
        tmp = -(cos(b) / (sin(b) / x))
    else
        tmp = (f * (1.0d0 / (f * b))) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -3.3e+209) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= -0.016) {
		tmp = (-1.0 - x) / Math.sin(B);
	} else if (F <= -7.6e-90) {
		tmp = Math.sqrt(0.5) * (F / Math.sin(B));
	} else if (F <= 2.7e-95) {
		tmp = -(Math.cos(B) / (Math.sin(B) / x));
	} else {
		tmp = (F * (1.0 / (F * B))) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -3.3e+209:
		tmp = (-1.0 / B) - t_0
	elif F <= -0.016:
		tmp = (-1.0 - x) / math.sin(B)
	elif F <= -7.6e-90:
		tmp = math.sqrt(0.5) * (F / math.sin(B))
	elif F <= 2.7e-95:
		tmp = -(math.cos(B) / (math.sin(B) / x))
	else:
		tmp = (F * (1.0 / (F * B))) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -3.3e+209)
		tmp = Float64(Float64(-1.0 / B) - t_0);
	elseif (F <= -0.016)
		tmp = Float64(Float64(-1.0 - x) / sin(B));
	elseif (F <= -7.6e-90)
		tmp = Float64(sqrt(0.5) * Float64(F / sin(B)));
	elseif (F <= 2.7e-95)
		tmp = Float64(-Float64(cos(B) / Float64(sin(B) / x)));
	else
		tmp = Float64(Float64(F * Float64(1.0 / Float64(F * B))) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -3.3e+209)
		tmp = (-1.0 / B) - t_0;
	elseif (F <= -0.016)
		tmp = (-1.0 - x) / sin(B);
	elseif (F <= -7.6e-90)
		tmp = sqrt(0.5) * (F / sin(B));
	elseif (F <= 2.7e-95)
		tmp = -(cos(B) / (sin(B) / x));
	else
		tmp = (F * (1.0 / (F * B))) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -3.3e+209], N[(N[(-1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -0.016], N[(N[(-1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -7.6e-90], N[(N[Sqrt[0.5], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.7e-95], (-N[(N[Cos[B], $MachinePrecision] / N[(N[Sin[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), N[(N[(F * N[(1.0 / N[(F * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -3.2999999999999998e209

   1. Initial program 37.1%

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

   2. Taylor expanded in F around -inf 99.8%

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

      1. *-un-lft-identity 99.8%

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

      2. +-commutative 99.8%

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

      3. div-inv 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. *-lft-identity 99.9%

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

      2. unsub-neg 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in B around 0 82.2%

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

   if -3.2999999999999998e209 < F < -0.016

   1. Initial program 62.5%

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

   2. Taylor expanded in F around -inf 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. +-commutative 99.7%

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

      3. div-inv 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. *-lft-identity 99.8%

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

      2. unsub-neg 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in B around inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. distribute-neg-in 99.8%

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

      4. distribute-neg-frac 99.8%

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

      5. metadata-eval 99.8%

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

      6. sub-neg 99.8%

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

      7. *-commutative 99.8%

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

      8. div-sub 99.8%

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

   9. Simplified 99.8%

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

   10. Taylor expanded in B around 0 88.2%

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

   if -0.016 < F < -7.6e-90

   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. Step-by-step derivation

      1. +-commutative 99.5%

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

      2. unsub-neg 99.5%

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

      3. associate-*l/ 99.3%

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

      4. associate-*r/ 99.4%

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

      5. *-commutative 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in F around 0 99.4%

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

   5. Taylor expanded in F around inf 61.1%

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

   6. Taylor expanded in x around 0 61.2%

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

   if -7.6e-90 < F < 2.7e-95

   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. Taylor expanded in F around -inf 37.9%

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

      1. *-un-lft-identity 37.9%

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

      2. +-commutative 37.9%

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

      3. div-inv 37.9%

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

   4. Applied egg-rr 37.9%

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

      1. *-lft-identity 37.9%

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

      2. unsub-neg 37.9%

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

   6. Simplified 37.9%

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

   7. Taylor expanded in B around inf 37.8%

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

      1. mul-1-neg 37.8%

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

      2. +-commutative 37.8%

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

      3. distribute-neg-in 37.8%

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

      4. distribute-neg-frac 37.8%

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

      5. metadata-eval 37.8%

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

      6. sub-neg 37.8%

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

      7. *-commutative 37.8%

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

      8. div-sub 37.8%

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

   9. Simplified 37.8%

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

   10. Taylor expanded in x around inf 83.7%

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

       1. mul-1-neg 83.7%

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

       2. associate-/l* 83.6%

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

   12. Simplified 83.6%

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

   if 2.7e-95 < F

   1. Initial program 67.7%

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

      1. +-commutative 67.7%

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

      2. unsub-neg 67.7%

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

      3. associate-*l/ 78.9%

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

      4. associate-*r/ 78.7%

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

      5. *-commutative 78.7%

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

   3. Simplified 78.8%

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

   4. Taylor expanded in F around inf 89.4%

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

      1. *-commutative 89.4%

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

      2. associate-/r* 89.3%

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

   6. Simplified 89.3%

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

   7. Taylor expanded in B around 0 71.8%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.3 \cdot 10^{+209}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -0.016:\\ \;\;\;\;\frac{-1 - x}{\sin B}\\ \mathbf{elif}\;F \leq -7.6 \cdot 10^{-90}:\\ \;\;\;\;\sqrt{0.5} \cdot \frac{F}{\sin B}\\ \mathbf{elif}\;F \leq 2.7 \cdot 10^{-95}:\\ \;\;\;\;-\frac{\cos B}{\frac{\sin B}{x}}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{F \cdot B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 8: 70.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -6.6 \cdot 10^{+208}:\\ \;\;\;\;\frac{-1}{B} - t_0\\ \mathbf{elif}\;F \leq -0.011:\\ \;\;\;\;\frac{-1 - x}{\sin B}\\ \mathbf{elif}\;F \leq -2.15 \cdot 10^{-89}:\\ \;\;\;\;\sqrt{0.5} \cdot \frac{F}{\sin B}\\ \mathbf{elif}\;F \leq 2.7 \cdot 10^{-95}:\\ \;\;\;\;\frac{-x}{\sin B} \cdot \cos B\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{F \cdot B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -6.6e+208)
     (- (/ -1.0 B) t_0)
     (if (<= F -0.011)
       (/ (- -1.0 x) (sin B))
       (if (<= F -2.15e-89)
         (* (sqrt 0.5) (/ F (sin B)))
         (if (<= F 2.7e-95)
           (* (/ (- x) (sin B)) (cos B))
           (- (* F (/ 1.0 (* F B))) t_0)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -6.6e+208) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= -0.011) {
		tmp = (-1.0 - x) / sin(B);
	} else if (F <= -2.15e-89) {
		tmp = sqrt(0.5) * (F / sin(B));
	} else if (F <= 2.7e-95) {
		tmp = (-x / sin(B)) * cos(B);
	} else {
		tmp = (F * (1.0 / (F * B))) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-6.6d+208)) then
        tmp = ((-1.0d0) / b) - t_0
    else if (f <= (-0.011d0)) then
        tmp = ((-1.0d0) - x) / sin(b)
    else if (f <= (-2.15d-89)) then
        tmp = sqrt(0.5d0) * (f / sin(b))
    else if (f <= 2.7d-95) then
        tmp = (-x / sin(b)) * cos(b)
    else
        tmp = (f * (1.0d0 / (f * b))) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -6.6e+208) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= -0.011) {
		tmp = (-1.0 - x) / Math.sin(B);
	} else if (F <= -2.15e-89) {
		tmp = Math.sqrt(0.5) * (F / Math.sin(B));
	} else if (F <= 2.7e-95) {
		tmp = (-x / Math.sin(B)) * Math.cos(B);
	} else {
		tmp = (F * (1.0 / (F * B))) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -6.6e+208:
		tmp = (-1.0 / B) - t_0
	elif F <= -0.011:
		tmp = (-1.0 - x) / math.sin(B)
	elif F <= -2.15e-89:
		tmp = math.sqrt(0.5) * (F / math.sin(B))
	elif F <= 2.7e-95:
		tmp = (-x / math.sin(B)) * math.cos(B)
	else:
		tmp = (F * (1.0 / (F * B))) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -6.6e+208)
		tmp = Float64(Float64(-1.0 / B) - t_0);
	elseif (F <= -0.011)
		tmp = Float64(Float64(-1.0 - x) / sin(B));
	elseif (F <= -2.15e-89)
		tmp = Float64(sqrt(0.5) * Float64(F / sin(B)));
	elseif (F <= 2.7e-95)
		tmp = Float64(Float64(Float64(-x) / sin(B)) * cos(B));
	else
		tmp = Float64(Float64(F * Float64(1.0 / Float64(F * B))) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -6.6e+208)
		tmp = (-1.0 / B) - t_0;
	elseif (F <= -0.011)
		tmp = (-1.0 - x) / sin(B);
	elseif (F <= -2.15e-89)
		tmp = sqrt(0.5) * (F / sin(B));
	elseif (F <= 2.7e-95)
		tmp = (-x / sin(B)) * cos(B);
	else
		tmp = (F * (1.0 / (F * B))) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -6.6e+208], N[(N[(-1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -0.011], N[(N[(-1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -2.15e-89], N[(N[Sqrt[0.5], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.7e-95], N[(N[((-x) / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Cos[B], $MachinePrecision]), $MachinePrecision], N[(N[(F * N[(1.0 / N[(F * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -6.6e208

   1. Initial program 37.1%

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

   2. Taylor expanded in F around -inf 99.8%

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

      1. *-un-lft-identity 99.8%

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

      2. +-commutative 99.8%

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

      3. div-inv 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. *-lft-identity 99.9%

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

      2. unsub-neg 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in B around 0 82.2%

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

   if -6.6e208 < F < -0.010999999999999999

   1. Initial program 62.5%

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

   2. Taylor expanded in F around -inf 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. +-commutative 99.7%

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

      3. div-inv 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. *-lft-identity 99.8%

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

      2. unsub-neg 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in B around inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. distribute-neg-in 99.8%

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

      4. distribute-neg-frac 99.8%

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

      5. metadata-eval 99.8%

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

      6. sub-neg 99.8%

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

      7. *-commutative 99.8%

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

      8. div-sub 99.8%

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

   9. Simplified 99.8%

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

   10. Taylor expanded in B around 0 88.2%

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

   if -0.010999999999999999 < F < -2.14999999999999993e-89

   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. Step-by-step derivation

      1. +-commutative 99.5%

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

      2. unsub-neg 99.5%

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

      3. associate-*l/ 99.3%

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

      4. associate-*r/ 99.4%

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

      5. *-commutative 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in F around 0 99.4%

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

   5. Taylor expanded in F around inf 61.1%

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

   6. Taylor expanded in x around 0 61.2%

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

   if -2.14999999999999993e-89 < F < 2.7e-95

   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. Taylor expanded in F around -inf 37.9%

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

      1. *-un-lft-identity 37.9%

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

      2. +-commutative 37.9%

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

      3. div-inv 37.9%

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

   4. Applied egg-rr 37.9%

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

      1. *-lft-identity 37.9%

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

      2. unsub-neg 37.9%

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

   6. Simplified 37.9%

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

   7. Taylor expanded in x around inf 83.7%

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

      1. associate-*r/ 83.7%

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

      2. *-commutative 83.7%

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

      3. neg-mul-1 83.7%

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

      4. distribute-lft-neg-in 83.7%

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

      5. *-rgt-identity 83.7%

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

      6. times-frac 83.7%

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

      7. rem-square-sqrt 64.9%

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

      8. associate-*r/ 64.9%

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

      9. /-rgt-identity 64.9%

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

      10. rem-square-sqrt 83.7%

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

   9. Simplified 83.7%

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

   if 2.7e-95 < F

   1. Initial program 67.7%

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

      1. +-commutative 67.7%

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

      2. unsub-neg 67.7%

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

      3. associate-*l/ 78.9%

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

      4. associate-*r/ 78.7%

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

      5. *-commutative 78.7%

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

   3. Simplified 78.8%

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

   4. Taylor expanded in F around inf 89.4%

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

      1. *-commutative 89.4%

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

      2. associate-/r* 89.3%

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

   6. Simplified 89.3%

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

   7. Taylor expanded in B around 0 71.8%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6.6 \cdot 10^{+208}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -0.011:\\ \;\;\;\;\frac{-1 - x}{\sin B}\\ \mathbf{elif}\;F \leq -2.15 \cdot 10^{-89}:\\ \;\;\;\;\sqrt{0.5} \cdot \frac{F}{\sin B}\\ \mathbf{elif}\;F \leq 2.7 \cdot 10^{-95}:\\ \;\;\;\;\frac{-x}{\sin B} \cdot \cos B\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{F \cdot B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 9: 84.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -5.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq -1.9 \cdot 10^{-89}:\\ \;\;\;\;\sqrt{0.5} \cdot \frac{F}{\sin B}\\ \mathbf{elif}\;F \leq 2.35 \cdot 10^{-10}:\\ \;\;\;\;\frac{-x}{\sin B} \cdot \cos B\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -5.5e-5)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F -1.9e-89)
       (* (sqrt 0.5) (/ F (sin B)))
       (if (<= F 2.35e-10)
         (* (/ (- x) (sin B)) (cos 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 <= -5.5e-5) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= -1.9e-89) {
		tmp = sqrt(0.5) * (F / sin(B));
	} else if (F <= 2.35e-10) {
		tmp = (-x / sin(B)) * cos(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 <= (-5.5d-5)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= (-1.9d-89)) then
        tmp = sqrt(0.5d0) * (f / sin(b))
    else if (f <= 2.35d-10) then
        tmp = (-x / sin(b)) * cos(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 <= -5.5e-5) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= -1.9e-89) {
		tmp = Math.sqrt(0.5) * (F / Math.sin(B));
	} else if (F <= 2.35e-10) {
		tmp = (-x / Math.sin(B)) * Math.cos(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 <= -5.5e-5:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= -1.9e-89:
		tmp = math.sqrt(0.5) * (F / math.sin(B))
	elif F <= 2.35e-10:
		tmp = (-x / math.sin(B)) * math.cos(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 <= -5.5e-5)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= -1.9e-89)
		tmp = Float64(sqrt(0.5) * Float64(F / sin(B)));
	elseif (F <= 2.35e-10)
		tmp = Float64(Float64(Float64(-x) / sin(B)) * cos(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 <= -5.5e-5)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= -1.9e-89)
		tmp = sqrt(0.5) * (F / sin(B));
	elseif (F <= 2.35e-10)
		tmp = (-x / sin(B)) * cos(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, -5.5e-5], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -1.9e-89], N[(N[Sqrt[0.5], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.35e-10], N[(N[((-x) / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Cos[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -5.5000000000000002e-5

   1. Initial program 51.3%

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

   2. Taylor expanded in F around -inf 99.8%

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

      1. *-un-lft-identity 99.8%

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

      2. +-commutative 99.8%

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

      3. div-inv 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. *-lft-identity 99.8%

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

      2. unsub-neg 99.8%

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

   6. Simplified 99.8%

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

   if -5.5000000000000002e-5 < F < -1.9000000000000001e-89

   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. Step-by-step derivation

      1. +-commutative 99.5%

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

      2. unsub-neg 99.5%

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

      3. associate-*l/ 99.3%

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

      4. associate-*r/ 99.4%

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

      5. *-commutative 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in F around 0 99.4%

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

   5. Taylor expanded in F around inf 61.1%

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

   6. Taylor expanded in x around 0 61.2%

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

   if -1.9000000000000001e-89 < F < 2.3500000000000002e-10

   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. Taylor expanded in F around -inf 41.0%

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

      1. *-un-lft-identity 41.0%

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

      2. +-commutative 41.0%

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

      3. div-inv 41.0%

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

   4. Applied egg-rr 41.0%

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

      1. *-lft-identity 41.0%

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

      2. unsub-neg 41.0%

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

   6. Simplified 41.0%

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

   7. Taylor expanded in x around inf 80.4%

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

      1. associate-*r/ 80.4%

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

      2. *-commutative 80.4%

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

      3. neg-mul-1 80.4%

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

      4. distribute-lft-neg-in 80.4%

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

      5. *-rgt-identity 80.4%

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

      6. times-frac 80.5%

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

      7. rem-square-sqrt 61.6%

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

      8. associate-*r/ 61.6%

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

      9. /-rgt-identity 61.6%

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

      10. rem-square-sqrt 80.5%

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

   9. Simplified 80.5%

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

   if 2.3500000000000002e-10 < F

   1. Initial program 58.3%

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

      1. +-commutative 58.3%

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

      2. unsub-neg 58.3%

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

      3. associate-*l/ 72.8%

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

      4. associate-*r/ 72.7%

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

      5. *-commutative 72.7%

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

   3. Simplified 72.7%

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

   4. Taylor expanded in F around inf 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-/r* 99.2%

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

   6. Simplified 99.2%

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

      1. sub-neg 99.2%

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

      2. associate-/l/ 99.3%

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

   8. Applied egg-rr 99.3%

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

      1. sub-neg 99.3%

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

      2. associate-*r/ 99.3%

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

      3. *-commutative 99.3%

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

      4. times-frac 99.5%

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

      5. *-rgt-identity 99.5%

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

      6. associate-*r/ 99.4%

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

      7. rgt-mult-inverse 99.5%

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

      8. *-lft-identity 99.5%

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

   10. Simplified 99.5%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.9 \cdot 10^{-89}:\\ \;\;\;\;\sqrt{0.5} \cdot \frac{F}{\sin B}\\ \mathbf{elif}\;F \leq 2.35 \cdot 10^{-10}:\\ \;\;\;\;\frac{-x}{\sin B} \cdot \cos B\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 10: 76.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq -9.2 \cdot 10^{-90}:\\ \;\;\;\;\sqrt{0.5} \cdot \frac{F}{\sin B}\\ \mathbf{elif}\;F \leq 2.7 \cdot 10^{-95}:\\ \;\;\;\;\frac{-x}{\sin B} \cdot \cos B\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{F \cdot B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -2.5e-6)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F -9.2e-90)
       (* (sqrt 0.5) (/ F (sin B)))
       (if (<= F 2.7e-95)
         (* (/ (- x) (sin B)) (cos B))
         (- (* F (/ 1.0 (* F B))) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -2.5e-6) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= -9.2e-90) {
		tmp = sqrt(0.5) * (F / sin(B));
	} else if (F <= 2.7e-95) {
		tmp = (-x / sin(B)) * cos(B);
	} else {
		tmp = (F * (1.0 / (F * B))) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-2.5d-6)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= (-9.2d-90)) then
        tmp = sqrt(0.5d0) * (f / sin(b))
    else if (f <= 2.7d-95) then
        tmp = (-x / sin(b)) * cos(b)
    else
        tmp = (f * (1.0d0 / (f * b))) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -2.5e-6) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= -9.2e-90) {
		tmp = Math.sqrt(0.5) * (F / Math.sin(B));
	} else if (F <= 2.7e-95) {
		tmp = (-x / Math.sin(B)) * Math.cos(B);
	} else {
		tmp = (F * (1.0 / (F * B))) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -2.5e-6:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= -9.2e-90:
		tmp = math.sqrt(0.5) * (F / math.sin(B))
	elif F <= 2.7e-95:
		tmp = (-x / math.sin(B)) * math.cos(B)
	else:
		tmp = (F * (1.0 / (F * B))) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -2.5e-6)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= -9.2e-90)
		tmp = Float64(sqrt(0.5) * Float64(F / sin(B)));
	elseif (F <= 2.7e-95)
		tmp = Float64(Float64(Float64(-x) / sin(B)) * cos(B));
	else
		tmp = Float64(Float64(F * Float64(1.0 / Float64(F * B))) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -2.5e-6)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= -9.2e-90)
		tmp = sqrt(0.5) * (F / sin(B));
	elseif (F <= 2.7e-95)
		tmp = (-x / sin(B)) * cos(B);
	else
		tmp = (F * (1.0 / (F * B))) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.5e-6], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -9.2e-90], N[(N[Sqrt[0.5], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.7e-95], N[(N[((-x) / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Cos[B], $MachinePrecision]), $MachinePrecision], N[(N[(F * N[(1.0 / N[(F * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.5000000000000002e-6

   1. Initial program 51.3%

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

   2. Taylor expanded in F around -inf 99.8%

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

      1. *-un-lft-identity 99.8%

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

      2. +-commutative 99.8%

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

      3. div-inv 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. *-lft-identity 99.8%

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

      2. unsub-neg 99.8%

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

   6. Simplified 99.8%

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

   if -2.5000000000000002e-6 < F < -9.1999999999999992e-90

   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. Step-by-step derivation

      1. +-commutative 99.5%

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

      2. unsub-neg 99.5%

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

      3. associate-*l/ 99.3%

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

      4. associate-*r/ 99.4%

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

      5. *-commutative 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in F around 0 99.4%

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

   5. Taylor expanded in F around inf 61.1%

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

   6. Taylor expanded in x around 0 61.2%

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

   if -9.1999999999999992e-90 < F < 2.7e-95

   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. Taylor expanded in F around -inf 37.9%

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

      1. *-un-lft-identity 37.9%

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

      2. +-commutative 37.9%

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

      3. div-inv 37.9%

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

   4. Applied egg-rr 37.9%

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

      1. *-lft-identity 37.9%

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

      2. unsub-neg 37.9%

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

   6. Simplified 37.9%

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

   7. Taylor expanded in x around inf 83.7%

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

      1. associate-*r/ 83.7%

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

      2. *-commutative 83.7%

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

      3. neg-mul-1 83.7%

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

      4. distribute-lft-neg-in 83.7%

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

      5. *-rgt-identity 83.7%

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

      6. times-frac 83.7%

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

      7. rem-square-sqrt 64.9%

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

      8. associate-*r/ 64.9%

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

      9. /-rgt-identity 64.9%

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

      10. rem-square-sqrt 83.7%

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

   9. Simplified 83.7%

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

   if 2.7e-95 < F

   1. Initial program 67.7%

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

      1. +-commutative 67.7%

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

      2. unsub-neg 67.7%

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

      3. associate-*l/ 78.9%

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

      4. associate-*r/ 78.7%

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

      5. *-commutative 78.7%

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

   3. Simplified 78.8%

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

   4. Taylor expanded in F around inf 89.4%

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

      1. *-commutative 89.4%

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

      2. associate-/r* 89.3%

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

   6. Simplified 89.3%

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

   7. Taylor expanded in B around 0 71.8%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -9.2 \cdot 10^{-90}:\\ \;\;\;\;\sqrt{0.5} \cdot \frac{F}{\sin B}\\ \mathbf{elif}\;F \leq 2.7 \cdot 10^{-95}:\\ \;\;\;\;\frac{-x}{\sin B} \cdot \cos B\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{F \cdot B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 11: 63.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ t_1 := \frac{-1}{B} - t_0\\ t_2 := F \cdot \sqrt{0.5} - x\\ \mathbf{if}\;F \leq -2.42 \cdot 10^{+210}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq -0.0032:\\ \;\;\;\;\frac{-1 - x}{\sin B}\\ \mathbf{elif}\;F \leq -2.15 \cdot 10^{-89}:\\ \;\;\;\;\sqrt{0.5} \cdot \frac{F}{\sin B}\\ \mathbf{elif}\;F \leq 1.32 \cdot 10^{-236}:\\ \;\;\;\;\frac{t_2}{B}\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{-101}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 5 \cdot 10^{-87}:\\ \;\;\;\;\frac{1}{\frac{B}{t_2}}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{F \cdot B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B)))
        (t_1 (- (/ -1.0 B) t_0))
        (t_2 (- (* F (sqrt 0.5)) x)))
   (if (<= F -2.42e+210)
     t_1
     (if (<= F -0.0032)
       (/ (- -1.0 x) (sin B))
       (if (<= F -2.15e-89)
         (* (sqrt 0.5) (/ F (sin B)))
         (if (<= F 1.32e-236)
           (/ t_2 B)
           (if (<= F 6.2e-101)
             t_1
             (if (<= F 5e-87)
               (/ 1.0 (/ B t_2))
               (- (* F (/ 1.0 (* F B))) t_0)))))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double t_1 = (-1.0 / B) - t_0;
	double t_2 = (F * sqrt(0.5)) - x;
	double tmp;
	if (F <= -2.42e+210) {
		tmp = t_1;
	} else if (F <= -0.0032) {
		tmp = (-1.0 - x) / sin(B);
	} else if (F <= -2.15e-89) {
		tmp = sqrt(0.5) * (F / sin(B));
	} else if (F <= 1.32e-236) {
		tmp = t_2 / B;
	} else if (F <= 6.2e-101) {
		tmp = t_1;
	} else if (F <= 5e-87) {
		tmp = 1.0 / (B / t_2);
	} else {
		tmp = (F * (1.0 / (F * B))) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = x / tan(b)
    t_1 = ((-1.0d0) / b) - t_0
    t_2 = (f * sqrt(0.5d0)) - x
    if (f <= (-2.42d+210)) then
        tmp = t_1
    else if (f <= (-0.0032d0)) then
        tmp = ((-1.0d0) - x) / sin(b)
    else if (f <= (-2.15d-89)) then
        tmp = sqrt(0.5d0) * (f / sin(b))
    else if (f <= 1.32d-236) then
        tmp = t_2 / b
    else if (f <= 6.2d-101) then
        tmp = t_1
    else if (f <= 5d-87) then
        tmp = 1.0d0 / (b / t_2)
    else
        tmp = (f * (1.0d0 / (f * b))) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double t_1 = (-1.0 / B) - t_0;
	double t_2 = (F * Math.sqrt(0.5)) - x;
	double tmp;
	if (F <= -2.42e+210) {
		tmp = t_1;
	} else if (F <= -0.0032) {
		tmp = (-1.0 - x) / Math.sin(B);
	} else if (F <= -2.15e-89) {
		tmp = Math.sqrt(0.5) * (F / Math.sin(B));
	} else if (F <= 1.32e-236) {
		tmp = t_2 / B;
	} else if (F <= 6.2e-101) {
		tmp = t_1;
	} else if (F <= 5e-87) {
		tmp = 1.0 / (B / t_2);
	} else {
		tmp = (F * (1.0 / (F * B))) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	t_1 = (-1.0 / B) - t_0
	t_2 = (F * math.sqrt(0.5)) - x
	tmp = 0
	if F <= -2.42e+210:
		tmp = t_1
	elif F <= -0.0032:
		tmp = (-1.0 - x) / math.sin(B)
	elif F <= -2.15e-89:
		tmp = math.sqrt(0.5) * (F / math.sin(B))
	elif F <= 1.32e-236:
		tmp = t_2 / B
	elif F <= 6.2e-101:
		tmp = t_1
	elif F <= 5e-87:
		tmp = 1.0 / (B / t_2)
	else:
		tmp = (F * (1.0 / (F * B))) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	t_1 = Float64(Float64(-1.0 / B) - t_0)
	t_2 = Float64(Float64(F * sqrt(0.5)) - x)
	tmp = 0.0
	if (F <= -2.42e+210)
		tmp = t_1;
	elseif (F <= -0.0032)
		tmp = Float64(Float64(-1.0 - x) / sin(B));
	elseif (F <= -2.15e-89)
		tmp = Float64(sqrt(0.5) * Float64(F / sin(B)));
	elseif (F <= 1.32e-236)
		tmp = Float64(t_2 / B);
	elseif (F <= 6.2e-101)
		tmp = t_1;
	elseif (F <= 5e-87)
		tmp = Float64(1.0 / Float64(B / t_2));
	else
		tmp = Float64(Float64(F * Float64(1.0 / Float64(F * B))) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	t_1 = (-1.0 / B) - t_0;
	t_2 = (F * sqrt(0.5)) - x;
	tmp = 0.0;
	if (F <= -2.42e+210)
		tmp = t_1;
	elseif (F <= -0.0032)
		tmp = (-1.0 - x) / sin(B);
	elseif (F <= -2.15e-89)
		tmp = sqrt(0.5) * (F / sin(B));
	elseif (F <= 1.32e-236)
		tmp = t_2 / B;
	elseif (F <= 6.2e-101)
		tmp = t_1;
	elseif (F <= 5e-87)
		tmp = 1.0 / (B / t_2);
	else
		tmp = (F * (1.0 / (F * B))) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[F, -2.42e+210], t$95$1, If[LessEqual[F, -0.0032], N[(N[(-1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -2.15e-89], N[(N[Sqrt[0.5], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.32e-236], N[(t$95$2 / B), $MachinePrecision], If[LessEqual[F, 6.2e-101], t$95$1, If[LessEqual[F, 5e-87], N[(1.0 / N[(B / t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(F * N[(1.0 / N[(F * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if F < -2.42e210 or 1.32e-236 < F < 6.19999999999999946e-101

   1. Initial program 65.0%

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

   2. Taylor expanded in F around -inf 73.3%

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

      1. *-un-lft-identity 73.3%

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

      2. +-commutative 73.3%

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

      3. div-inv 73.4%

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

   4. Applied egg-rr 73.4%

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

      1. *-lft-identity 73.4%

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

      2. unsub-neg 73.4%

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

   6. Simplified 73.4%

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

   7. Taylor expanded in B around 0 73.6%

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

   if -2.42e210 < F < -0.00320000000000000015

   1. Initial program 62.5%

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

   2. Taylor expanded in F around -inf 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. +-commutative 99.7%

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

      3. div-inv 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. *-lft-identity 99.8%

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

      2. unsub-neg 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in B around inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. distribute-neg-in 99.8%

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

      4. distribute-neg-frac 99.8%

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

      5. metadata-eval 99.8%

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

      6. sub-neg 99.8%

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

      7. *-commutative 99.8%

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

      8. div-sub 99.8%

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

   9. Simplified 99.8%

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

   10. Taylor expanded in B around 0 88.2%

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

   if -0.00320000000000000015 < F < -2.14999999999999993e-89

   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. Step-by-step derivation

      1. +-commutative 99.5%

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

      2. unsub-neg 99.5%

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

      3. associate-*l/ 99.3%

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

      4. associate-*r/ 99.4%

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

      5. *-commutative 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in F around 0 99.4%

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

   5. Taylor expanded in F around inf 61.1%

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

   6. Taylor expanded in x around 0 61.2%

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

   if -2.14999999999999993e-89 < F < 1.32e-236

   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. Step-by-step derivation

      1. +-commutative 99.5%

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

      2. unsub-neg 99.5%

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

      3. associate-*l/ 99.5%

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

      4. associate-*r/ 99.5%

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

      5. *-commutative 99.5%

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

   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. Taylor expanded in F around 0 99.8%

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

   5. Taylor expanded in B around 0 64.5%

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

   6. Taylor expanded in x around 0 64.5%

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

   if 6.19999999999999946e-101 < F < 5.00000000000000042e-87

   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. Step-by-step derivation

      1. +-commutative 99.4%

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

      2. unsub-neg 99.4%

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

      3. associate-*l/ 99.1%

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

      4. associate-*r/ 98.9%

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

      5. *-commutative 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in F around 0 98.9%

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

   5. Taylor expanded in B around 0 79.9%

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

      1. clear-num 80.2%

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

      2. inv-pow 80.2%

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

      3. inv-pow 80.2%

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

      4. sqrt-pow1 80.2%

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

      5. *-commutative 80.2%

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

      6. fma-def 80.2%

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

      7. metadata-eval 80.2%

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

   7. Applied egg-rr 80.2%

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

      1. unpow-1 80.2%

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

   9. Simplified 80.2%

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

   10. Taylor expanded in x around 0 80.2%

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

   if 5.00000000000000042e-87 < F

   1. Initial program 66.5%

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

      1. +-commutative 66.5%

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

      2. unsub-neg 66.5%

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

      3. associate-*l/ 78.1%

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

      4. associate-*r/ 78.0%

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

      5. *-commutative 78.0%

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

   3. Simplified 78.1%

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

   4. Taylor expanded in F around inf 92.5%

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

      1. *-commutative 92.5%

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

      2. associate-/r* 92.5%

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

   6. Simplified 92.5%

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

   7. Taylor expanded in B around 0 74.3%

      \[\leadsto F \cdot \color{blue}{\frac{1}{F \cdot B}} - \frac{x}{\tan B} \]
3. Recombined 6 regimes into one program.

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.42 \cdot 10^{+210}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -0.0032:\\ \;\;\;\;\frac{-1 - x}{\sin B}\\ \mathbf{elif}\;F \leq -2.15 \cdot 10^{-89}:\\ \;\;\;\;\sqrt{0.5} \cdot \frac{F}{\sin B}\\ \mathbf{elif}\;F \leq 1.32 \cdot 10^{-236}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{-101}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{-87}:\\ \;\;\;\;\frac{1}{\frac{B}{F \cdot \sqrt{0.5} - x}}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{F \cdot B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 12: 63.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := F \cdot \sqrt{0.5} - x\\ t_1 := \frac{x}{\tan B}\\ t_2 := \frac{-1}{B} - t_1\\ \mathbf{if}\;F \leq -7.8 \cdot 10^{+208}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;F \leq -1.1 \cdot 10^{-9}:\\ \;\;\;\;\frac{-1 - x}{\sin B}\\ \mathbf{elif}\;F \leq 1.25 \cdot 10^{-230}:\\ \;\;\;\;\frac{t_0}{B}\\ \mathbf{elif}\;F \leq 7 \cdot 10^{-103}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;F \leq 1.35 \cdot 10^{-89}:\\ \;\;\;\;\frac{1}{\frac{B}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{F \cdot B} - t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (* F (sqrt 0.5)) x))
        (t_1 (/ x (tan B)))
        (t_2 (- (/ -1.0 B) t_1)))
   (if (<= F -7.8e+208)
     t_2
     (if (<= F -1.1e-9)
       (/ (- -1.0 x) (sin B))
       (if (<= F 1.25e-230)
         (/ t_0 B)
         (if (<= F 7e-103)
           t_2
           (if (<= F 1.35e-89)
             (/ 1.0 (/ B t_0))
             (- (* F (/ 1.0 (* F B))) t_1))))))))

C

double code(double F, double B, double x) {
	double t_0 = (F * sqrt(0.5)) - x;
	double t_1 = x / tan(B);
	double t_2 = (-1.0 / B) - t_1;
	double tmp;
	if (F <= -7.8e+208) {
		tmp = t_2;
	} else if (F <= -1.1e-9) {
		tmp = (-1.0 - x) / sin(B);
	} else if (F <= 1.25e-230) {
		tmp = t_0 / B;
	} else if (F <= 7e-103) {
		tmp = t_2;
	} else if (F <= 1.35e-89) {
		tmp = 1.0 / (B / t_0);
	} else {
		tmp = (F * (1.0 / (F * 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) :: t_2
    real(8) :: tmp
    t_0 = (f * sqrt(0.5d0)) - x
    t_1 = x / tan(b)
    t_2 = ((-1.0d0) / b) - t_1
    if (f <= (-7.8d+208)) then
        tmp = t_2
    else if (f <= (-1.1d-9)) then
        tmp = ((-1.0d0) - x) / sin(b)
    else if (f <= 1.25d-230) then
        tmp = t_0 / b
    else if (f <= 7d-103) then
        tmp = t_2
    else if (f <= 1.35d-89) then
        tmp = 1.0d0 / (b / t_0)
    else
        tmp = (f * (1.0d0 / (f * b))) - t_1
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (F * Math.sqrt(0.5)) - x;
	double t_1 = x / Math.tan(B);
	double t_2 = (-1.0 / B) - t_1;
	double tmp;
	if (F <= -7.8e+208) {
		tmp = t_2;
	} else if (F <= -1.1e-9) {
		tmp = (-1.0 - x) / Math.sin(B);
	} else if (F <= 1.25e-230) {
		tmp = t_0 / B;
	} else if (F <= 7e-103) {
		tmp = t_2;
	} else if (F <= 1.35e-89) {
		tmp = 1.0 / (B / t_0);
	} else {
		tmp = (F * (1.0 / (F * B))) - t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (F * math.sqrt(0.5)) - x
	t_1 = x / math.tan(B)
	t_2 = (-1.0 / B) - t_1
	tmp = 0
	if F <= -7.8e+208:
		tmp = t_2
	elif F <= -1.1e-9:
		tmp = (-1.0 - x) / math.sin(B)
	elif F <= 1.25e-230:
		tmp = t_0 / B
	elif F <= 7e-103:
		tmp = t_2
	elif F <= 1.35e-89:
		tmp = 1.0 / (B / t_0)
	else:
		tmp = (F * (1.0 / (F * B))) - t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(F * sqrt(0.5)) - x)
	t_1 = Float64(x / tan(B))
	t_2 = Float64(Float64(-1.0 / B) - t_1)
	tmp = 0.0
	if (F <= -7.8e+208)
		tmp = t_2;
	elseif (F <= -1.1e-9)
		tmp = Float64(Float64(-1.0 - x) / sin(B));
	elseif (F <= 1.25e-230)
		tmp = Float64(t_0 / B);
	elseif (F <= 7e-103)
		tmp = t_2;
	elseif (F <= 1.35e-89)
		tmp = Float64(1.0 / Float64(B / t_0));
	else
		tmp = Float64(Float64(F * Float64(1.0 / Float64(F * B))) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (F * sqrt(0.5)) - x;
	t_1 = x / tan(B);
	t_2 = (-1.0 / B) - t_1;
	tmp = 0.0;
	if (F <= -7.8e+208)
		tmp = t_2;
	elseif (F <= -1.1e-9)
		tmp = (-1.0 - x) / sin(B);
	elseif (F <= 1.25e-230)
		tmp = t_0 / B;
	elseif (F <= 7e-103)
		tmp = t_2;
	elseif (F <= 1.35e-89)
		tmp = 1.0 / (B / t_0);
	else
		tmp = (F * (1.0 / (F * B))) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-1.0 / B), $MachinePrecision] - t$95$1), $MachinePrecision]}, If[LessEqual[F, -7.8e+208], t$95$2, If[LessEqual[F, -1.1e-9], N[(N[(-1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.25e-230], N[(t$95$0 / B), $MachinePrecision], If[LessEqual[F, 7e-103], t$95$2, If[LessEqual[F, 1.35e-89], N[(1.0 / N[(B / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(F * N[(1.0 / N[(F * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -7.8000000000000001e208 or 1.25000000000000009e-230 < F < 7.00000000000000032e-103

   1. Initial program 65.0%

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

   2. Taylor expanded in F around -inf 73.3%

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

      1. *-un-lft-identity 73.3%

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

      2. +-commutative 73.3%

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

      3. div-inv 73.4%

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

   4. Applied egg-rr 73.4%

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

      1. *-lft-identity 73.4%

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

      2. unsub-neg 73.4%

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

   6. Simplified 73.4%

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

   7. Taylor expanded in B around 0 73.6%

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

   if -7.8000000000000001e208 < F < -1.0999999999999999e-9

   1. Initial program 64.2%

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

   2. Taylor expanded in F around -inf 95.6%

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

      1. *-un-lft-identity 95.6%

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

      2. +-commutative 95.6%

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

      3. div-inv 95.6%

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

   4. Applied egg-rr 95.6%

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

      1. *-lft-identity 95.6%

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

      2. unsub-neg 95.6%

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

   6. Simplified 95.6%

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

   7. Taylor expanded in B around inf 95.6%

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

      1. mul-1-neg 95.6%

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

      2. +-commutative 95.6%

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

      3. distribute-neg-in 95.6%

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

      4. distribute-neg-frac 95.6%

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

      5. metadata-eval 95.6%

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

      6. sub-neg 95.6%

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

      7. *-commutative 95.6%

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

      8. div-sub 95.7%

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

   9. Simplified 95.7%

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

   10. Taylor expanded in B around 0 84.6%

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

   if -1.0999999999999999e-9 < F < 1.25000000000000009e-230

   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. Step-by-step derivation

      1. +-commutative 99.5%

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

      2. unsub-neg 99.5%

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

      3. associate-*l/ 99.4%

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

      4. associate-*r/ 99.5%

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

      5. *-commutative 99.5%

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

   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. Taylor expanded in F around 0 99.7%

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

   5. Taylor expanded in B around 0 57.0%

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

   6. Taylor expanded in x around 0 57.0%

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

   if 7.00000000000000032e-103 < F < 1.34999999999999994e-89

   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. Step-by-step derivation

      1. +-commutative 99.4%

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

      2. unsub-neg 99.4%

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

      3. associate-*l/ 99.1%

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

      4. associate-*r/ 98.9%

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

      5. *-commutative 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in F around 0 98.9%

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

   5. Taylor expanded in B around 0 79.9%

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

      1. clear-num 80.2%

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

      2. inv-pow 80.2%

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

      3. inv-pow 80.2%

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

      4. sqrt-pow1 80.2%

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

      5. *-commutative 80.2%

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

      6. fma-def 80.2%

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

      7. metadata-eval 80.2%

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

   7. Applied egg-rr 80.2%

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

      1. unpow-1 80.2%

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

   9. Simplified 80.2%

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

   10. Taylor expanded in x around 0 80.2%

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

   if 1.34999999999999994e-89 < F

   1. Initial program 66.5%

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

      1. +-commutative 66.5%

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

      2. unsub-neg 66.5%

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

      3. associate-*l/ 78.1%

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

      4. associate-*r/ 78.0%

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

      5. *-commutative 78.0%

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

   3. Simplified 78.1%

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

   4. Taylor expanded in F around inf 92.5%

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

      1. *-commutative 92.5%

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

      2. associate-/r* 92.5%

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

   6. Simplified 92.5%

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

   7. Taylor expanded in B around 0 74.3%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7.8 \cdot 10^{+208}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.1 \cdot 10^{-9}:\\ \;\;\;\;\frac{-1 - x}{\sin B}\\ \mathbf{elif}\;F \leq 1.25 \cdot 10^{-230}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{elif}\;F \leq 7 \cdot 10^{-103}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.35 \cdot 10^{-89}:\\ \;\;\;\;\frac{1}{\frac{B}{F \cdot \sqrt{0.5} - x}}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{F \cdot B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 13: 54.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2.75 \cdot 10^{+210}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -6.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{-1 - x}{\sin B}\\ \mathbf{elif}\;F \leq -6.2 \cdot 10^{-270}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 1.25 \cdot 10^{-237}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 8.2 \cdot 10^{+172}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ -1.0 B) (/ x (tan B)))))
   (if (<= F -2.75e+210)
     t_0
     (if (<= F -6.5e-11)
       (/ (- -1.0 x) (sin B))
       (if (<= F -6.2e-270)
         t_0
         (if (<= F 1.25e-237)
           (/ (- x) B)
           (if (<= F 8.2e+172)
             t_0
             (+
              (* B (+ 0.16666666666666666 (* x 0.3333333333333333)))
              (/ (- 1.0 x) B)))))))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 / B) - (x / tan(B));
	double tmp;
	if (F <= -2.75e+210) {
		tmp = t_0;
	} else if (F <= -6.5e-11) {
		tmp = (-1.0 - x) / sin(B);
	} else if (F <= -6.2e-270) {
		tmp = t_0;
	} else if (F <= 1.25e-237) {
		tmp = -x / B;
	} else if (F <= 8.2e+172) {
		tmp = t_0;
	} else {
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((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) :: t_0
    real(8) :: tmp
    t_0 = ((-1.0d0) / b) - (x / tan(b))
    if (f <= (-2.75d+210)) then
        tmp = t_0
    else if (f <= (-6.5d-11)) then
        tmp = ((-1.0d0) - x) / sin(b)
    else if (f <= (-6.2d-270)) then
        tmp = t_0
    else if (f <= 1.25d-237) then
        tmp = -x / b
    else if (f <= 8.2d+172) then
        tmp = t_0
    else
        tmp = (b * (0.16666666666666666d0 + (x * 0.3333333333333333d0))) + ((1.0d0 - x) / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (-1.0 / B) - (x / Math.tan(B));
	double tmp;
	if (F <= -2.75e+210) {
		tmp = t_0;
	} else if (F <= -6.5e-11) {
		tmp = (-1.0 - x) / Math.sin(B);
	} else if (F <= -6.2e-270) {
		tmp = t_0;
	} else if (F <= 1.25e-237) {
		tmp = -x / B;
	} else if (F <= 8.2e+172) {
		tmp = t_0;
	} else {
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (-1.0 / B) - (x / math.tan(B))
	tmp = 0
	if F <= -2.75e+210:
		tmp = t_0
	elif F <= -6.5e-11:
		tmp = (-1.0 - x) / math.sin(B)
	elif F <= -6.2e-270:
		tmp = t_0
	elif F <= 1.25e-237:
		tmp = -x / B
	elif F <= 8.2e+172:
		tmp = t_0
	else:
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 / B) - Float64(x / tan(B)))
	tmp = 0.0
	if (F <= -2.75e+210)
		tmp = t_0;
	elseif (F <= -6.5e-11)
		tmp = Float64(Float64(-1.0 - x) / sin(B));
	elseif (F <= -6.2e-270)
		tmp = t_0;
	elseif (F <= 1.25e-237)
		tmp = Float64(Float64(-x) / B);
	elseif (F <= 8.2e+172)
		tmp = t_0;
	else
		tmp = Float64(Float64(B * Float64(0.16666666666666666 + Float64(x * 0.3333333333333333))) + Float64(Float64(1.0 - x) / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (-1.0 / B) - (x / tan(B));
	tmp = 0.0;
	if (F <= -2.75e+210)
		tmp = t_0;
	elseif (F <= -6.5e-11)
		tmp = (-1.0 - x) / sin(B);
	elseif (F <= -6.2e-270)
		tmp = t_0;
	elseif (F <= 1.25e-237)
		tmp = -x / B;
	elseif (F <= 8.2e+172)
		tmp = t_0;
	else
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.75e+210], t$95$0, If[LessEqual[F, -6.5e-11], N[(N[(-1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -6.2e-270], t$95$0, If[LessEqual[F, 1.25e-237], N[((-x) / B), $MachinePrecision], If[LessEqual[F, 8.2e+172], t$95$0, N[(N[(B * N[(0.16666666666666666 + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.75000000000000006e210 or -6.49999999999999953e-11 < F < -6.2e-270 or 1.2500000000000001e-237 < F < 8.200000000000001e172

   1. Initial program 84.2%

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

   2. Taylor expanded in F around -inf 54.4%

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

      1. *-un-lft-identity 54.4%

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

      2. +-commutative 54.4%

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

      3. div-inv 54.4%

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

   4. Applied egg-rr 54.4%

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

      1. *-lft-identity 54.4%

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

      2. unsub-neg 54.4%

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

   6. Simplified 54.4%

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

   7. Taylor expanded in B around 0 59.2%

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

   if -2.75000000000000006e210 < F < -6.49999999999999953e-11

   1. Initial program 64.2%

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

   2. Taylor expanded in F around -inf 95.6%

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

      1. *-un-lft-identity 95.6%

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

      2. +-commutative 95.6%

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

      3. div-inv 95.6%

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

   4. Applied egg-rr 95.6%

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

      1. *-lft-identity 95.6%

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

      2. unsub-neg 95.6%

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

   6. Simplified 95.6%

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

   7. Taylor expanded in B around inf 95.6%

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

      1. mul-1-neg 95.6%

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

      2. +-commutative 95.6%

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

      3. distribute-neg-in 95.6%

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

      4. distribute-neg-frac 95.6%

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

      5. metadata-eval 95.6%

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

      6. sub-neg 95.6%

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

      7. *-commutative 95.6%

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

      8. div-sub 95.7%

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

   9. Simplified 95.7%

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

   10. Taylor expanded in B around 0 84.6%

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

   if -6.2e-270 < F < 1.2500000000000001e-237

   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. Step-by-step derivation

      1. +-commutative 99.6%

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

      2. unsub-neg 99.6%

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

      3. associate-*l/ 99.6%

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

      4. associate-*r/ 99.6%

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

      5. *-commutative 99.6%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in F around inf 7.3%

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

      1. *-commutative 7.3%

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

      2. associate-/r* 7.3%

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

   6. Simplified 7.3%

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

   7. Taylor expanded in B around 0 23.4%

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

   8. Taylor expanded in x around inf 81.5%

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

      1. associate-*r/ 81.5%

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

      2. neg-mul-1 81.5%

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

   10. Simplified 81.5%

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

   if 8.200000000000001e172 < F

   1. Initial program 29.9%

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

      1. +-commutative 29.9%

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

      2. unsub-neg 29.9%

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

      3. associate-*l/ 47.3%

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

      4. associate-*r/ 47.3%

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

      5. *-commutative 47.3%

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

   3. Simplified 47.3%

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

   4. Taylor expanded in F around inf 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-/r* 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in B around 0 59.4%

      \[\leadsto \color{blue}{\left(\left(0.16666666666666666 + 0.3333333333333333 \cdot x\right) \cdot B + \frac{1}{B}\right) - \frac{x}{B}} \]
   8. Step-by-step derivation

      1. associate--l+ 59.4%

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

      2. *-commutative 59.4%

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

      3. *-commutative 59.4%

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

      4. div-sub 59.4%

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

   9. Simplified 59.4%

      \[\leadsto \color{blue}{B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.75 \cdot 10^{+210}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -6.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{-1 - x}{\sin B}\\ \mathbf{elif}\;F \leq -6.2 \cdot 10^{-270}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.25 \cdot 10^{-237}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 8.2 \cdot 10^{+172}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \end{array} \]

Alternative 14: 56.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2.25 \cdot 10^{+208}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -1.1 \cdot 10^{-9}:\\ \;\;\;\;\frac{-1 - x}{\sin B}\\ \mathbf{elif}\;F \leq 1.1 \cdot 10^{-233}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{elif}\;F \leq 8.2 \cdot 10^{+172}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ -1.0 B) (/ x (tan B)))))
   (if (<= F -2.25e+208)
     t_0
     (if (<= F -1.1e-9)
       (/ (- -1.0 x) (sin B))
       (if (<= F 1.1e-233)
         (/ (- (* F (sqrt 0.5)) x) B)
         (if (<= F 8.2e+172)
           t_0
           (+
            (* B (+ 0.16666666666666666 (* x 0.3333333333333333)))
            (/ (- 1.0 x) B))))))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 / B) - (x / tan(B));
	double tmp;
	if (F <= -2.25e+208) {
		tmp = t_0;
	} else if (F <= -1.1e-9) {
		tmp = (-1.0 - x) / sin(B);
	} else if (F <= 1.1e-233) {
		tmp = ((F * sqrt(0.5)) - x) / B;
	} else if (F <= 8.2e+172) {
		tmp = t_0;
	} else {
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((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) :: t_0
    real(8) :: tmp
    t_0 = ((-1.0d0) / b) - (x / tan(b))
    if (f <= (-2.25d+208)) then
        tmp = t_0
    else if (f <= (-1.1d-9)) then
        tmp = ((-1.0d0) - x) / sin(b)
    else if (f <= 1.1d-233) then
        tmp = ((f * sqrt(0.5d0)) - x) / b
    else if (f <= 8.2d+172) then
        tmp = t_0
    else
        tmp = (b * (0.16666666666666666d0 + (x * 0.3333333333333333d0))) + ((1.0d0 - x) / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (-1.0 / B) - (x / Math.tan(B));
	double tmp;
	if (F <= -2.25e+208) {
		tmp = t_0;
	} else if (F <= -1.1e-9) {
		tmp = (-1.0 - x) / Math.sin(B);
	} else if (F <= 1.1e-233) {
		tmp = ((F * Math.sqrt(0.5)) - x) / B;
	} else if (F <= 8.2e+172) {
		tmp = t_0;
	} else {
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (-1.0 / B) - (x / math.tan(B))
	tmp = 0
	if F <= -2.25e+208:
		tmp = t_0
	elif F <= -1.1e-9:
		tmp = (-1.0 - x) / math.sin(B)
	elif F <= 1.1e-233:
		tmp = ((F * math.sqrt(0.5)) - x) / B
	elif F <= 8.2e+172:
		tmp = t_0
	else:
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 / B) - Float64(x / tan(B)))
	tmp = 0.0
	if (F <= -2.25e+208)
		tmp = t_0;
	elseif (F <= -1.1e-9)
		tmp = Float64(Float64(-1.0 - x) / sin(B));
	elseif (F <= 1.1e-233)
		tmp = Float64(Float64(Float64(F * sqrt(0.5)) - x) / B);
	elseif (F <= 8.2e+172)
		tmp = t_0;
	else
		tmp = Float64(Float64(B * Float64(0.16666666666666666 + Float64(x * 0.3333333333333333))) + Float64(Float64(1.0 - x) / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (-1.0 / B) - (x / tan(B));
	tmp = 0.0;
	if (F <= -2.25e+208)
		tmp = t_0;
	elseif (F <= -1.1e-9)
		tmp = (-1.0 - x) / sin(B);
	elseif (F <= 1.1e-233)
		tmp = ((F * sqrt(0.5)) - x) / B;
	elseif (F <= 8.2e+172)
		tmp = t_0;
	else
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.25e+208], t$95$0, If[LessEqual[F, -1.1e-9], N[(N[(-1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.1e-233], N[(N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 8.2e+172], t$95$0, N[(N[(B * N[(0.16666666666666666 + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.25000000000000007e208 or 1.1e-233 < F < 8.200000000000001e172

   1. Initial program 77.4%

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

   2. Taylor expanded in F around -inf 60.6%

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

      1. *-un-lft-identity 60.6%

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

      2. +-commutative 60.6%

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

      3. div-inv 60.7%

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

   4. Applied egg-rr 60.7%

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

      1. *-lft-identity 60.7%

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

      2. unsub-neg 60.7%

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

   6. Simplified 60.7%

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

   7. Taylor expanded in B around 0 63.4%

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

   if -2.25000000000000007e208 < F < -1.0999999999999999e-9

   1. Initial program 64.2%

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

   2. Taylor expanded in F around -inf 95.6%

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

      1. *-un-lft-identity 95.6%

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

      2. +-commutative 95.6%

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

      3. div-inv 95.6%

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

   4. Applied egg-rr 95.6%

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

      1. *-lft-identity 95.6%

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

      2. unsub-neg 95.6%

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

   6. Simplified 95.6%

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

   7. Taylor expanded in B around inf 95.6%

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

      1. mul-1-neg 95.6%

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

      2. +-commutative 95.6%

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

      3. distribute-neg-in 95.6%

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

      4. distribute-neg-frac 95.6%

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

      5. metadata-eval 95.6%

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

      6. sub-neg 95.6%

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

      7. *-commutative 95.6%

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

      8. div-sub 95.7%

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

   9. Simplified 95.7%

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

   10. Taylor expanded in B around 0 84.6%

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

   if -1.0999999999999999e-9 < F < 1.1e-233

   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. Step-by-step derivation

      1. +-commutative 99.5%

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

      2. unsub-neg 99.5%

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

      3. associate-*l/ 99.4%

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

      4. associate-*r/ 99.5%

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

      5. *-commutative 99.5%

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

   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. Taylor expanded in F around 0 99.7%

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

   5. Taylor expanded in B around 0 57.0%

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

   6. Taylor expanded in x around 0 57.0%

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

   if 8.200000000000001e172 < F

   1. Initial program 29.9%

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

      1. +-commutative 29.9%

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

      2. unsub-neg 29.9%

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

      3. associate-*l/ 47.3%

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

      4. associate-*r/ 47.3%

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

      5. *-commutative 47.3%

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

   3. Simplified 47.3%

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

   4. Taylor expanded in F around inf 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-/r* 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in B around 0 59.4%

      \[\leadsto \color{blue}{\left(\left(0.16666666666666666 + 0.3333333333333333 \cdot x\right) \cdot B + \frac{1}{B}\right) - \frac{x}{B}} \]
   8. Step-by-step derivation

      1. associate--l+ 59.4%

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

      2. *-commutative 59.4%

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

      3. *-commutative 59.4%

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

      4. div-sub 59.4%

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

   9. Simplified 59.4%

      \[\leadsto \color{blue}{B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.25 \cdot 10^{+208}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.1 \cdot 10^{-9}:\\ \;\;\;\;\frac{-1 - x}{\sin B}\\ \mathbf{elif}\;F \leq 1.1 \cdot 10^{-233}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{elif}\;F \leq 8.2 \cdot 10^{+172}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \end{array} \]

Alternative 15: 51.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -8.8 \cdot 10^{-29}:\\ \;\;\;\;\frac{-1 - x}{\sin B}\\ \mathbf{elif}\;F \leq 1.22 \cdot 10^{-60}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -8.8e-29)
   (/ (- -1.0 x) (sin B))
   (if (<= F 1.22e-60)
     (/ (- x) B)
     (+
      (* B (+ 0.16666666666666666 (* x 0.3333333333333333)))
      (/ (- 1.0 x) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -8.8e-29) {
		tmp = (-1.0 - x) / sin(B);
	} else if (F <= 1.22e-60) {
		tmp = -x / B;
	} else {
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((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 <= (-8.8d-29)) then
        tmp = ((-1.0d0) - x) / sin(b)
    else if (f <= 1.22d-60) then
        tmp = -x / b
    else
        tmp = (b * (0.16666666666666666d0 + (x * 0.3333333333333333d0))) + ((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 <= -8.8e-29) {
		tmp = (-1.0 - x) / Math.sin(B);
	} else if (F <= 1.22e-60) {
		tmp = -x / B;
	} else {
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -8.8e-29:
		tmp = (-1.0 - x) / math.sin(B)
	elif F <= 1.22e-60:
		tmp = -x / B
	else:
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -8.8e-29)
		tmp = Float64(Float64(-1.0 - x) / sin(B));
	elseif (F <= 1.22e-60)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(B * Float64(0.16666666666666666 + Float64(x * 0.3333333333333333))) + Float64(Float64(1.0 - x) / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -8.8e-29)
		tmp = (-1.0 - x) / sin(B);
	elseif (F <= 1.22e-60)
		tmp = -x / B;
	else
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -8.8e-29], N[(N[(-1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.22e-60], N[((-x) / B), $MachinePrecision], N[(N[(B * N[(0.16666666666666666 + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -8.79999999999999961e-29

   1. Initial program 54.4%

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

   2. Taylor expanded in F around -inf 93.9%

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

      1. *-un-lft-identity 93.9%

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

      2. +-commutative 93.9%

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

      3. div-inv 93.9%

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

   4. Applied egg-rr 93.9%

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

      1. *-lft-identity 93.9%

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

      2. unsub-neg 93.9%

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

   6. Simplified 93.9%

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

   7. Taylor expanded in B around inf 93.9%

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

      1. mul-1-neg 93.9%

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

      2. +-commutative 93.9%

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

      3. distribute-neg-in 93.9%

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

      4. distribute-neg-frac 93.9%

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

      5. metadata-eval 93.9%

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

      6. sub-neg 93.9%

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

      7. *-commutative 93.9%

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

      8. div-sub 93.9%

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

   9. Simplified 93.9%

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

   10. Taylor expanded in B around 0 75.2%

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

   if -8.79999999999999961e-29 < F < 1.22e-60

   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. Step-by-step derivation

      1. +-commutative 99.5%

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

      2. unsub-neg 99.5%

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

      3. associate-*l/ 99.5%

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

      4. associate-*r/ 99.4%

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

      5. *-commutative 99.4%

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

   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. Taylor expanded in F around inf 27.6%

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

      1. *-commutative 27.6%

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

      2. associate-/r* 27.6%

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

   6. Simplified 27.6%

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

   7. Taylor expanded in B around 0 21.9%

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

   8. Taylor expanded in x around inf 42.4%

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

      1. associate-*r/ 42.4%

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

      2. neg-mul-1 42.4%

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

   10. Simplified 42.4%

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

   if 1.22e-60 < F

   1. Initial program 64.8%

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

      1. +-commutative 64.8%

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

      2. unsub-neg 64.8%

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

      3. associate-*l/ 77.0%

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

      4. associate-*r/ 76.9%

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

      5. *-commutative 76.9%

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

   3. Simplified 76.9%

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

   4. Taylor expanded in F around inf 92.1%

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

      1. *-commutative 92.1%

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

      2. associate-/r* 92.1%

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

   6. Simplified 92.1%

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

   7. Taylor expanded in B around 0 46.1%

      \[\leadsto \color{blue}{\left(\left(0.16666666666666666 + 0.3333333333333333 \cdot x\right) \cdot B + \frac{1}{B}\right) - \frac{x}{B}} \]
   8. Step-by-step derivation

      1. associate--l+ 46.1%

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

      2. *-commutative 46.1%

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

      3. *-commutative 46.1%

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

      4. div-sub 46.1%

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

   9. Simplified 46.1%

      \[\leadsto \color{blue}{B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -8.8 \cdot 10^{-29}:\\ \;\;\;\;\frac{-1 - x}{\sin B}\\ \mathbf{elif}\;F \leq 1.22 \cdot 10^{-60}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \end{array} \]

Alternative 16: 44.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -420000:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.22 \cdot 10^{-60}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -420000.0)
   (/ -1.0 (sin B))
   (if (<= F 1.22e-60)
     (/ (- x) B)
     (+
      (* B (+ 0.16666666666666666 (* x 0.3333333333333333)))
      (/ (- 1.0 x) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -420000.0) {
		tmp = -1.0 / sin(B);
	} else if (F <= 1.22e-60) {
		tmp = -x / B;
	} else {
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((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 <= (-420000.0d0)) then
        tmp = (-1.0d0) / sin(b)
    else if (f <= 1.22d-60) then
        tmp = -x / b
    else
        tmp = (b * (0.16666666666666666d0 + (x * 0.3333333333333333d0))) + ((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 <= -420000.0) {
		tmp = -1.0 / Math.sin(B);
	} else if (F <= 1.22e-60) {
		tmp = -x / B;
	} else {
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -420000.0:
		tmp = -1.0 / math.sin(B)
	elif F <= 1.22e-60:
		tmp = -x / B
	else:
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -420000.0)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 1.22e-60)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(B * Float64(0.16666666666666666 + Float64(x * 0.3333333333333333))) + Float64(Float64(1.0 - x) / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -420000.0)
		tmp = -1.0 / sin(B);
	elseif (F <= 1.22e-60)
		tmp = -x / B;
	else
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -420000.0], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.22e-60], N[((-x) / B), $MachinePrecision], N[(N[(B * N[(0.16666666666666666 + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.2e5

   1. Initial program 50.7%

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

   2. Taylor expanded in F around -inf 99.8%

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

      1. *-un-lft-identity 99.8%

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

      2. +-commutative 99.8%

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

      3. div-inv 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. *-lft-identity 99.8%

         \[\leadsto \color{blue}{\frac{-1}{\sin B} + \left(-\frac{x}{\tan B}\right)} \]

      2. unsub-neg 99.8%

         \[\leadsto \color{blue}{\frac{-1}{\sin B} - \frac{x}{\tan B}} \]

   6. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{-1}{\sin B} - \frac{x}{\tan B}} \]

   7. Taylor expanded in x around 0 61.7%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   if -4.2e5 < F < 1.22e-60

   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. Step-by-step derivation

      1. +-commutative 99.5%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      2. unsub-neg 99.5%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      3. associate-*l/ 99.4%

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      4. associate-*r/ 99.4%

         \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      5. *-commutative 99.4%

         \[\leadsto \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \cdot F} - x \cdot \frac{1}{\tan B} \]

   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. Taylor expanded in F around inf 27.2%

      \[\leadsto F \cdot \color{blue}{\frac{1}{\sin B \cdot F}} - \frac{x}{\tan B} \]
   5. Step-by-step derivation

      1. *-commutative 27.2%

         \[\leadsto F \cdot \frac{1}{\color{blue}{F \cdot \sin B}} - \frac{x}{\tan B} \]

      2. associate-/r* 27.2%

         \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   6. Simplified 27.2%

      \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   7. Taylor expanded in B around 0 21.7%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

   8. Taylor expanded in x around inf 41.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   9. Step-by-step derivation

      1. associate-*r/ 41.0%

         \[\leadsto \color{blue}{\frac{-1 \cdot x}{B}} \]

      2. neg-mul-1 41.0%

         \[\leadsto \frac{\color{blue}{-x}}{B} \]

   10. Simplified 41.0%

       \[\leadsto \color{blue}{\frac{-x}{B}} \]

   if 1.22e-60 < F

   1. Initial program 64.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative 64.8%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      2. unsub-neg 64.8%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      3. associate-*l/ 77.0%

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      4. associate-*r/ 76.9%

         \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      5. *-commutative 76.9%

         \[\leadsto \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \cdot F} - x \cdot \frac{1}{\tan B} \]

   3. Simplified 76.9%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]

   4. Taylor expanded in F around inf 92.1%

      \[\leadsto F \cdot \color{blue}{\frac{1}{\sin B \cdot F}} - \frac{x}{\tan B} \]
   5. Step-by-step derivation

      1. *-commutative 92.1%

         \[\leadsto F \cdot \frac{1}{\color{blue}{F \cdot \sin B}} - \frac{x}{\tan B} \]

      2. associate-/r* 92.1%

         \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   6. Simplified 92.1%

      \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   7. Taylor expanded in B around 0 46.1%

      \[\leadsto \color{blue}{\left(\left(0.16666666666666666 + 0.3333333333333333 \cdot x\right) \cdot B + \frac{1}{B}\right) - \frac{x}{B}} \]
   8. Step-by-step derivation

      1. associate--l+ 46.1%

         \[\leadsto \color{blue}{\left(0.16666666666666666 + 0.3333333333333333 \cdot x\right) \cdot B + \left(\frac{1}{B} - \frac{x}{B}\right)} \]

      2. *-commutative 46.1%

         \[\leadsto \color{blue}{B \cdot \left(0.16666666666666666 + 0.3333333333333333 \cdot x\right)} + \left(\frac{1}{B} - \frac{x}{B}\right) \]

      3. *-commutative 46.1%

         \[\leadsto B \cdot \left(0.16666666666666666 + \color{blue}{x \cdot 0.3333333333333333}\right) + \left(\frac{1}{B} - \frac{x}{B}\right) \]

      4. div-sub 46.1%

         \[\leadsto B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \color{blue}{\frac{1 - x}{B}} \]

   9. Simplified 46.1%

      \[\leadsto \color{blue}{B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -420000:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.22 \cdot 10^{-60}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \end{array} \]

Alternative 17: 43.9% accurate, 18.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.6 \cdot 10^{-28}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.18 \cdot 10^{-60}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.6e-28)
   (/ (- -1.0 x) B)
   (if (<= F 1.18e-60)
     (/ (- x) B)
     (+
      (* B (+ 0.16666666666666666 (* x 0.3333333333333333)))
      (/ (- 1.0 x) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.6e-28) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.18e-60) {
		tmp = -x / B;
	} else {
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((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 <= (-1.6d-28)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 1.18d-60) then
        tmp = -x / b
    else
        tmp = (b * (0.16666666666666666d0 + (x * 0.3333333333333333d0))) + ((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 <= -1.6e-28) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.18e-60) {
		tmp = -x / B;
	} else {
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.6e-28:
		tmp = (-1.0 - x) / B
	elif F <= 1.18e-60:
		tmp = -x / B
	else:
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.6e-28)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 1.18e-60)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(B * Float64(0.16666666666666666 + Float64(x * 0.3333333333333333))) + Float64(Float64(1.0 - x) / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.6e-28)
		tmp = (-1.0 - x) / B;
	elseif (F <= 1.18e-60)
		tmp = -x / B;
	else
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.6e-28], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.18e-60], N[((-x) / B), $MachinePrecision], N[(N[(B * N[(0.16666666666666666 + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.59999999999999991e-28

   1. Initial program 54.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 93.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. *-un-lft-identity 93.9%

         \[\leadsto \color{blue}{1 \cdot \left(\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}\right)} \]

      2. +-commutative 93.9%

         \[\leadsto 1 \cdot \color{blue}{\left(\frac{-1}{\sin B} + \left(-x \cdot \frac{1}{\tan B}\right)\right)} \]

      3. div-inv 93.9%

         \[\leadsto 1 \cdot \left(\frac{-1}{\sin B} + \left(-\color{blue}{\frac{x}{\tan B}}\right)\right) \]

   4. Applied egg-rr 93.9%

      \[\leadsto \color{blue}{1 \cdot \left(\frac{-1}{\sin B} + \left(-\frac{x}{\tan B}\right)\right)} \]
   5. Step-by-step derivation

      1. *-lft-identity 93.9%

         \[\leadsto \color{blue}{\frac{-1}{\sin B} + \left(-\frac{x}{\tan B}\right)} \]

      2. unsub-neg 93.9%

         \[\leadsto \color{blue}{\frac{-1}{\sin B} - \frac{x}{\tan B}} \]

   6. Simplified 93.9%

      \[\leadsto \color{blue}{\frac{-1}{\sin B} - \frac{x}{\tan B}} \]

   7. Taylor expanded in B around inf 93.9%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\cos B \cdot x}{\sin B} + \frac{1}{\sin B}\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg 93.9%

         \[\leadsto \color{blue}{-\left(\frac{\cos B \cdot x}{\sin B} + \frac{1}{\sin B}\right)} \]

      2. +-commutative 93.9%

         \[\leadsto -\color{blue}{\left(\frac{1}{\sin B} + \frac{\cos B \cdot x}{\sin B}\right)} \]

      3. distribute-neg-in 93.9%

         \[\leadsto \color{blue}{\left(-\frac{1}{\sin B}\right) + \left(-\frac{\cos B \cdot x}{\sin B}\right)} \]

      4. distribute-neg-frac 93.9%

         \[\leadsto \color{blue}{\frac{-1}{\sin B}} + \left(-\frac{\cos B \cdot x}{\sin B}\right) \]

      5. metadata-eval 93.9%

         \[\leadsto \frac{\color{blue}{-1}}{\sin B} + \left(-\frac{\cos B \cdot x}{\sin B}\right) \]

      6. sub-neg 93.9%

         \[\leadsto \color{blue}{\frac{-1}{\sin B} - \frac{\cos B \cdot x}{\sin B}} \]

      7. *-commutative 93.9%

         \[\leadsto \frac{-1}{\sin B} - \frac{\color{blue}{x \cdot \cos B}}{\sin B} \]

      8. div-sub 93.9%

         \[\leadsto \color{blue}{\frac{-1 - x \cdot \cos B}{\sin B}} \]

   9. Simplified 93.9%

      \[\leadsto \color{blue}{\frac{-1 - x \cdot \cos B}{\sin B}} \]

   10. Taylor expanded in B around 0 46.3%

       \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   11. 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%

          \[\leadsto \frac{-1 + \color{blue}{\left(-x\right)}}{B} \]

       5. sub-neg 46.3%

          \[\leadsto \frac{\color{blue}{-1 - x}}{B} \]

   12. Simplified 46.3%

       \[\leadsto \color{blue}{\frac{-1 - x}{B}} \]

   if -1.59999999999999991e-28 < F < 1.17999999999999994e-60

   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. Step-by-step derivation

      1. +-commutative 99.5%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      2. unsub-neg 99.5%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      3. associate-*l/ 99.5%

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      4. associate-*r/ 99.4%

         \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      5. *-commutative 99.4%

         \[\leadsto \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \cdot F} - x \cdot \frac{1}{\tan B} \]

   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. Taylor expanded in F around inf 27.6%

      \[\leadsto F \cdot \color{blue}{\frac{1}{\sin B \cdot F}} - \frac{x}{\tan B} \]
   5. Step-by-step derivation

      1. *-commutative 27.6%

         \[\leadsto F \cdot \frac{1}{\color{blue}{F \cdot \sin B}} - \frac{x}{\tan B} \]

      2. associate-/r* 27.6%

         \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   6. Simplified 27.6%

      \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   7. Taylor expanded in B around 0 21.9%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

   8. Taylor expanded in x around inf 42.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   9. Step-by-step derivation

      1. associate-*r/ 42.4%

         \[\leadsto \color{blue}{\frac{-1 \cdot x}{B}} \]

      2. neg-mul-1 42.4%

         \[\leadsto \frac{\color{blue}{-x}}{B} \]

   10. Simplified 42.4%

       \[\leadsto \color{blue}{\frac{-x}{B}} \]

   if 1.17999999999999994e-60 < F

   1. Initial program 64.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative 64.8%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      2. unsub-neg 64.8%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      3. associate-*l/ 77.0%

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      4. associate-*r/ 76.9%

         \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      5. *-commutative 76.9%

         \[\leadsto \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \cdot F} - x \cdot \frac{1}{\tan B} \]

   3. Simplified 76.9%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]

   4. Taylor expanded in F around inf 92.1%

      \[\leadsto F \cdot \color{blue}{\frac{1}{\sin B \cdot F}} - \frac{x}{\tan B} \]
   5. Step-by-step derivation

      1. *-commutative 92.1%

         \[\leadsto F \cdot \frac{1}{\color{blue}{F \cdot \sin B}} - \frac{x}{\tan B} \]

      2. associate-/r* 92.1%

         \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   6. Simplified 92.1%

      \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   7. Taylor expanded in B around 0 46.1%

      \[\leadsto \color{blue}{\left(\left(0.16666666666666666 + 0.3333333333333333 \cdot x\right) \cdot B + \frac{1}{B}\right) - \frac{x}{B}} \]
   8. Step-by-step derivation

      1. associate--l+ 46.1%

         \[\leadsto \color{blue}{\left(0.16666666666666666 + 0.3333333333333333 \cdot x\right) \cdot B + \left(\frac{1}{B} - \frac{x}{B}\right)} \]

      2. *-commutative 46.1%

         \[\leadsto \color{blue}{B \cdot \left(0.16666666666666666 + 0.3333333333333333 \cdot x\right)} + \left(\frac{1}{B} - \frac{x}{B}\right) \]

      3. *-commutative 46.1%

         \[\leadsto B \cdot \left(0.16666666666666666 + \color{blue}{x \cdot 0.3333333333333333}\right) + \left(\frac{1}{B} - \frac{x}{B}\right) \]

      4. div-sub 46.1%

         \[\leadsto B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \color{blue}{\frac{1 - x}{B}} \]

   9. Simplified 46.1%

      \[\leadsto \color{blue}{B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.6 \cdot 10^{-28}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.18 \cdot 10^{-60}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \end{array} \]

Alternative 18: 44.0% accurate, 18.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{-28}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333 - 0.16666666666666666\right) - \frac{x + 1}{B}\\ \mathbf{elif}\;F \leq 1.22 \cdot 10^{-60}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -5e-28)
   (- (* B (- (* x 0.3333333333333333) 0.16666666666666666)) (/ (+ x 1.0) B))
   (if (<= F 1.22e-60)
     (/ (- x) B)
     (+
      (* B (+ 0.16666666666666666 (* x 0.3333333333333333)))
      (/ (- 1.0 x) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -5e-28) {
		tmp = (B * ((x * 0.3333333333333333) - 0.16666666666666666)) - ((x + 1.0) / B);
	} else if (F <= 1.22e-60) {
		tmp = -x / B;
	} else {
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((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 <= (-5d-28)) then
        tmp = (b * ((x * 0.3333333333333333d0) - 0.16666666666666666d0)) - ((x + 1.0d0) / b)
    else if (f <= 1.22d-60) then
        tmp = -x / b
    else
        tmp = (b * (0.16666666666666666d0 + (x * 0.3333333333333333d0))) + ((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 <= -5e-28) {
		tmp = (B * ((x * 0.3333333333333333) - 0.16666666666666666)) - ((x + 1.0) / B);
	} else if (F <= 1.22e-60) {
		tmp = -x / B;
	} else {
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -5e-28:
		tmp = (B * ((x * 0.3333333333333333) - 0.16666666666666666)) - ((x + 1.0) / B)
	elif F <= 1.22e-60:
		tmp = -x / B
	else:
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -5e-28)
		tmp = Float64(Float64(B * Float64(Float64(x * 0.3333333333333333) - 0.16666666666666666)) - Float64(Float64(x + 1.0) / B));
	elseif (F <= 1.22e-60)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(B * Float64(0.16666666666666666 + Float64(x * 0.3333333333333333))) + Float64(Float64(1.0 - x) / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -5e-28)
		tmp = (B * ((x * 0.3333333333333333) - 0.16666666666666666)) - ((x + 1.0) / B);
	elseif (F <= 1.22e-60)
		tmp = -x / B;
	else
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -5e-28], N[(N[(B * N[(N[(x * 0.3333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision] - N[(N[(x + 1.0), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.22e-60], N[((-x) / B), $MachinePrecision], N[(N[(B * N[(0.16666666666666666 + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.0000000000000002e-28

   1. Initial program 54.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 93.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 46.4%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot x - 0.16666666666666666\right) \cdot B + -1 \cdot \frac{1 + x}{B}} \]

   if -5.0000000000000002e-28 < F < 1.22e-60

   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. Step-by-step derivation

      1. +-commutative 99.5%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      2. unsub-neg 99.5%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      3. associate-*l/ 99.5%

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      4. associate-*r/ 99.4%

         \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      5. *-commutative 99.4%

         \[\leadsto \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \cdot F} - x \cdot \frac{1}{\tan B} \]

   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. Taylor expanded in F around inf 27.6%

      \[\leadsto F \cdot \color{blue}{\frac{1}{\sin B \cdot F}} - \frac{x}{\tan B} \]
   5. Step-by-step derivation

      1. *-commutative 27.6%

         \[\leadsto F \cdot \frac{1}{\color{blue}{F \cdot \sin B}} - \frac{x}{\tan B} \]

      2. associate-/r* 27.6%

         \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   6. Simplified 27.6%

      \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   7. Taylor expanded in B around 0 21.9%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

   8. Taylor expanded in x around inf 42.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   9. Step-by-step derivation

      1. associate-*r/ 42.4%

         \[\leadsto \color{blue}{\frac{-1 \cdot x}{B}} \]

      2. neg-mul-1 42.4%

         \[\leadsto \frac{\color{blue}{-x}}{B} \]

   10. Simplified 42.4%

       \[\leadsto \color{blue}{\frac{-x}{B}} \]

   if 1.22e-60 < F

   1. Initial program 64.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative 64.8%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      2. unsub-neg 64.8%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      3. associate-*l/ 77.0%

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      4. associate-*r/ 76.9%

         \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      5. *-commutative 76.9%

         \[\leadsto \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \cdot F} - x \cdot \frac{1}{\tan B} \]

   3. Simplified 76.9%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]

   4. Taylor expanded in F around inf 92.1%

      \[\leadsto F \cdot \color{blue}{\frac{1}{\sin B \cdot F}} - \frac{x}{\tan B} \]
   5. Step-by-step derivation

      1. *-commutative 92.1%

         \[\leadsto F \cdot \frac{1}{\color{blue}{F \cdot \sin B}} - \frac{x}{\tan B} \]

      2. associate-/r* 92.1%

         \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   6. Simplified 92.1%

      \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   7. Taylor expanded in B around 0 46.1%

      \[\leadsto \color{blue}{\left(\left(0.16666666666666666 + 0.3333333333333333 \cdot x\right) \cdot B + \frac{1}{B}\right) - \frac{x}{B}} \]
   8. Step-by-step derivation

      1. associate--l+ 46.1%

         \[\leadsto \color{blue}{\left(0.16666666666666666 + 0.3333333333333333 \cdot x\right) \cdot B + \left(\frac{1}{B} - \frac{x}{B}\right)} \]

      2. *-commutative 46.1%

         \[\leadsto \color{blue}{B \cdot \left(0.16666666666666666 + 0.3333333333333333 \cdot x\right)} + \left(\frac{1}{B} - \frac{x}{B}\right) \]

      3. *-commutative 46.1%

         \[\leadsto B \cdot \left(0.16666666666666666 + \color{blue}{x \cdot 0.3333333333333333}\right) + \left(\frac{1}{B} - \frac{x}{B}\right) \]

      4. div-sub 46.1%

         \[\leadsto B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \color{blue}{\frac{1 - x}{B}} \]

   9. Simplified 46.1%

      \[\leadsto \color{blue}{B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{-28}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333 - 0.16666666666666666\right) - \frac{x + 1}{B}\\ \mathbf{elif}\;F \leq 1.22 \cdot 10^{-60}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \end{array} \]

Alternative 19: 43.7% accurate, 35.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.2 \cdot 10^{-27}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.65 \cdot 10^{-95}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.2e-27)
   (/ (- -1.0 x) B)
   (if (<= F 2.65e-95) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.2e-27) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 2.65e-95) {
		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 <= (-1.2d-27)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 2.65d-95) 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 <= -1.2e-27) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 2.65e-95) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.2e-27:
		tmp = (-1.0 - x) / B
	elif F <= 2.65e-95:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.2e-27)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 2.65e-95)
		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 <= -1.2e-27)
		tmp = (-1.0 - x) / B;
	elseif (F <= 2.65e-95)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.2e-27], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 2.65e-95], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.20000000000000001e-27

   1. Initial program 54.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 93.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. *-un-lft-identity 93.9%

         \[\leadsto \color{blue}{1 \cdot \left(\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}\right)} \]

      2. +-commutative 93.9%

         \[\leadsto 1 \cdot \color{blue}{\left(\frac{-1}{\sin B} + \left(-x \cdot \frac{1}{\tan B}\right)\right)} \]

      3. div-inv 93.9%

         \[\leadsto 1 \cdot \left(\frac{-1}{\sin B} + \left(-\color{blue}{\frac{x}{\tan B}}\right)\right) \]

   4. Applied egg-rr 93.9%

      \[\leadsto \color{blue}{1 \cdot \left(\frac{-1}{\sin B} + \left(-\frac{x}{\tan B}\right)\right)} \]
   5. Step-by-step derivation

      1. *-lft-identity 93.9%

         \[\leadsto \color{blue}{\frac{-1}{\sin B} + \left(-\frac{x}{\tan B}\right)} \]

      2. unsub-neg 93.9%

         \[\leadsto \color{blue}{\frac{-1}{\sin B} - \frac{x}{\tan B}} \]

   6. Simplified 93.9%

      \[\leadsto \color{blue}{\frac{-1}{\sin B} - \frac{x}{\tan B}} \]

   7. Taylor expanded in B around inf 93.9%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\cos B \cdot x}{\sin B} + \frac{1}{\sin B}\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg 93.9%

         \[\leadsto \color{blue}{-\left(\frac{\cos B \cdot x}{\sin B} + \frac{1}{\sin B}\right)} \]

      2. +-commutative 93.9%

         \[\leadsto -\color{blue}{\left(\frac{1}{\sin B} + \frac{\cos B \cdot x}{\sin B}\right)} \]

      3. distribute-neg-in 93.9%

         \[\leadsto \color{blue}{\left(-\frac{1}{\sin B}\right) + \left(-\frac{\cos B \cdot x}{\sin B}\right)} \]

      4. distribute-neg-frac 93.9%

         \[\leadsto \color{blue}{\frac{-1}{\sin B}} + \left(-\frac{\cos B \cdot x}{\sin B}\right) \]

      5. metadata-eval 93.9%

         \[\leadsto \frac{\color{blue}{-1}}{\sin B} + \left(-\frac{\cos B \cdot x}{\sin B}\right) \]

      6. sub-neg 93.9%

         \[\leadsto \color{blue}{\frac{-1}{\sin B} - \frac{\cos B \cdot x}{\sin B}} \]

      7. *-commutative 93.9%

         \[\leadsto \frac{-1}{\sin B} - \frac{\color{blue}{x \cdot \cos B}}{\sin B} \]

      8. div-sub 93.9%

         \[\leadsto \color{blue}{\frac{-1 - x \cdot \cos B}{\sin B}} \]

   9. Simplified 93.9%

      \[\leadsto \color{blue}{\frac{-1 - x \cdot \cos B}{\sin B}} \]

   10. Taylor expanded in B around 0 46.3%

       \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   11. 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%

          \[\leadsto \frac{-1 + \color{blue}{\left(-x\right)}}{B} \]

       5. sub-neg 46.3%

          \[\leadsto \frac{\color{blue}{-1 - x}}{B} \]

   12. Simplified 46.3%

       \[\leadsto \color{blue}{\frac{-1 - x}{B}} \]

   if -1.20000000000000001e-27 < F < 2.6499999999999999e-95

   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. Step-by-step derivation

      1. +-commutative 99.5%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      2. unsub-neg 99.5%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      3. associate-*l/ 99.5%

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      4. associate-*r/ 99.4%

         \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      5. *-commutative 99.4%

         \[\leadsto \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \cdot F} - x \cdot \frac{1}{\tan B} \]

   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. Taylor expanded in F around inf 25.3%

      \[\leadsto F \cdot \color{blue}{\frac{1}{\sin B \cdot F}} - \frac{x}{\tan B} \]
   5. Step-by-step derivation

      1. *-commutative 25.3%

         \[\leadsto F \cdot \frac{1}{\color{blue}{F \cdot \sin B}} - \frac{x}{\tan B} \]

      2. associate-/r* 25.3%

         \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   6. Simplified 25.3%

      \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   7. Taylor expanded in B around 0 21.1%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

   8. Taylor expanded in x around inf 43.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   9. Step-by-step derivation

      1. associate-*r/ 43.2%

         \[\leadsto \color{blue}{\frac{-1 \cdot x}{B}} \]

      2. neg-mul-1 43.2%

         \[\leadsto \frac{\color{blue}{-x}}{B} \]

   10. Simplified 43.2%

       \[\leadsto \color{blue}{\frac{-x}{B}} \]

   if 2.6499999999999999e-95 < F

   1. Initial program 67.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative 67.7%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      2. unsub-neg 67.7%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      3. associate-*l/ 78.9%

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      4. associate-*r/ 78.7%

         \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      5. *-commutative 78.7%

         \[\leadsto \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \cdot F} - x \cdot \frac{1}{\tan B} \]

   3. Simplified 78.8%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]

   4. Taylor expanded in F around inf 89.4%

      \[\leadsto F \cdot \color{blue}{\frac{1}{\sin B \cdot F}} - \frac{x}{\tan B} \]
   5. Step-by-step derivation

      1. *-commutative 89.4%

         \[\leadsto F \cdot \frac{1}{\color{blue}{F \cdot \sin B}} - \frac{x}{\tan B} \]

      2. associate-/r* 89.3%

         \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   6. Simplified 89.3%

      \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   7. Taylor expanded in B around 0 43.9%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.2 \cdot 10^{-27}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.65 \cdot 10^{-95}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 20: 37.1% accurate, 39.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{-28}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.32 \cdot 10^{+146}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1e-28)
   (/ (- -1.0 x) B)
   (if (<= F 1.32e+146) (/ (- x) B) (/ 1.0 B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1e-28) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.32e+146) {
		tmp = -x / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1d-28)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 1.32d+146) then
        tmp = -x / b
    else
        tmp = 1.0d0 / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1e-28) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.32e+146) {
		tmp = -x / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1e-28:
		tmp = (-1.0 - x) / B
	elif F <= 1.32e+146:
		tmp = -x / B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1e-28)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 1.32e+146)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(1.0 / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1e-28)
		tmp = (-1.0 - x) / B;
	elseif (F <= 1.32e+146)
		tmp = -x / B;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1e-28], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.32e+146], N[((-x) / B), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -9.99999999999999971e-29

   1. Initial program 54.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 93.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. *-un-lft-identity 93.9%

         \[\leadsto \color{blue}{1 \cdot \left(\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}\right)} \]

      2. +-commutative 93.9%

         \[\leadsto 1 \cdot \color{blue}{\left(\frac{-1}{\sin B} + \left(-x \cdot \frac{1}{\tan B}\right)\right)} \]

      3. div-inv 93.9%

         \[\leadsto 1 \cdot \left(\frac{-1}{\sin B} + \left(-\color{blue}{\frac{x}{\tan B}}\right)\right) \]

   4. Applied egg-rr 93.9%

      \[\leadsto \color{blue}{1 \cdot \left(\frac{-1}{\sin B} + \left(-\frac{x}{\tan B}\right)\right)} \]
   5. Step-by-step derivation

      1. *-lft-identity 93.9%

         \[\leadsto \color{blue}{\frac{-1}{\sin B} + \left(-\frac{x}{\tan B}\right)} \]

      2. unsub-neg 93.9%

         \[\leadsto \color{blue}{\frac{-1}{\sin B} - \frac{x}{\tan B}} \]

   6. Simplified 93.9%

      \[\leadsto \color{blue}{\frac{-1}{\sin B} - \frac{x}{\tan B}} \]

   7. Taylor expanded in B around inf 93.9%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\cos B \cdot x}{\sin B} + \frac{1}{\sin B}\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg 93.9%

         \[\leadsto \color{blue}{-\left(\frac{\cos B \cdot x}{\sin B} + \frac{1}{\sin B}\right)} \]

      2. +-commutative 93.9%

         \[\leadsto -\color{blue}{\left(\frac{1}{\sin B} + \frac{\cos B \cdot x}{\sin B}\right)} \]

      3. distribute-neg-in 93.9%

         \[\leadsto \color{blue}{\left(-\frac{1}{\sin B}\right) + \left(-\frac{\cos B \cdot x}{\sin B}\right)} \]

      4. distribute-neg-frac 93.9%

         \[\leadsto \color{blue}{\frac{-1}{\sin B}} + \left(-\frac{\cos B \cdot x}{\sin B}\right) \]

      5. metadata-eval 93.9%

         \[\leadsto \frac{\color{blue}{-1}}{\sin B} + \left(-\frac{\cos B \cdot x}{\sin B}\right) \]

      6. sub-neg 93.9%

         \[\leadsto \color{blue}{\frac{-1}{\sin B} - \frac{\cos B \cdot x}{\sin B}} \]

      7. *-commutative 93.9%

         \[\leadsto \frac{-1}{\sin B} - \frac{\color{blue}{x \cdot \cos B}}{\sin B} \]

      8. div-sub 93.9%

         \[\leadsto \color{blue}{\frac{-1 - x \cdot \cos B}{\sin B}} \]

   9. Simplified 93.9%

      \[\leadsto \color{blue}{\frac{-1 - x \cdot \cos B}{\sin B}} \]

   10. Taylor expanded in B around 0 46.3%

       \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   11. 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%

          \[\leadsto \frac{-1 + \color{blue}{\left(-x\right)}}{B} \]

       5. sub-neg 46.3%

          \[\leadsto \frac{\color{blue}{-1 - x}}{B} \]

   12. Simplified 46.3%

       \[\leadsto \color{blue}{\frac{-1 - x}{B}} \]

   if -9.99999999999999971e-29 < F < 1.32000000000000011e146

   1. Initial program 96.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative 96.8%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      2. unsub-neg 96.8%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      3. associate-*l/ 99.5%

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      4. associate-*r/ 99.4%

         \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      5. *-commutative 99.4%

         \[\leadsto \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \cdot F} - x \cdot \frac{1}{\tan B} \]

   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. Taylor expanded in F around inf 44.3%

      \[\leadsto F \cdot \color{blue}{\frac{1}{\sin B \cdot F}} - \frac{x}{\tan B} \]
   5. Step-by-step derivation

      1. *-commutative 44.3%

         \[\leadsto F \cdot \frac{1}{\color{blue}{F \cdot \sin B}} - \frac{x}{\tan B} \]

      2. associate-/r* 44.2%

         \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   6. Simplified 44.2%

      \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   7. Taylor expanded in B around 0 26.1%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

   8. Taylor expanded in x around inf 37.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   9. Step-by-step derivation

      1. associate-*r/ 37.1%

         \[\leadsto \color{blue}{\frac{-1 \cdot x}{B}} \]

      2. neg-mul-1 37.1%

         \[\leadsto \frac{\color{blue}{-x}}{B} \]

   10. Simplified 37.1%

       \[\leadsto \color{blue}{\frac{-x}{B}} \]

   if 1.32000000000000011e146 < F

   1. Initial program 35.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative 35.0%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      2. unsub-neg 35.0%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      3. associate-*l/ 50.5%

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      4. associate-*r/ 50.5%

         \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      5. *-commutative 50.5%

         \[\leadsto \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \cdot F} - x \cdot \frac{1}{\tan B} \]

   3. Simplified 50.5%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]

   4. Taylor expanded in F around inf 99.7%

      \[\leadsto F \cdot \color{blue}{\frac{1}{\sin B \cdot F}} - \frac{x}{\tan B} \]
   5. Step-by-step derivation

      1. *-commutative 99.7%

         \[\leadsto F \cdot \frac{1}{\color{blue}{F \cdot \sin B}} - \frac{x}{\tan B} \]

      2. associate-/r* 99.6%

         \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   6. Simplified 99.6%

      \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   7. Taylor expanded in B around 0 54.8%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

   8. Taylor expanded in x around 0 35.7%

      \[\leadsto \color{blue}{\frac{1}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 39.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{-28}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.32 \cdot 10^{+146}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]

Alternative 21: 29.8% accurate, 53.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 5.8 \cdot 10^{+146}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (if (<= F 5.8e+146) (/ (- x) B) (/ 1.0 B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= 5.8e+146) {
		tmp = -x / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= 5.8d+146) then
        tmp = -x / b
    else
        tmp = 1.0d0 / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= 5.8e+146) {
		tmp = -x / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= 5.8e+146:
		tmp = -x / B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= 5.8e+146)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(1.0 / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= 5.8e+146)
		tmp = -x / B;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, 5.8e+146], N[((-x) / B), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 5.7999999999999997e146

   1. Initial program 81.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative 81.9%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      2. unsub-neg 81.9%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      3. associate-*l/ 87.2%

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      4. associate-*r/ 87.2%

         \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      5. *-commutative 87.2%

         \[\leadsto \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \cdot F} - x \cdot \frac{1}{\tan B} \]

   3. Simplified 87.3%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]

   4. Taylor expanded in F around inf 41.4%

      \[\leadsto F \cdot \color{blue}{\frac{1}{\sin B \cdot F}} - \frac{x}{\tan B} \]
   5. Step-by-step derivation

      1. *-commutative 41.4%

         \[\leadsto F \cdot \frac{1}{\color{blue}{F \cdot \sin B}} - \frac{x}{\tan B} \]

      2. associate-/r* 41.4%

         \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   6. Simplified 41.4%

      \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   7. Taylor expanded in B around 0 23.6%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

   8. Taylor expanded in x around inf 30.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   9. Step-by-step derivation

      1. associate-*r/ 30.8%

         \[\leadsto \color{blue}{\frac{-1 \cdot x}{B}} \]

      2. neg-mul-1 30.8%

         \[\leadsto \frac{\color{blue}{-x}}{B} \]

   10. Simplified 30.8%

       \[\leadsto \color{blue}{\frac{-x}{B}} \]

   if 5.7999999999999997e146 < F

   1. Initial program 35.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative 35.0%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      2. unsub-neg 35.0%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      3. associate-*l/ 50.5%

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      4. associate-*r/ 50.5%

         \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      5. *-commutative 50.5%

         \[\leadsto \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \cdot F} - x \cdot \frac{1}{\tan B} \]

   3. Simplified 50.5%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]

   4. Taylor expanded in F around inf 99.7%

      \[\leadsto F \cdot \color{blue}{\frac{1}{\sin B \cdot F}} - \frac{x}{\tan B} \]
   5. Step-by-step derivation

      1. *-commutative 99.7%

         \[\leadsto F \cdot \frac{1}{\color{blue}{F \cdot \sin B}} - \frac{x}{\tan B} \]

      2. associate-/r* 99.6%

         \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   6. Simplified 99.6%

      \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   7. Taylor expanded in B around 0 54.8%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

   8. Taylor expanded in x around 0 35.7%

      \[\leadsto \color{blue}{\frac{1}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 5.8 \cdot 10^{+146}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]

Alternative 22: 9.7% accurate, 108.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (/ 1.0 B))

C

double code(double F, double B, double x) {
	return 1.0 / B;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = 1.0d0 / b
end function

Java

public static double code(double F, double B, double x) {
	return 1.0 / B;
}

Python

def code(F, B, x):
	return 1.0 / B

Julia

function code(F, B, x)
	return Float64(1.0 / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = 1.0 / B;
end

Wolfram

code[F_, B_, x_] := N[(1.0 / B), $MachinePrecision]

Derivation

1. Initial program 75.5%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
2. Step-by-step derivation

   1. +-commutative 75.5%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

   2. unsub-neg 75.5%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

   3. associate-*l/ 82.2%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

   4. associate-*r/ 82.2%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

   5. *-commutative 82.2%

      \[\leadsto \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \cdot F} - x \cdot \frac{1}{\tan B} \]

3. Simplified 82.3%

   \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]

4. Taylor expanded in F around inf 49.4%

   \[\leadsto F \cdot \color{blue}{\frac{1}{\sin B \cdot F}} - \frac{x}{\tan B} \]
5. Step-by-step derivation

   1. *-commutative 49.4%

      \[\leadsto F \cdot \frac{1}{\color{blue}{F \cdot \sin B}} - \frac{x}{\tan B} \]

   2. associate-/r* 49.4%

      \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

6. Simplified 49.4%

   \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

7. Taylor expanded in B around 0 27.8%

   \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

8. Taylor expanded in x around 0 9.3%

   \[\leadsto \color{blue}{\frac{1}{B}} \]

9. Final simplification 9.3%

   \[\leadsto \frac{1}{B} \]

Reproduce

herbie shell --seed 2023257 
(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))))))