VandenBroeck and Keller, Equation (23)

Percentage Accurate: 77.1% → 99.7%

Time: 18.2s

Alternatives: 27

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: 77.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2 \cdot 10^{+35}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 5.4 \cdot 10^{+28}:\\ \;\;\;\;\frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\frac{\sin B}{F}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -2e+35)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 5.4e+28)
       (- (/ (pow (+ (* F F) (+ 2.0 (* x 2.0))) -0.5) (/ (sin B) F)) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -2e+35) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 5.4e+28) {
		tmp = (pow(((F * F) + (2.0 + (x * 2.0))), -0.5) / (sin(B) / F)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-2d+35)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 5.4d+28) then
        tmp = ((((f * f) + (2.0d0 + (x * 2.0d0))) ** (-0.5d0)) / (sin(b) / f)) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -2e+35) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 5.4e+28) {
		tmp = (Math.pow(((F * F) + (2.0 + (x * 2.0))), -0.5) / (Math.sin(B) / F)) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -2e+35:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 5.4e+28:
		tmp = (math.pow(((F * F) + (2.0 + (x * 2.0))), -0.5) / (math.sin(B) / F)) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -2e+35)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 5.4e+28)
		tmp = Float64(Float64((Float64(Float64(F * F) + Float64(2.0 + Float64(x * 2.0))) ^ -0.5) / Float64(sin(B) / F)) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -2e+35)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 5.4e+28)
		tmp = ((((F * F) + (2.0 + (x * 2.0))) ^ -0.5) / (sin(B) / F)) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2e+35], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 5.4e+28], N[(N[(N[Power[N[(N[(F * F), $MachinePrecision] + N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[(N[Sin[B], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.9999999999999999e35

   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. Step-by-step derivation

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 74.8%

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

   5. Taylor expanded in F around -inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{-1}{\sin B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \sin B\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. sin-lowering-sin.f64 99.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 99.8%

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

   if -1.9999999999999999e35 < F < 5.4000000000000003e28

   1. Initial program 99.4%

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

      1. +-commutative N/A

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

      2. div-inv N/A

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

      3. sub-neg N/A

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

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(\frac{x}{\tan B}\right)}\right) \]

   4. Applied egg-rr 99.7%

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

   if 5.4000000000000003e28 < F

   1. Initial program 57.6%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 74.1%

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

   5. Taylor expanded in F around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{\sin B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \sin B\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. sin-lowering-sin.f64 99.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 99.8%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2 \cdot 10^{+152}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 10^{+25}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{F \cdot F + 2}}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -2e+152)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1e+25)
       (- (/ (* F (sqrt (/ 1.0 (+ (* F F) 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 <= -2e+152) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1e+25) {
		tmp = ((F * sqrt((1.0 / ((F * F) + 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 <= (-2d+152)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1d+25) then
        tmp = ((f * sqrt((1.0d0 / ((f * f) + 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 <= -2e+152) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1e+25) {
		tmp = ((F * Math.sqrt((1.0 / ((F * F) + 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 <= -2e+152:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1e+25:
		tmp = ((F * math.sqrt((1.0 / ((F * F) + 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 <= -2e+152)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1e+25)
		tmp = Float64(Float64(Float64(F * sqrt(Float64(1.0 / Float64(Float64(F * F) + 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 <= -2e+152)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1e+25)
		tmp = ((F * sqrt((1.0 / ((F * F) + 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, -2e+152], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1e+25], N[(N[(N[(F * N[Sqrt[N[(1.0 / N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[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 < -2.0000000000000001e152

   1. Initial program 42.6%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 60.4%

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

   5. Taylor expanded in F around -inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{-1}{\sin B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \sin B\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. sin-lowering-sin.f64 99.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 99.8%

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

   if -2.0000000000000001e152 < F < 1.00000000000000009e25

   1. Initial program 95.6%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + {F}^{2}}}\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. *-lowering-*.f64 99.7%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 99.7%

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

   if 1.00000000000000009e25 < F

   1. Initial program 58.8%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 74.8%

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

   5. Taylor expanded in F around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{\sin B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \sin B\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. sin-lowering-sin.f64 99.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{+152}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 10^{+25}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{F \cdot F + 2}}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1.42:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.4)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.42)
       (- (/ (* 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 <= -1.4) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.42) {
		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 <= (-1.4d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.42d0) 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 <= -1.4) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.42) {
		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 <= -1.4:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.42:
		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 <= -1.4)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.42)
		tmp = Float64(Float64(Float64(F * 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 <= -1.4)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.42)
		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, -1.4], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.42], N[(N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.3999999999999999

   1. Initial program 57.1%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 76.3%

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

   5. Taylor expanded in F around -inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{-1}{\sin B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \sin B\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. sin-lowering-sin.f64 98.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 98.9%

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

   if -1.3999999999999999 < F < 1.4199999999999999

   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 N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + {F}^{2}}}\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. *-lowering-*.f64 99.7%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 99.7%

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

   8. Taylor expanded in F around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{F \cdot \sqrt{\frac{1}{2}}}{\sin B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2}}\right), \sin B\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2}}\right)\right), \sin B\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right), \sin B\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. sin-lowering-sin.f64 99.1%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   10. Simplified 99.1%

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

   if 1.4199999999999999 < F

   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. Step-by-step derivation

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 77.1%

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

   5. Taylor expanded in F around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{\sin B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \sin B\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. sin-lowering-sin.f64 98.5%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 98.5%

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

Alternative 4: 86.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B \cdot \frac{\sqrt{F \cdot F + 2}}{F}}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -310000000000:\\ \;\;\;\;\frac{-1}{\sin B} - t\_1\\ \mathbf{elif}\;F \leq -3 \cdot 10^{-103}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 2.35 \cdot 10^{-56}:\\ \;\;\;\;0 - t\_1\\ \mathbf{elif}\;F \leq 430000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* (sin B) (/ (sqrt (+ (* F F) 2.0)) F))))
        (t_1 (/ x (tan B))))
   (if (<= F -310000000000.0)
     (- (/ -1.0 (sin B)) t_1)
     (if (<= F -3e-103)
       t_0
       (if (<= F 2.35e-56)
         (- 0.0 t_1)
         (if (<= F 430000.0) t_0 (- (/ 1.0 (sin B)) t_1)))))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / (sin(B) * (sqrt(((F * F) + 2.0)) / F));
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -310000000000.0) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= -3e-103) {
		tmp = t_0;
	} else if (F <= 2.35e-56) {
		tmp = 0.0 - t_1;
	} else if (F <= 430000.0) {
		tmp = t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_1;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 / (sin(b) * (sqrt(((f * f) + 2.0d0)) / f))
    t_1 = x / tan(b)
    if (f <= (-310000000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_1
    else if (f <= (-3d-103)) then
        tmp = t_0
    else if (f <= 2.35d-56) then
        tmp = 0.0d0 - t_1
    else if (f <= 430000.0d0) then
        tmp = t_0
    else
        tmp = (1.0d0 / sin(b)) - t_1
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = 1.0 / (Math.sin(B) * (Math.sqrt(((F * F) + 2.0)) / F));
	double t_1 = x / Math.tan(B);
	double tmp;
	if (F <= -310000000000.0) {
		tmp = (-1.0 / Math.sin(B)) - t_1;
	} else if (F <= -3e-103) {
		tmp = t_0;
	} else if (F <= 2.35e-56) {
		tmp = 0.0 - t_1;
	} else if (F <= 430000.0) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = 1.0 / (math.sin(B) * (math.sqrt(((F * F) + 2.0)) / F))
	t_1 = x / math.tan(B)
	tmp = 0
	if F <= -310000000000.0:
		tmp = (-1.0 / math.sin(B)) - t_1
	elif F <= -3e-103:
		tmp = t_0
	elif F <= 2.35e-56:
		tmp = 0.0 - t_1
	elif F <= 430000.0:
		tmp = t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / Float64(sin(B) * Float64(sqrt(Float64(Float64(F * F) + 2.0)) / F)))
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -310000000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= -3e-103)
		tmp = t_0;
	elseif (F <= 2.35e-56)
		tmp = Float64(0.0 - t_1);
	elseif (F <= 430000.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = 1.0 / (sin(B) * (sqrt(((F * F) + 2.0)) / F));
	t_1 = x / tan(B);
	tmp = 0.0;
	if (F <= -310000000000.0)
		tmp = (-1.0 / sin(B)) - t_1;
	elseif (F <= -3e-103)
		tmp = t_0;
	elseif (F <= 2.35e-56)
		tmp = 0.0 - t_1;
	elseif (F <= 430000.0)
		tmp = t_0;
	else
		tmp = (1.0 / sin(B)) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[(N[Sin[B], $MachinePrecision] * N[(N[Sqrt[N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -310000000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, -3e-103], t$95$0, If[LessEqual[F, 2.35e-56], N[(0.0 - t$95$1), $MachinePrecision], If[LessEqual[F, 430000.0], t$95$0, N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.1e11

   1. Initial program 55.1%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 75.2%

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

   5. Taylor expanded in F around -inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{-1}{\sin B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \sin B\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. sin-lowering-sin.f64 99.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 99.8%

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

   if -3.1e11 < F < -3e-103 or 2.35e-56 < F < 4.3e5

   1. Initial program 99.2%

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

   3. Taylor expanded in x around 0

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \color{blue}{\left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \color{blue}{\sin B}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin \color{blue}{B}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right) \]

      10. sin-lowering-sin.f64 80.7%

          \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right) \]

   5. Simplified 80.7%

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

      1. associate-*r/ N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

      4. unpow1/2 N/A

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

      5. div-inv N/A

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

      6. clear-num N/A

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

      7. associate-/l/ N/A

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

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\sin B \cdot \frac{{\left(2 + F \cdot F\right)}^{\frac{1}{2}}}{F}\right)}\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\sin B, \color{blue}{\left(\frac{{\left(2 + F \cdot F\right)}^{\frac{1}{2}}}{F}\right)}\right)\right) \]

      10. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), \left(\frac{\color{blue}{{\left(2 + F \cdot F\right)}^{\frac{1}{2}}}}{F}\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), \mathsf{/.f64}\left(\left({\left(2 + F \cdot F\right)}^{\frac{1}{2}}\right), \color{blue}{F}\right)\right)\right) \]

      12. unpow1/2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), \mathsf{/.f64}\left(\left(\sqrt{2 + F \cdot F}\right), F\right)\right)\right) \]

      13. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(2 + F \cdot F\right)\right), F\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right), F\right)\right)\right) \]

      15. *-lowering-*.f64 80.8%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(B\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right), F\right)\right)\right) \]

   7. Applied egg-rr 80.8%

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

   if -3e-103 < F < 2.35e-56

   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 N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + {F}^{2}}}\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. *-lowering-*.f64 99.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 99.8%

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

   8. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right) \]

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \cos B\right), \color{blue}{\left(\mathsf{neg}\left(\sin B\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \cos B\right), \left(\mathsf{neg}\left(\color{blue}{\sin B}\right)\right)\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \left(\mathsf{neg}\left(\sin B\right)\right)\right) \]

      6. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \mathsf{neg.f64}\left(\sin B\right)\right) \]

      7. sin-lowering-sin.f64 86.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \mathsf{neg.f64}\left(\mathsf{sin.f64}\left(B\right)\right)\right) \]

   10. Simplified 86.3%

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

       1. distribute-frac-neg2 N/A

          \[\leadsto \mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right) \]

       2. clear-num N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\sin B}{x \cdot \cos B}}\right) \]

       3. associate-/l/ N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\frac{\sin B}{\cos B}}{x}}\right) \]

       4. tan-quot N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\tan B}{x}}\right) \]

       5. clear-num N/A

          \[\leadsto \mathsf{neg}\left(\frac{x}{\tan B}\right) \]

       6. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{x}{\tan B}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, \tan B\right)\right) \]

       8. tan-lowering-tan.f64 86.5%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   12. Applied egg-rr 86.5%

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

   if 4.3e5 < F

   1. Initial program 61.1%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 76.2%

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

   5. Taylor expanded in F around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{\sin B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \sin B\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. sin-lowering-sin.f64 99.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 99.8%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -310000000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -3 \cdot 10^{-103}:\\ \;\;\;\;\frac{1}{\sin B \cdot \frac{\sqrt{F \cdot F + 2}}{F}}\\ \mathbf{elif}\;F \leq 2.35 \cdot 10^{-56}:\\ \;\;\;\;0 - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 430000:\\ \;\;\;\;\frac{1}{\sin B \cdot \frac{\sqrt{F \cdot F + 2}}{F}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := F \cdot \frac{\sqrt{\frac{1}{F \cdot F + 2}}}{\sin B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -310000000000:\\ \;\;\;\;\frac{-1}{\sin B} - t\_1\\ \mathbf{elif}\;F \leq -5.6 \cdot 10^{-99}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 6.6 \cdot 10^{-73}:\\ \;\;\;\;0 - t\_1\\ \mathbf{elif}\;F \leq 430000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* F (/ (sqrt (/ 1.0 (+ (* F F) 2.0))) (sin B))))
        (t_1 (/ x (tan B))))
   (if (<= F -310000000000.0)
     (- (/ -1.0 (sin B)) t_1)
     (if (<= F -5.6e-99)
       t_0
       (if (<= F 6.6e-73)
         (- 0.0 t_1)
         (if (<= F 430000.0) t_0 (- (/ 1.0 (sin B)) t_1)))))))

C

double code(double F, double B, double x) {
	double t_0 = F * (sqrt((1.0 / ((F * F) + 2.0))) / sin(B));
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -310000000000.0) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= -5.6e-99) {
		tmp = t_0;
	} else if (F <= 6.6e-73) {
		tmp = 0.0 - t_1;
	} else if (F <= 430000.0) {
		tmp = t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_1;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = f * (sqrt((1.0d0 / ((f * f) + 2.0d0))) / sin(b))
    t_1 = x / tan(b)
    if (f <= (-310000000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_1
    else if (f <= (-5.6d-99)) then
        tmp = t_0
    else if (f <= 6.6d-73) then
        tmp = 0.0d0 - t_1
    else if (f <= 430000.0d0) then
        tmp = t_0
    else
        tmp = (1.0d0 / sin(b)) - t_1
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = F * (Math.sqrt((1.0 / ((F * F) + 2.0))) / Math.sin(B));
	double t_1 = x / Math.tan(B);
	double tmp;
	if (F <= -310000000000.0) {
		tmp = (-1.0 / Math.sin(B)) - t_1;
	} else if (F <= -5.6e-99) {
		tmp = t_0;
	} else if (F <= 6.6e-73) {
		tmp = 0.0 - t_1;
	} else if (F <= 430000.0) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = F * (math.sqrt((1.0 / ((F * F) + 2.0))) / math.sin(B))
	t_1 = x / math.tan(B)
	tmp = 0
	if F <= -310000000000.0:
		tmp = (-1.0 / math.sin(B)) - t_1
	elif F <= -5.6e-99:
		tmp = t_0
	elif F <= 6.6e-73:
		tmp = 0.0 - t_1
	elif F <= 430000.0:
		tmp = t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(F * Float64(sqrt(Float64(1.0 / Float64(Float64(F * F) + 2.0))) / sin(B)))
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -310000000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= -5.6e-99)
		tmp = t_0;
	elseif (F <= 6.6e-73)
		tmp = Float64(0.0 - t_1);
	elseif (F <= 430000.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = F * (sqrt((1.0 / ((F * F) + 2.0))) / sin(B));
	t_1 = x / tan(B);
	tmp = 0.0;
	if (F <= -310000000000.0)
		tmp = (-1.0 / sin(B)) - t_1;
	elseif (F <= -5.6e-99)
		tmp = t_0;
	elseif (F <= 6.6e-73)
		tmp = 0.0 - t_1;
	elseif (F <= 430000.0)
		tmp = t_0;
	else
		tmp = (1.0 / sin(B)) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(F * N[(N[Sqrt[N[(1.0 / N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -310000000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, -5.6e-99], t$95$0, If[LessEqual[F, 6.6e-73], N[(0.0 - t$95$1), $MachinePrecision], If[LessEqual[F, 430000.0], t$95$0, N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.1e11

   1. Initial program 55.1%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 75.2%

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

   5. Taylor expanded in F around -inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{-1}{\sin B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \sin B\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. sin-lowering-sin.f64 99.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 99.8%

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

   if -3.1e11 < F < -5.6000000000000001e-99 or 6.60000000000000007e-73 < F < 4.3e5

   1. Initial program 99.2%

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

   3. Taylor expanded in x around 0

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \color{blue}{\left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \color{blue}{\sin B}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin \color{blue}{B}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right) \]

      10. sin-lowering-sin.f64 79.1%

          \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right) \]

   5. Simplified 79.1%

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

   if -5.6000000000000001e-99 < F < 6.60000000000000007e-73

   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 N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + {F}^{2}}}\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. *-lowering-*.f64 99.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 99.8%

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

   8. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right) \]

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \cos B\right), \color{blue}{\left(\mathsf{neg}\left(\sin B\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \cos B\right), \left(\mathsf{neg}\left(\color{blue}{\sin B}\right)\right)\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \left(\mathsf{neg}\left(\sin B\right)\right)\right) \]

      6. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \mathsf{neg.f64}\left(\sin B\right)\right) \]

      7. sin-lowering-sin.f64 87.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \mathsf{neg.f64}\left(\mathsf{sin.f64}\left(B\right)\right)\right) \]

   10. Simplified 87.0%

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

       1. distribute-frac-neg2 N/A

          \[\leadsto \mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right) \]

       2. clear-num N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\sin B}{x \cdot \cos B}}\right) \]

       3. associate-/l/ N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\frac{\sin B}{\cos B}}{x}}\right) \]

       4. tan-quot N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\tan B}{x}}\right) \]

       5. clear-num N/A

          \[\leadsto \mathsf{neg}\left(\frac{x}{\tan B}\right) \]

       6. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{x}{\tan B}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, \tan B\right)\right) \]

       8. tan-lowering-tan.f64 87.2%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   12. Applied egg-rr 87.2%

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

   if 4.3e5 < F

   1. Initial program 61.1%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 76.2%

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

   5. Taylor expanded in F around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{\sin B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \sin B\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. sin-lowering-sin.f64 99.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 99.8%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -310000000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -5.6 \cdot 10^{-99}:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{F \cdot F + 2}}}{\sin B}\\ \mathbf{elif}\;F \leq 6.6 \cdot 10^{-73}:\\ \;\;\;\;0 - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 430000:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{F \cdot F + 2}}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F}{\frac{\sin B}{{\left(F \cdot F + 2\right)}^{-0.5}}}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -310000000000:\\ \;\;\;\;\frac{-1}{\sin B} - t\_1\\ \mathbf{elif}\;F \leq -4.9 \cdot 10^{-100}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 1.25 \cdot 10^{-69}:\\ \;\;\;\;0 - t\_1\\ \mathbf{elif}\;F \leq 430000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ F (/ (sin B) (pow (+ (* F F) 2.0) -0.5)))) (t_1 (/ x (tan B))))
   (if (<= F -310000000000.0)
     (- (/ -1.0 (sin B)) t_1)
     (if (<= F -4.9e-100)
       t_0
       (if (<= F 1.25e-69)
         (- 0.0 t_1)
         (if (<= F 430000.0) t_0 (- (/ 1.0 (sin B)) t_1)))))))

C

double code(double F, double B, double x) {
	double t_0 = F / (sin(B) / pow(((F * F) + 2.0), -0.5));
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -310000000000.0) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= -4.9e-100) {
		tmp = t_0;
	} else if (F <= 1.25e-69) {
		tmp = 0.0 - t_1;
	} else if (F <= 430000.0) {
		tmp = t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_1;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = f / (sin(b) / (((f * f) + 2.0d0) ** (-0.5d0)))
    t_1 = x / tan(b)
    if (f <= (-310000000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_1
    else if (f <= (-4.9d-100)) then
        tmp = t_0
    else if (f <= 1.25d-69) then
        tmp = 0.0d0 - t_1
    else if (f <= 430000.0d0) then
        tmp = t_0
    else
        tmp = (1.0d0 / sin(b)) - t_1
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = F / (Math.sin(B) / Math.pow(((F * F) + 2.0), -0.5));
	double t_1 = x / Math.tan(B);
	double tmp;
	if (F <= -310000000000.0) {
		tmp = (-1.0 / Math.sin(B)) - t_1;
	} else if (F <= -4.9e-100) {
		tmp = t_0;
	} else if (F <= 1.25e-69) {
		tmp = 0.0 - t_1;
	} else if (F <= 430000.0) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = F / (math.sin(B) / math.pow(((F * F) + 2.0), -0.5))
	t_1 = x / math.tan(B)
	tmp = 0
	if F <= -310000000000.0:
		tmp = (-1.0 / math.sin(B)) - t_1
	elif F <= -4.9e-100:
		tmp = t_0
	elif F <= 1.25e-69:
		tmp = 0.0 - t_1
	elif F <= 430000.0:
		tmp = t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(F / Float64(sin(B) / (Float64(Float64(F * F) + 2.0) ^ -0.5)))
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -310000000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= -4.9e-100)
		tmp = t_0;
	elseif (F <= 1.25e-69)
		tmp = Float64(0.0 - t_1);
	elseif (F <= 430000.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = F / (sin(B) / (((F * F) + 2.0) ^ -0.5));
	t_1 = x / tan(B);
	tmp = 0.0;
	if (F <= -310000000000.0)
		tmp = (-1.0 / sin(B)) - t_1;
	elseif (F <= -4.9e-100)
		tmp = t_0;
	elseif (F <= 1.25e-69)
		tmp = 0.0 - t_1;
	elseif (F <= 430000.0)
		tmp = t_0;
	else
		tmp = (1.0 / sin(B)) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(F / N[(N[Sin[B], $MachinePrecision] / N[Power[N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -310000000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, -4.9e-100], t$95$0, If[LessEqual[F, 1.25e-69], N[(0.0 - t$95$1), $MachinePrecision], If[LessEqual[F, 430000.0], t$95$0, N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.1e11

   1. Initial program 55.1%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 75.2%

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

   5. Taylor expanded in F around -inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{-1}{\sin B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \sin B\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. sin-lowering-sin.f64 99.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 99.8%

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

   if -3.1e11 < F < -4.9000000000000003e-100 or 1.25000000000000008e-69 < F < 4.3e5

   1. Initial program 99.2%

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

   3. Taylor expanded in x around 0

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \color{blue}{\left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \color{blue}{\sin B}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin \color{blue}{B}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right) \]

      10. sin-lowering-sin.f64 79.1%

          \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right) \]

   5. Simplified 79.1%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(F, \color{blue}{\left(\frac{\sin B}{\sqrt{\frac{1}{2 + F \cdot F}}}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(F, \mathsf{/.f64}\left(\sin B, \color{blue}{\left(\sqrt{\frac{1}{2 + F \cdot F}}\right)}\right)\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(B\right), \left(\sqrt{\color{blue}{\frac{1}{2 + F \cdot F}}}\right)\right)\right) \]

      6. pow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(B\right), \left({\left(\frac{1}{2 + F \cdot F}\right)}^{\color{blue}{\frac{1}{2}}}\right)\right)\right) \]

      7. inv-pow N/A

         \[\leadsto \mathsf{/.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(B\right), \left({\left({\left(2 + F \cdot F\right)}^{-1}\right)}^{\frac{1}{2}}\right)\right)\right) \]

      8. pow-pow N/A

         \[\leadsto \mathsf{/.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(B\right), \left({\left(2 + F \cdot F\right)}^{\color{blue}{\left(-1 \cdot \frac{1}{2}\right)}}\right)\right)\right) \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(B\right), \mathsf{pow.f64}\left(\left(2 + F \cdot F\right), \color{blue}{\left(-1 \cdot \frac{1}{2}\right)}\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(B\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(2, \left(F \cdot F\right)\right), \left(\color{blue}{-1} \cdot \frac{1}{2}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(B\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right), \left(-1 \cdot \frac{1}{2}\right)\right)\right)\right) \]

      12. metadata-eval 79.0%

          \[\leadsto \mathsf{/.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(B\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right), \frac{-1}{2}\right)\right)\right) \]

   7. Applied egg-rr 79.0%

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

   if -4.9000000000000003e-100 < F < 1.25000000000000008e-69

   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 N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + {F}^{2}}}\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. *-lowering-*.f64 99.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 99.8%

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

   8. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right) \]

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \cos B\right), \color{blue}{\left(\mathsf{neg}\left(\sin B\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \cos B\right), \left(\mathsf{neg}\left(\color{blue}{\sin B}\right)\right)\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \left(\mathsf{neg}\left(\sin B\right)\right)\right) \]

      6. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \mathsf{neg.f64}\left(\sin B\right)\right) \]

      7. sin-lowering-sin.f64 87.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \mathsf{neg.f64}\left(\mathsf{sin.f64}\left(B\right)\right)\right) \]

   10. Simplified 87.0%

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

       1. distribute-frac-neg2 N/A

          \[\leadsto \mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right) \]

       2. clear-num N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\sin B}{x \cdot \cos B}}\right) \]

       3. associate-/l/ N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\frac{\sin B}{\cos B}}{x}}\right) \]

       4. tan-quot N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\tan B}{x}}\right) \]

       5. clear-num N/A

          \[\leadsto \mathsf{neg}\left(\frac{x}{\tan B}\right) \]

       6. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{x}{\tan B}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, \tan B\right)\right) \]

       8. tan-lowering-tan.f64 87.2%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   12. Applied egg-rr 87.2%

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

   if 4.3e5 < F

   1. Initial program 61.1%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 76.2%

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

   5. Taylor expanded in F around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{\sin B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \sin B\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. sin-lowering-sin.f64 99.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 99.8%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -310000000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -4.9 \cdot 10^{-100}:\\ \;\;\;\;\frac{F}{\frac{\sin B}{{\left(F \cdot F + 2\right)}^{-0.5}}}\\ \mathbf{elif}\;F \leq 1.25 \cdot 10^{-69}:\\ \;\;\;\;0 - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 430000:\\ \;\;\;\;\frac{F}{\frac{\sin B}{{\left(F \cdot F + 2\right)}^{-0.5}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -310000000000:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq -8.5 \cdot 10^{-106}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \frac{1}{\sqrt{F \cdot F + \left(2 + x \cdot 2\right)}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 18000:\\ \;\;\;\;\sqrt{\frac{1}{F \cdot F + 2}} \cdot \frac{F}{B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -310000000000.0)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F -8.5e-106)
       (-
        (* (/ F (sin B)) (/ 1.0 (sqrt (+ (* F F) (+ 2.0 (* x 2.0))))))
        (/ x B))
       (if (<= F 18000.0)
         (- (* (sqrt (/ 1.0 (+ (* F F) 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 <= -310000000000.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= -8.5e-106) {
		tmp = ((F / sin(B)) * (1.0 / sqrt(((F * F) + (2.0 + (x * 2.0)))))) - (x / B);
	} else if (F <= 18000.0) {
		tmp = (sqrt((1.0 / ((F * F) + 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 <= (-310000000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= (-8.5d-106)) then
        tmp = ((f / sin(b)) * (1.0d0 / sqrt(((f * f) + (2.0d0 + (x * 2.0d0)))))) - (x / b)
    else if (f <= 18000.0d0) then
        tmp = (sqrt((1.0d0 / ((f * f) + 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 <= -310000000000.0) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= -8.5e-106) {
		tmp = ((F / Math.sin(B)) * (1.0 / Math.sqrt(((F * F) + (2.0 + (x * 2.0)))))) - (x / B);
	} else if (F <= 18000.0) {
		tmp = (Math.sqrt((1.0 / ((F * F) + 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 <= -310000000000.0:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= -8.5e-106:
		tmp = ((F / math.sin(B)) * (1.0 / math.sqrt(((F * F) + (2.0 + (x * 2.0)))))) - (x / B)
	elif F <= 18000.0:
		tmp = (math.sqrt((1.0 / ((F * F) + 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 <= -310000000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= -8.5e-106)
		tmp = Float64(Float64(Float64(F / sin(B)) * Float64(1.0 / sqrt(Float64(Float64(F * F) + Float64(2.0 + Float64(x * 2.0)))))) - Float64(x / B));
	elseif (F <= 18000.0)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(Float64(F * F) + 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 <= -310000000000.0)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= -8.5e-106)
		tmp = ((F / sin(B)) * (1.0 / sqrt(((F * F) + (2.0 + (x * 2.0)))))) - (x / B);
	elseif (F <= 18000.0)
		tmp = (sqrt((1.0 / ((F * F) + 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, -310000000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -8.5e-106], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(N[(F * F), $MachinePrecision] + N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 18000.0], N[(N[(N[Sqrt[N[(1.0 / N[(N[(F * F), $MachinePrecision] + 2.0), $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 4 regimes

2. if F < -3.1e11

   1. Initial program 55.1%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 75.2%

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

   5. Taylor expanded in F around -inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{-1}{\sin B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \sin B\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. sin-lowering-sin.f64 99.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 99.8%

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

   if -3.1e11 < F < -8.4999999999999998e-106

   1. Initial program 99.2%

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

   3. Taylor expanded in B around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\color{blue}{\left(\frac{x}{B}\right)}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{sin.f64}\left(B\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 92.9%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{F}, \mathsf{sin.f64}\left(B\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]

   5. Simplified 92.9%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{sin.f64}\left(B\right)\right), \left({\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{\left(\mathsf{neg}\left(\frac{1}{\color{blue}{2}}\right)\right)}\right)\right)\right) \]

      2. associate-+r+ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{sin.f64}\left(B\right)\right), \left({\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)}\right)\right)\right) \]

      3. pow-flip N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{sin.f64}\left(B\right)\right), \left(\frac{1}{\color{blue}{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\left(\frac{1}{2}\right)}}}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{sin.f64}\left(B\right)\right), \left(\frac{1}{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{\frac{1}{2}}}\right)\right)\right) \]

      5. pow1/2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{sin.f64}\left(B\right)\right), \left(\frac{1}{\sqrt{F \cdot F + \left(2 + x \cdot 2\right)}}\right)\right)\right) \]

      6. associate-+r+ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{sin.f64}\left(B\right)\right), \left(\frac{1}{\sqrt{\left(F \cdot F + 2\right) + x \cdot 2}}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{sin.f64}\left(B\right)\right), \left(\frac{1}{\sqrt{\left(2 + F \cdot F\right) + x \cdot 2}}\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{sin.f64}\left(B\right)\right), \left(\frac{1}{\sqrt{\left(2 + F \cdot F\right) + 2 \cdot x}}\right)\right)\right) \]

      9. associate-+r+ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{sin.f64}\left(B\right)\right), \left(\frac{1}{\sqrt{2 + \left(F \cdot F + 2 \cdot x\right)}}\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\sqrt{2 + \left(F \cdot F + 2 \cdot x\right)}\right)}\right)\right)\right) \]

      11. associate-+r+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(1, \left(\sqrt{\left(2 + F \cdot F\right) + 2 \cdot x}\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(1, \left(\sqrt{\left(F \cdot F + 2\right) + 2 \cdot x}\right)\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(1, \left(\sqrt{\left(F \cdot F + 2\right) + x \cdot 2}\right)\right)\right)\right) \]

      14. associate-+r+ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(1, \left(\sqrt{F \cdot F + \left(2 + x \cdot 2\right)}\right)\right)\right)\right) \]

      15. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(F \cdot F + \left(2 + x \cdot 2\right)\right)\right)\right)\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(F \cdot F\right), \left(2 + x \cdot 2\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \left(2 + x \cdot 2\right)\right)\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \left(2 + 2 \cdot x\right)\right)\right)\right)\right)\right) \]

      19. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{+.f64}\left(2, \left(2 \cdot x\right)\right)\right)\right)\right)\right)\right) \]

      20. *-lowering-*.f64 93.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 93.0%

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

   if -8.4999999999999998e-106 < F < 18000

   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 N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + {F}^{2}}}\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. *-lowering-*.f64 99.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 99.8%

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

   8. Taylor expanded in B around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{F}{B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{F}{B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \left(\frac{F}{B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \left(\frac{F}{B}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \left(\frac{F}{B}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \left(\frac{F}{B}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \left(\frac{F}{B}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \left(\frac{F}{B}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. /-lowering-/.f64 94.0%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{/.f64}\left(F, B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   10. Simplified 94.0%

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

   if 18000 < F

   1. Initial program 62.1%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 76.8%

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

   5. Taylor expanded in F around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{\sin B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \sin B\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. sin-lowering-sin.f64 99.0%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 99.0%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -310000000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -8.5 \cdot 10^{-106}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \frac{1}{\sqrt{F \cdot F + \left(2 + x \cdot 2\right)}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 18000:\\ \;\;\;\;\sqrt{\frac{1}{F \cdot F + 2}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1000000000000:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq -4.6 \cdot 10^{-105}:\\ \;\;\;\;\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 20000:\\ \;\;\;\;\sqrt{\frac{1}{F \cdot F + 2}} \cdot \frac{F}{B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1000000000000.0)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F -4.6e-105)
       (- (/ (* F (pow (+ (* F F) (+ 2.0 (* x 2.0))) -0.5)) (sin B)) (/ x B))
       (if (<= F 20000.0)
         (- (* (sqrt (/ 1.0 (+ (* F F) 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 <= -1000000000000.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= -4.6e-105) {
		tmp = ((F * pow(((F * F) + (2.0 + (x * 2.0))), -0.5)) / sin(B)) - (x / B);
	} else if (F <= 20000.0) {
		tmp = (sqrt((1.0 / ((F * F) + 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 <= (-1000000000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= (-4.6d-105)) then
        tmp = ((f * (((f * f) + (2.0d0 + (x * 2.0d0))) ** (-0.5d0))) / sin(b)) - (x / b)
    else if (f <= 20000.0d0) then
        tmp = (sqrt((1.0d0 / ((f * f) + 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 <= -1000000000000.0) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= -4.6e-105) {
		tmp = ((F * Math.pow(((F * F) + (2.0 + (x * 2.0))), -0.5)) / Math.sin(B)) - (x / B);
	} else if (F <= 20000.0) {
		tmp = (Math.sqrt((1.0 / ((F * F) + 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 <= -1000000000000.0:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= -4.6e-105:
		tmp = ((F * math.pow(((F * F) + (2.0 + (x * 2.0))), -0.5)) / math.sin(B)) - (x / B)
	elif F <= 20000.0:
		tmp = (math.sqrt((1.0 / ((F * F) + 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 <= -1000000000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= -4.6e-105)
		tmp = Float64(Float64(Float64(F * (Float64(Float64(F * F) + Float64(2.0 + Float64(x * 2.0))) ^ -0.5)) / sin(B)) - Float64(x / B));
	elseif (F <= 20000.0)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(Float64(F * F) + 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 <= -1000000000000.0)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= -4.6e-105)
		tmp = ((F * (((F * F) + (2.0 + (x * 2.0))) ^ -0.5)) / sin(B)) - (x / B);
	elseif (F <= 20000.0)
		tmp = (sqrt((1.0 / ((F * F) + 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, -1000000000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -4.6e-105], N[(N[(N[(F * N[Power[N[(N[(F * F), $MachinePrecision] + N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 20000.0], N[(N[(N[Sqrt[N[(1.0 / N[(N[(F * F), $MachinePrecision] + 2.0), $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 4 regimes

2. if F < -1e12

   1. Initial program 55.1%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 75.2%

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

   5. Taylor expanded in F around -inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{-1}{\sin B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \sin B\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. sin-lowering-sin.f64 99.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 99.8%

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

   if -1e12 < F < -4.6000000000000002e-105

   1. Initial program 99.2%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 99.5%

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

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, 2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{sin.f64}\left(B\right)\right), \color{blue}{\left(\frac{x}{B}\right)}\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 92.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, 2\right)\right)\right), \frac{-1}{2}\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \color{blue}{B}\right)\right) \]

   7. Simplified 92.9%

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

   if -4.6000000000000002e-105 < F < 2e4

   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 N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + {F}^{2}}}\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. *-lowering-*.f64 99.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 99.8%

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

   8. Taylor expanded in B around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{F}{B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{F}{B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \left(\frac{F}{B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \left(\frac{F}{B}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \left(\frac{F}{B}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \left(\frac{F}{B}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \left(\frac{F}{B}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \left(\frac{F}{B}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. /-lowering-/.f64 94.0%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{/.f64}\left(F, B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   10. Simplified 94.0%

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

   if 2e4 < F

   1. Initial program 62.1%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 76.8%

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

   5. Taylor expanded in F around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{\sin B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \sin B\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. sin-lowering-sin.f64 99.0%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 99.0%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1000000000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -4.6 \cdot 10^{-105}:\\ \;\;\;\;\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 20000:\\ \;\;\;\;\sqrt{\frac{1}{F \cdot F + 2}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3.8 \cdot 10^{+31}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 17000:\\ \;\;\;\;\sqrt{\frac{1}{F \cdot F + 2}} \cdot \frac{F}{B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -3.8e+31)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 17000.0)
       (- (* (sqrt (/ 1.0 (+ (* F F) 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 <= -3.8e+31) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 17000.0) {
		tmp = (sqrt((1.0 / ((F * F) + 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 <= (-3.8d+31)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 17000.0d0) then
        tmp = (sqrt((1.0d0 / ((f * f) + 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 <= -3.8e+31) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 17000.0) {
		tmp = (Math.sqrt((1.0 / ((F * F) + 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 <= -3.8e+31:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 17000.0:
		tmp = (math.sqrt((1.0 / ((F * F) + 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 <= -3.8e+31)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 17000.0)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(Float64(F * F) + 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 <= -3.8e+31)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 17000.0)
		tmp = (sqrt((1.0 / ((F * F) + 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, -3.8e+31], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 17000.0], N[(N[(N[Sqrt[N[(1.0 / N[(N[(F * F), $MachinePrecision] + 2.0), $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 < -3.8000000000000001e31

   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. Step-by-step derivation

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 74.8%

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

   5. Taylor expanded in F around -inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{-1}{\sin B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \sin B\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. sin-lowering-sin.f64 99.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 99.8%

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

   if -3.8000000000000001e31 < F < 17000

   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 N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + {F}^{2}}}\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. *-lowering-*.f64 99.7%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 99.7%

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

   8. Taylor expanded in B around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{F}{B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{F}{B}\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \left(\frac{F}{B}\right)\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \left(\frac{F}{B}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \left(\frac{F}{B}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \left(\frac{F}{B}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \left(\frac{F}{B}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \left(\frac{F}{B}\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      8. /-lowering-/.f64 88.5%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{/.f64}\left(F, B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   10. Simplified 88.5%

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

   if 17000 < F

   1. Initial program 62.1%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 76.8%

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

   5. Taylor expanded in F around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{\sin B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \sin B\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. sin-lowering-sin.f64 99.0%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 99.0%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.8 \cdot 10^{+31}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 17000:\\ \;\;\;\;\sqrt{\frac{1}{F \cdot F + 2}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 85.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.035:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq -4.45 \cdot 10^{-100}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 9.6 \cdot 10^{-54}:\\ \;\;\;\;0 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -0.035)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F -4.45e-100)
       (/ (* F (sqrt 0.5)) (sin B))
       (if (<= F 9.6e-54) (- 0.0 t_0) (- (/ 1.0 (sin B)) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -0.035) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= -4.45e-100) {
		tmp = (F * sqrt(0.5)) / sin(B);
	} else if (F <= 9.6e-54) {
		tmp = 0.0 - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-0.035d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= (-4.45d-100)) then
        tmp = (f * sqrt(0.5d0)) / sin(b)
    else if (f <= 9.6d-54) then
        tmp = 0.0d0 - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -0.035) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= -4.45e-100) {
		tmp = (F * Math.sqrt(0.5)) / Math.sin(B);
	} else if (F <= 9.6e-54) {
		tmp = 0.0 - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -0.035:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= -4.45e-100:
		tmp = (F * math.sqrt(0.5)) / math.sin(B)
	elif F <= 9.6e-54:
		tmp = 0.0 - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -0.035)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= -4.45e-100)
		tmp = Float64(Float64(F * sqrt(0.5)) / sin(B));
	elseif (F <= 9.6e-54)
		tmp = Float64(0.0 - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -0.035)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= -4.45e-100)
		tmp = (F * sqrt(0.5)) / sin(B);
	elseif (F <= 9.6e-54)
		tmp = 0.0 - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.035], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -4.45e-100], N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 9.6e-54], N[(0.0 - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.035000000000000003

   1. Initial program 57.1%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 76.3%

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

   5. Taylor expanded in F around -inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{-1}{\sin B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \sin B\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. sin-lowering-sin.f64 98.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 98.9%

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

   if -0.035000000000000003 < F < -4.4500000000000002e-100

   1. Initial program 99.1%

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

   3. Taylor expanded in x around 0

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \color{blue}{\left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \color{blue}{\sin B}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin \color{blue}{B}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right) \]

      10. sin-lowering-sin.f64 73.5%

          \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right) \]

   5. Simplified 73.5%

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

   6. Taylor expanded in F around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2}}\right), \color{blue}{\sin B}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2}}\right)\right), \sin \color{blue}{B}\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right), \sin B\right) \]

      4. sin-lowering-sin.f64 73.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right), \mathsf{sin.f64}\left(B\right)\right) \]

   8. Simplified 73.4%

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

   if -4.4500000000000002e-100 < F < 9.60000000000000053e-54

   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 N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + {F}^{2}}}\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. *-lowering-*.f64 99.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 99.8%

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

   8. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right) \]

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \cos B\right), \color{blue}{\left(\mathsf{neg}\left(\sin B\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \cos B\right), \left(\mathsf{neg}\left(\color{blue}{\sin B}\right)\right)\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \left(\mathsf{neg}\left(\sin B\right)\right)\right) \]

      6. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \mathsf{neg.f64}\left(\sin B\right)\right) \]

      7. sin-lowering-sin.f64 86.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \mathsf{neg.f64}\left(\mathsf{sin.f64}\left(B\right)\right)\right) \]

   10. Simplified 86.3%

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

       1. distribute-frac-neg2 N/A

          \[\leadsto \mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right) \]

       2. clear-num N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\sin B}{x \cdot \cos B}}\right) \]

       3. associate-/l/ N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\frac{\sin B}{\cos B}}{x}}\right) \]

       4. tan-quot N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\tan B}{x}}\right) \]

       5. clear-num N/A

          \[\leadsto \mathsf{neg}\left(\frac{x}{\tan B}\right) \]

       6. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{x}{\tan B}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, \tan B\right)\right) \]

       8. tan-lowering-tan.f64 86.5%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   12. Applied egg-rr 86.5%

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

   if 9.60000000000000053e-54 < F

   1. Initial program 65.2%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 78.7%

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

   5. Taylor expanded in F around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{\sin B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \sin B\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. sin-lowering-sin.f64 93.1%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 93.1%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.035:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -4.45 \cdot 10^{-100}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 9.6 \cdot 10^{-54}:\\ \;\;\;\;0 - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 78.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.042:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq -3 \cdot 10^{-100}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 1.46 \cdot 10^{-53}:\\ \;\;\;\;0 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -0.042)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F -3e-100)
       (/ (* F (sqrt 0.5)) (sin B))
       (if (<= F 1.46e-53) (- 0.0 t_0) (- (/ 1.0 (sin B)) (/ x B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -0.042) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= -3e-100) {
		tmp = (F * sqrt(0.5)) / sin(B);
	} else if (F <= 1.46e-53) {
		tmp = 0.0 - t_0;
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-0.042d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= (-3d-100)) then
        tmp = (f * sqrt(0.5d0)) / sin(b)
    else if (f <= 1.46d-53) then
        tmp = 0.0d0 - t_0
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -0.042) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= -3e-100) {
		tmp = (F * Math.sqrt(0.5)) / Math.sin(B);
	} else if (F <= 1.46e-53) {
		tmp = 0.0 - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -0.042:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= -3e-100:
		tmp = (F * math.sqrt(0.5)) / math.sin(B)
	elif F <= 1.46e-53:
		tmp = 0.0 - t_0
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -0.042)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= -3e-100)
		tmp = Float64(Float64(F * sqrt(0.5)) / sin(B));
	elseif (F <= 1.46e-53)
		tmp = Float64(0.0 - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -0.042)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= -3e-100)
		tmp = (F * sqrt(0.5)) / sin(B);
	elseif (F <= 1.46e-53)
		tmp = 0.0 - t_0;
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.042], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -3e-100], N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.46e-53], N[(0.0 - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.0420000000000000026

   1. Initial program 57.1%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 76.3%

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

   5. Taylor expanded in F around -inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{-1}{\sin B}\right)}, \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \sin B\right), \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. sin-lowering-sin.f64 98.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 98.9%

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

   if -0.0420000000000000026 < F < -3.0000000000000001e-100

   1. Initial program 99.1%

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

   3. Taylor expanded in x around 0

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \color{blue}{\left(\frac{\sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \color{blue}{\sin B}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \sin \color{blue}{B}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \sin B\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \sin B\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \sin B\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \sin B\right)\right) \]

      10. sin-lowering-sin.f64 73.5%

          \[\leadsto \mathsf{*.f64}\left(F, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right)\right) \]

   5. Simplified 73.5%

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

   6. Taylor expanded in F around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2}}\right), \color{blue}{\sin B}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2}}\right)\right), \sin \color{blue}{B}\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right), \sin B\right) \]

      4. sin-lowering-sin.f64 73.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right), \mathsf{sin.f64}\left(B\right)\right) \]

   8. Simplified 73.4%

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

   if -3.0000000000000001e-100 < F < 1.45999999999999989e-53

   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 N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + {F}^{2}}}\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. *-lowering-*.f64 99.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 99.8%

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

   8. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right) \]

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \cos B\right), \color{blue}{\left(\mathsf{neg}\left(\sin B\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \cos B\right), \left(\mathsf{neg}\left(\color{blue}{\sin B}\right)\right)\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \left(\mathsf{neg}\left(\sin B\right)\right)\right) \]

      6. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \mathsf{neg.f64}\left(\sin B\right)\right) \]

      7. sin-lowering-sin.f64 86.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \mathsf{neg.f64}\left(\mathsf{sin.f64}\left(B\right)\right)\right) \]

   10. Simplified 86.3%

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

       1. distribute-frac-neg2 N/A

          \[\leadsto \mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right) \]

       2. clear-num N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\sin B}{x \cdot \cos B}}\right) \]

       3. associate-/l/ N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\frac{\sin B}{\cos B}}{x}}\right) \]

       4. tan-quot N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\tan B}{x}}\right) \]

       5. clear-num N/A

          \[\leadsto \mathsf{neg}\left(\frac{x}{\tan B}\right) \]

       6. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{x}{\tan B}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, \tan B\right)\right) \]

       8. tan-lowering-tan.f64 86.5%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   12. Applied egg-rr 86.5%

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

   if 1.45999999999999989e-53 < F

   1. Initial program 65.2%

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

   3. Taylor expanded in B around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\color{blue}{\left(\frac{x}{B}\right)}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{sin.f64}\left(B\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 44.2%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{F}, \mathsf{sin.f64}\left(B\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]

   5. Simplified 44.2%

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

   6. Taylor expanded in F around inf

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{1}{\sin B}\right), \color{blue}{\left(\frac{x}{B}\right)}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \sin B\right), \left(\frac{\color{blue}{x}}{B}\right)\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \left(\frac{x}{B}\right)\right) \]

      4. /-lowering-/.f64 71.0%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \color{blue}{B}\right)\right) \]

   8. Simplified 71.0%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.042:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -3 \cdot 10^{-100}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 1.46 \cdot 10^{-53}:\\ \;\;\;\;0 - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 56.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.0038:\\ \;\;\;\;\frac{\left(\sqrt{\frac{1}{2 + \left(F \cdot F + x \cdot 2\right)}} \cdot \left(F + 0.16666666666666666 \cdot \left(F \cdot \left(B \cdot B\right)\right)\right) + B \cdot \left(B \cdot \left(x \cdot 0.3333333333333333\right)\right)\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 0.0038)
   (/
    (-
     (+
      (*
       (sqrt (/ 1.0 (+ 2.0 (+ (* F F) (* x 2.0)))))
       (+ F (* 0.16666666666666666 (* F (* B B)))))
      (* B (* B (* x 0.3333333333333333))))
     x)
    B)
   (- 0.0 (/ x (tan B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 0.0038) {
		tmp = (((sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0))))) * (F + (0.16666666666666666 * (F * (B * B))))) + (B * (B * (x * 0.3333333333333333)))) - x) / B;
	} else {
		tmp = 0.0 - (x / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (b <= 0.0038d0) then
        tmp = (((sqrt((1.0d0 / (2.0d0 + ((f * f) + (x * 2.0d0))))) * (f + (0.16666666666666666d0 * (f * (b * b))))) + (b * (b * (x * 0.3333333333333333d0)))) - x) / b
    else
        tmp = 0.0d0 - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (B <= 0.0038) {
		tmp = (((Math.sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0))))) * (F + (0.16666666666666666 * (F * (B * B))))) + (B * (B * (x * 0.3333333333333333)))) - x) / B;
	} else {
		tmp = 0.0 - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if B <= 0.0038:
		tmp = (((math.sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0))))) * (F + (0.16666666666666666 * (F * (B * B))))) + (B * (B * (x * 0.3333333333333333)))) - x) / B
	else:
		tmp = 0.0 - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 0.0038)
		tmp = Float64(Float64(Float64(Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(Float64(F * F) + Float64(x * 2.0))))) * Float64(F + Float64(0.16666666666666666 * Float64(F * Float64(B * B))))) + Float64(B * Float64(B * Float64(x * 0.3333333333333333)))) - x) / B);
	else
		tmp = Float64(0.0 - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (B <= 0.0038)
		tmp = (((sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0))))) * (F + (0.16666666666666666 * (F * (B * B))))) + (B * (B * (x * 0.3333333333333333)))) - x) / B;
	else
		tmp = 0.0 - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 0.0038], N[(N[(N[(N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(N[(F * F), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F + N[(0.16666666666666666 * N[(F * N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(B * N[(B * N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(0.0 - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 0.00379999999999999999

   1. Initial program 76.3%

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

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \frac{1}{3} \cdot x\right)\right) - x}{B}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \frac{1}{3} \cdot x\right)\right) - x\right), \color{blue}{B}\right) \]

   5. Simplified 62.2%

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

   if 0.00379999999999999999 < B

   1. Initial program 80.5%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 80.6%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + {F}^{2}}}\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. *-lowering-*.f64 80.6%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 80.6%

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

   8. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right) \]

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \cos B\right), \color{blue}{\left(\mathsf{neg}\left(\sin B\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \cos B\right), \left(\mathsf{neg}\left(\color{blue}{\sin B}\right)\right)\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \left(\mathsf{neg}\left(\sin B\right)\right)\right) \]

      6. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \mathsf{neg.f64}\left(\sin B\right)\right) \]

      7. sin-lowering-sin.f64 60.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \mathsf{neg.f64}\left(\mathsf{sin.f64}\left(B\right)\right)\right) \]

   10. Simplified 60.2%

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

       1. distribute-frac-neg2 N/A

          \[\leadsto \mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right) \]

       2. clear-num N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\sin B}{x \cdot \cos B}}\right) \]

       3. associate-/l/ N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\frac{\sin B}{\cos B}}{x}}\right) \]

       4. tan-quot N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\tan B}{x}}\right) \]

       5. clear-num N/A

          \[\leadsto \mathsf{neg}\left(\frac{x}{\tan B}\right) \]

       6. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{x}{\tan B}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, \tan B\right)\right) \]

       8. tan-lowering-tan.f64 60.3%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   12. Applied egg-rr 60.3%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 0.0038:\\ \;\;\;\;\frac{\left(\sqrt{\frac{1}{2 + \left(F \cdot F + x \cdot 2\right)}} \cdot \left(F + 0.16666666666666666 \cdot \left(F \cdot \left(B \cdot B\right)\right)\right) + B \cdot \left(B \cdot \left(x \cdot 0.3333333333333333\right)\right)\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 55.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.2 \cdot 10^{+153}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}}\\ \mathbf{elif}\;F \leq -1.4 \cdot 10^{-45}:\\ \;\;\;\;\frac{\left(-1 + \frac{1 + \left(x \cdot 2\right) \cdot 0.5}{F \cdot F}\right) - x}{B}\\ \mathbf{elif}\;F \leq 2.55 \cdot 10^{+88}:\\ \;\;\;\;0 - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.2e+153)
   (/ -1.0 (/ (tan B) x))
   (if (<= F -1.4e-45)
     (/ (- (+ -1.0 (/ (+ 1.0 (* (* x 2.0) 0.5)) (* F F))) x) B)
     (if (<= F 2.55e+88) (- 0.0 (/ x (tan B))) (/ 1.0 (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.2e+153) {
		tmp = -1.0 / (tan(B) / x);
	} else if (F <= -1.4e-45) {
		tmp = ((-1.0 + ((1.0 + ((x * 2.0) * 0.5)) / (F * F))) - x) / B;
	} else if (F <= 2.55e+88) {
		tmp = 0.0 - (x / tan(B));
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-3.2d+153)) then
        tmp = (-1.0d0) / (tan(b) / x)
    else if (f <= (-1.4d-45)) then
        tmp = (((-1.0d0) + ((1.0d0 + ((x * 2.0d0) * 0.5d0)) / (f * f))) - x) / b
    else if (f <= 2.55d+88) then
        tmp = 0.0d0 - (x / tan(b))
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.2e+153) {
		tmp = -1.0 / (Math.tan(B) / x);
	} else if (F <= -1.4e-45) {
		tmp = ((-1.0 + ((1.0 + ((x * 2.0) * 0.5)) / (F * F))) - x) / B;
	} else if (F <= 2.55e+88) {
		tmp = 0.0 - (x / Math.tan(B));
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.2e+153:
		tmp = -1.0 / (math.tan(B) / x)
	elif F <= -1.4e-45:
		tmp = ((-1.0 + ((1.0 + ((x * 2.0) * 0.5)) / (F * F))) - x) / B
	elif F <= 2.55e+88:
		tmp = 0.0 - (x / math.tan(B))
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.2e+153)
		tmp = Float64(-1.0 / Float64(tan(B) / x));
	elseif (F <= -1.4e-45)
		tmp = Float64(Float64(Float64(-1.0 + Float64(Float64(1.0 + Float64(Float64(x * 2.0) * 0.5)) / Float64(F * F))) - x) / B);
	elseif (F <= 2.55e+88)
		tmp = Float64(0.0 - Float64(x / tan(B)));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -3.2e+153)
		tmp = -1.0 / (tan(B) / x);
	elseif (F <= -1.4e-45)
		tmp = ((-1.0 + ((1.0 + ((x * 2.0) * 0.5)) / (F * F))) - x) / B;
	elseif (F <= 2.55e+88)
		tmp = 0.0 - (x / tan(B));
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.2e+153], N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.4e-45], N[(N[(N[(-1.0 + N[(N[(1.0 + N[(N[(x * 2.0), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 2.55e+88], N[(0.0 - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.2000000000000001e153

   1. Initial program 42.6%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 60.4%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + {F}^{2}}}\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. *-lowering-*.f64 60.4%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 60.4%

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

   8. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right) \]

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \cos B\right), \color{blue}{\left(\mathsf{neg}\left(\sin B\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \cos B\right), \left(\mathsf{neg}\left(\color{blue}{\sin B}\right)\right)\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \left(\mathsf{neg}\left(\sin B\right)\right)\right) \]

      6. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \mathsf{neg.f64}\left(\sin B\right)\right) \]

      7. sin-lowering-sin.f64 60.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \mathsf{neg.f64}\left(\mathsf{sin.f64}\left(B\right)\right)\right) \]

   10. Simplified 60.4%

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

       1. distribute-frac-neg2 N/A

          \[\leadsto \mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right) \]

       2. clear-num N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\sin B}{x \cdot \cos B}}\right) \]

       3. associate-/l/ N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\frac{\sin B}{\cos B}}{x}}\right) \]

       4. tan-quot N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\tan B}{x}}\right) \]

       5. distribute-neg-frac N/A

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

       6. metadata-eval N/A

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

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(\frac{\tan B}{x}\right)}\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(\tan B, \color{blue}{x}\right)\right) \]

       9. tan-lowering-tan.f64 60.5%

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(B\right), x\right)\right) \]

   12. Applied egg-rr 60.5%

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

   if -3.2000000000000001e153 < F < -1.4000000000000001e-45

   1. Initial program 81.4%

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

   3. Taylor expanded in B around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x\right), \color{blue}{B}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right), x\right), B\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right), x\right), B\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right), x\right), B\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2} + 2 \cdot x\right)\right)\right)\right)\right), x\right), B\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left({F}^{2}\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left(F \cdot F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      11. *-lowering-*.f64 65.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   5. Simplified 65.0%

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

   6. Taylor expanded in F around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}} - 1\right)}, x\right), B\right) \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}} + \left(\mathsf{neg}\left(1\right)\right)\right), x\right), B\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}} + -1\right), x\right), B\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right), -1\right), x\right), B\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{2} \cdot \left(2 + 2 \cdot x\right)}{{F}^{2}}\right), -1\right), x\right), B\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(2 + 2 \cdot x\right)\right), \left({F}^{2}\right)\right), -1\right), x\right), B\right) \]

      6. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2} \cdot 2 + \frac{1}{2} \cdot \left(2 \cdot x\right)\right), \left({F}^{2}\right)\right), -1\right), x\right), B\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(1 + \frac{1}{2} \cdot \left(2 \cdot x\right)\right), \left({F}^{2}\right)\right), -1\right), x\right), B\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \left(2 \cdot x\right)\right)\right), \left({F}^{2}\right)\right), -1\right), x\right), B\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(2 \cdot x\right)\right)\right), \left({F}^{2}\right)\right), -1\right), x\right), B\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(x \cdot 2\right)\right)\right), \left({F}^{2}\right)\right), -1\right), x\right), B\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, 2\right)\right)\right), \left({F}^{2}\right)\right), -1\right), x\right), B\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, 2\right)\right)\right), \left(F \cdot F\right)\right), -1\right), x\right), B\right) \]

      13. *-lowering-*.f64 61.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, 2\right)\right)\right), \mathsf{*.f64}\left(F, F\right)\right), -1\right), x\right), B\right) \]

   8. Simplified 61.0%

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

   if -1.4000000000000001e-45 < F < 2.5499999999999999e88

   1. Initial program 98.7%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + {F}^{2}}}\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. *-lowering-*.f64 99.7%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 99.7%

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

   8. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right) \]

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \cos B\right), \color{blue}{\left(\mathsf{neg}\left(\sin B\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \cos B\right), \left(\mathsf{neg}\left(\color{blue}{\sin B}\right)\right)\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \left(\mathsf{neg}\left(\sin B\right)\right)\right) \]

      6. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \mathsf{neg.f64}\left(\sin B\right)\right) \]

      7. sin-lowering-sin.f64 72.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \mathsf{neg.f64}\left(\mathsf{sin.f64}\left(B\right)\right)\right) \]

   10. Simplified 72.8%

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

       1. distribute-frac-neg2 N/A

          \[\leadsto \mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right) \]

       2. clear-num N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\sin B}{x \cdot \cos B}}\right) \]

       3. associate-/l/ N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\frac{\sin B}{\cos B}}{x}}\right) \]

       4. tan-quot N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\tan B}{x}}\right) \]

       5. clear-num N/A

          \[\leadsto \mathsf{neg}\left(\frac{x}{\tan B}\right) \]

       6. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{x}{\tan B}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, \tan B\right)\right) \]

       8. tan-lowering-tan.f64 73.0%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   12. Applied egg-rr 73.0%

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

   if 2.5499999999999999e88 < F

   1. Initial program 46.4%

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

   3. Taylor expanded in F around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \color{blue}{\left(\frac{1}{\sin B}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\sin B}\right)\right) \]

      2. sin-lowering-sin.f64 99.7%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right)\right) \]

   5. Simplified 99.7%

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

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\sin B}\right) \]

      2. sin-lowering-sin.f64 56.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right) \]

   8. Simplified 56.0%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.2 \cdot 10^{+153}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}}\\ \mathbf{elif}\;F \leq -1.4 \cdot 10^{-45}:\\ \;\;\;\;\frac{\left(-1 + \frac{1 + \left(x \cdot 2\right) \cdot 0.5}{F \cdot F}\right) - x}{B}\\ \mathbf{elif}\;F \leq 2.55 \cdot 10^{+88}:\\ \;\;\;\;0 - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 55.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0 - \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1 \cdot 10^{+151}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq -1.4 \cdot 10^{-45}:\\ \;\;\;\;\frac{\left(-1 + \frac{1 + \left(x \cdot 2\right) \cdot 0.5}{F \cdot F}\right) - x}{B}\\ \mathbf{elif}\;F \leq 3.4 \cdot 10^{+88}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- 0.0 (/ x (tan B)))))
   (if (<= F -1e+151)
     t_0
     (if (<= F -1.4e-45)
       (/ (- (+ -1.0 (/ (+ 1.0 (* (* x 2.0) 0.5)) (* F F))) x) B)
       (if (<= F 3.4e+88) t_0 (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = 0.0 - (x / tan(B));
	double tmp;
	if (F <= -1e+151) {
		tmp = t_0;
	} else if (F <= -1.4e-45) {
		tmp = ((-1.0 + ((1.0 + ((x * 2.0) * 0.5)) / (F * F))) - x) / B;
	} else if (F <= 3.4e+88) {
		tmp = t_0;
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.0d0 - (x / tan(b))
    if (f <= (-1d+151)) then
        tmp = t_0
    else if (f <= (-1.4d-45)) then
        tmp = (((-1.0d0) + ((1.0d0 + ((x * 2.0d0) * 0.5d0)) / (f * f))) - x) / b
    else if (f <= 3.4d+88) then
        tmp = t_0
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = 0.0 - (x / Math.tan(B));
	double tmp;
	if (F <= -1e+151) {
		tmp = t_0;
	} else if (F <= -1.4e-45) {
		tmp = ((-1.0 + ((1.0 + ((x * 2.0) * 0.5)) / (F * F))) - x) / B;
	} else if (F <= 3.4e+88) {
		tmp = t_0;
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = 0.0 - (x / math.tan(B))
	tmp = 0
	if F <= -1e+151:
		tmp = t_0
	elif F <= -1.4e-45:
		tmp = ((-1.0 + ((1.0 + ((x * 2.0) * 0.5)) / (F * F))) - x) / B
	elif F <= 3.4e+88:
		tmp = t_0
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(0.0 - Float64(x / tan(B)))
	tmp = 0.0
	if (F <= -1e+151)
		tmp = t_0;
	elseif (F <= -1.4e-45)
		tmp = Float64(Float64(Float64(-1.0 + Float64(Float64(1.0 + Float64(Float64(x * 2.0) * 0.5)) / Float64(F * F))) - x) / B);
	elseif (F <= 3.4e+88)
		tmp = t_0;
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = 0.0 - (x / tan(B));
	tmp = 0.0;
	if (F <= -1e+151)
		tmp = t_0;
	elseif (F <= -1.4e-45)
		tmp = ((-1.0 + ((1.0 + ((x * 2.0) * 0.5)) / (F * F))) - x) / B;
	elseif (F <= 3.4e+88)
		tmp = t_0;
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(0.0 - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1e+151], t$95$0, If[LessEqual[F, -1.4e-45], N[(N[(N[(-1.0 + N[(N[(1.0 + N[(N[(x * 2.0), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 3.4e+88], t$95$0, N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.00000000000000002e151 or -1.4000000000000001e-45 < F < 3.40000000000000004e88

   1. Initial program 85.8%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 90.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + {F}^{2}}}\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. *-lowering-*.f64 90.7%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 90.7%

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

   8. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right) \]

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \cos B\right), \color{blue}{\left(\mathsf{neg}\left(\sin B\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \cos B\right), \left(\mathsf{neg}\left(\color{blue}{\sin B}\right)\right)\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \left(\mathsf{neg}\left(\sin B\right)\right)\right) \]

      6. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \mathsf{neg.f64}\left(\sin B\right)\right) \]

      7. sin-lowering-sin.f64 70.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \mathsf{neg.f64}\left(\mathsf{sin.f64}\left(B\right)\right)\right) \]

   10. Simplified 70.0%

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

       1. distribute-frac-neg2 N/A

          \[\leadsto \mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right) \]

       2. clear-num N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\sin B}{x \cdot \cos B}}\right) \]

       3. associate-/l/ N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\frac{\sin B}{\cos B}}{x}}\right) \]

       4. tan-quot N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\tan B}{x}}\right) \]

       5. clear-num N/A

          \[\leadsto \mathsf{neg}\left(\frac{x}{\tan B}\right) \]

       6. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{x}{\tan B}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, \tan B\right)\right) \]

       8. tan-lowering-tan.f64 70.1%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   12. Applied egg-rr 70.1%

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

   if -1.00000000000000002e151 < F < -1.4000000000000001e-45

   1. Initial program 81.4%

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

   3. Taylor expanded in B around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x\right), \color{blue}{B}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right), x\right), B\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right), x\right), B\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right), x\right), B\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2} + 2 \cdot x\right)\right)\right)\right)\right), x\right), B\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left({F}^{2}\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left(F \cdot F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      11. *-lowering-*.f64 65.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   5. Simplified 65.0%

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

   6. Taylor expanded in F around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}} - 1\right)}, x\right), B\right) \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}} + \left(\mathsf{neg}\left(1\right)\right)\right), x\right), B\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}} + -1\right), x\right), B\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right), -1\right), x\right), B\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{2} \cdot \left(2 + 2 \cdot x\right)}{{F}^{2}}\right), -1\right), x\right), B\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(2 + 2 \cdot x\right)\right), \left({F}^{2}\right)\right), -1\right), x\right), B\right) \]

      6. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2} \cdot 2 + \frac{1}{2} \cdot \left(2 \cdot x\right)\right), \left({F}^{2}\right)\right), -1\right), x\right), B\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(1 + \frac{1}{2} \cdot \left(2 \cdot x\right)\right), \left({F}^{2}\right)\right), -1\right), x\right), B\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \left(2 \cdot x\right)\right)\right), \left({F}^{2}\right)\right), -1\right), x\right), B\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(2 \cdot x\right)\right)\right), \left({F}^{2}\right)\right), -1\right), x\right), B\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(x \cdot 2\right)\right)\right), \left({F}^{2}\right)\right), -1\right), x\right), B\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, 2\right)\right)\right), \left({F}^{2}\right)\right), -1\right), x\right), B\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, 2\right)\right)\right), \left(F \cdot F\right)\right), -1\right), x\right), B\right) \]

      13. *-lowering-*.f64 61.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, 2\right)\right)\right), \mathsf{*.f64}\left(F, F\right)\right), -1\right), x\right), B\right) \]

   8. Simplified 61.0%

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

   if 3.40000000000000004e88 < F

   1. Initial program 46.4%

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

   3. Taylor expanded in F around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \color{blue}{\left(\frac{1}{\sin B}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\sin B}\right)\right) \]

      2. sin-lowering-sin.f64 99.7%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right)\right) \]

   5. Simplified 99.7%

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

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\sin B}\right) \]

      2. sin-lowering-sin.f64 56.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right) \]

   8. Simplified 56.0%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{+151}:\\ \;\;\;\;0 - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.4 \cdot 10^{-45}:\\ \;\;\;\;\frac{\left(-1 + \frac{1 + \left(x \cdot 2\right) \cdot 0.5}{F \cdot F}\right) - x}{B}\\ \mathbf{elif}\;F \leq 3.4 \cdot 10^{+88}:\\ \;\;\;\;0 - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 57.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 1.15 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{F \cdot F + \left(2 + x \cdot 2\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 1.15e-5)
   (/ (- (/ F (sqrt (+ (* F F) (+ 2.0 (* x 2.0))))) x) B)
   (- 0.0 (/ x (tan B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 1.15e-5) {
		tmp = ((F / sqrt(((F * F) + (2.0 + (x * 2.0))))) - x) / B;
	} else {
		tmp = 0.0 - (x / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (b <= 1.15d-5) then
        tmp = ((f / sqrt(((f * f) + (2.0d0 + (x * 2.0d0))))) - x) / b
    else
        tmp = 0.0d0 - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (B <= 1.15e-5) {
		tmp = ((F / Math.sqrt(((F * F) + (2.0 + (x * 2.0))))) - x) / B;
	} else {
		tmp = 0.0 - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if B <= 1.15e-5:
		tmp = ((F / math.sqrt(((F * F) + (2.0 + (x * 2.0))))) - x) / B
	else:
		tmp = 0.0 - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 1.15e-5)
		tmp = Float64(Float64(Float64(F / sqrt(Float64(Float64(F * F) + Float64(2.0 + Float64(x * 2.0))))) - x) / B);
	else
		tmp = Float64(0.0 - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (B <= 1.15e-5)
		tmp = ((F / sqrt(((F * F) + (2.0 + (x * 2.0))))) - x) / B;
	else
		tmp = 0.0 - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 1.15e-5], N[(N[(N[(F / N[Sqrt[N[(N[(F * F), $MachinePrecision] + N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(0.0 - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.15e-5

   1. Initial program 76.3%

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

   3. Taylor expanded in B around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x\right), \color{blue}{B}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right), x\right), B\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right), x\right), B\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right), x\right), B\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2} + 2 \cdot x\right)\right)\right)\right)\right), x\right), B\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left({F}^{2}\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left(F \cdot F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      11. *-lowering-*.f64 62.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   5. Simplified 62.6%

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

      1. sqrt-div N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(F \cdot \frac{\sqrt{1}}{\sqrt{2 + \left(F \cdot F + 2 \cdot x\right)}}\right), x\right), B\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(F \cdot \frac{1}{\sqrt{2 + \left(F \cdot F + 2 \cdot x\right)}}\right), x\right), B\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{F}{\sqrt{2 + \left(F \cdot F + 2 \cdot x\right)}}\right), x\right), B\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \left(\sqrt{2 + \left(F \cdot F + 2 \cdot x\right)}\right)\right), x\right), B\right) \]

      5. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \left(\sqrt{\left(2 + F \cdot F\right) + 2 \cdot x}\right)\right), x\right), B\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \left(\sqrt{\left(F \cdot F + 2\right) + 2 \cdot x}\right)\right), x\right), B\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \left(\sqrt{\left(F \cdot F + 2\right) + x \cdot 2}\right)\right), x\right), B\right) \]

      8. associate-+r+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \left(\sqrt{F \cdot F + \left(2 + x \cdot 2\right)}\right)\right), x\right), B\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{sqrt.f64}\left(\left(F \cdot F + \left(2 + x \cdot 2\right)\right)\right)\right), x\right), B\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(F \cdot F\right), \left(2 + x \cdot 2\right)\right)\right)\right), x\right), B\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \left(2 + x \cdot 2\right)\right)\right)\right), x\right), B\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \left(2 + 2 \cdot x\right)\right)\right)\right), x\right), B\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{+.f64}\left(2, \left(2 \cdot x\right)\right)\right)\right)\right), x\right), B\right) \]

      14. *-lowering-*.f64 62.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right), x\right), B\right) \]

   7. Applied egg-rr 62.6%

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

   if 1.15e-5 < B

   1. Initial program 80.5%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 80.6%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + {F}^{2}}}\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. *-lowering-*.f64 80.6%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 80.6%

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

   8. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right) \]

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \cos B\right), \color{blue}{\left(\mathsf{neg}\left(\sin B\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \cos B\right), \left(\mathsf{neg}\left(\color{blue}{\sin B}\right)\right)\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \left(\mathsf{neg}\left(\sin B\right)\right)\right) \]

      6. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \mathsf{neg.f64}\left(\sin B\right)\right) \]

      7. sin-lowering-sin.f64 60.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \mathsf{neg.f64}\left(\mathsf{sin.f64}\left(B\right)\right)\right) \]

   10. Simplified 60.2%

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

       1. distribute-frac-neg2 N/A

          \[\leadsto \mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right) \]

       2. clear-num N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\sin B}{x \cdot \cos B}}\right) \]

       3. associate-/l/ N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\frac{\sin B}{\cos B}}{x}}\right) \]

       4. tan-quot N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\tan B}{x}}\right) \]

       5. clear-num N/A

          \[\leadsto \mathsf{neg}\left(\frac{x}{\tan B}\right) \]

       6. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{x}{\tan B}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, \tan B\right)\right) \]

       8. tan-lowering-tan.f64 60.3%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   12. Applied egg-rr 60.3%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.15 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{F \cdot F + \left(2 + x \cdot 2\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 56.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 3.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{F \cdot F + 2}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 3.6e-6)
   (/ (- (* F (sqrt (/ 1.0 (+ (* F F) 2.0)))) x) B)
   (- 0.0 (/ x (tan B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 3.6e-6) {
		tmp = ((F * sqrt((1.0 / ((F * F) + 2.0)))) - x) / B;
	} else {
		tmp = 0.0 - (x / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (b <= 3.6d-6) then
        tmp = ((f * sqrt((1.0d0 / ((f * f) + 2.0d0)))) - x) / b
    else
        tmp = 0.0d0 - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (B <= 3.6e-6) {
		tmp = ((F * Math.sqrt((1.0 / ((F * F) + 2.0)))) - x) / B;
	} else {
		tmp = 0.0 - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if B <= 3.6e-6:
		tmp = ((F * math.sqrt((1.0 / ((F * F) + 2.0)))) - x) / B
	else:
		tmp = 0.0 - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 3.6e-6)
		tmp = Float64(Float64(Float64(F * sqrt(Float64(1.0 / Float64(Float64(F * F) + 2.0)))) - x) / B);
	else
		tmp = Float64(0.0 - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (B <= 3.6e-6)
		tmp = ((F * sqrt((1.0 / ((F * F) + 2.0)))) - x) / B;
	else
		tmp = 0.0 - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 3.6e-6], N[(N[(N[(F * N[Sqrt[N[(1.0 / N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(0.0 - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 3.59999999999999984e-6

   1. Initial program 76.3%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 89.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + {F}^{2}}}\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. *-lowering-*.f64 89.1%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 89.1%

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

   8. Taylor expanded in B around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + {F}^{2}}} - x\right), \color{blue}{B}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right), x\right), B\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + {F}^{2}}}\right)\right), x\right), B\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right)\right), x\right), B\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right)\right), x\right), B\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right)\right), x\right), B\right) \]

      8. *-lowering-*.f64 62.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right)\right), x\right), B\right) \]

   10. Simplified 62.6%

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

   if 3.59999999999999984e-6 < B

   1. Initial program 80.5%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 80.6%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + {F}^{2}}}\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. *-lowering-*.f64 80.6%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 80.6%

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

   8. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right) \]

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \cos B\right), \color{blue}{\left(\mathsf{neg}\left(\sin B\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \cos B\right), \left(\mathsf{neg}\left(\color{blue}{\sin B}\right)\right)\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \left(\mathsf{neg}\left(\sin B\right)\right)\right) \]

      6. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \mathsf{neg.f64}\left(\sin B\right)\right) \]

      7. sin-lowering-sin.f64 60.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \mathsf{neg.f64}\left(\mathsf{sin.f64}\left(B\right)\right)\right) \]

   10. Simplified 60.2%

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

       1. distribute-frac-neg2 N/A

          \[\leadsto \mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right) \]

       2. clear-num N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\sin B}{x \cdot \cos B}}\right) \]

       3. associate-/l/ N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\frac{\sin B}{\cos B}}{x}}\right) \]

       4. tan-quot N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\tan B}{x}}\right) \]

       5. clear-num N/A

          \[\leadsto \mathsf{neg}\left(\frac{x}{\tan B}\right) \]

       6. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{x}{\tan B}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, \tan B\right)\right) \]

       8. tan-lowering-tan.f64 60.3%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   12. Applied egg-rr 60.3%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{F \cdot F + 2}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 71.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.4 \cdot 10^{-45}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.46 \cdot 10^{-53}:\\ \;\;\;\;0 - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.4e-45)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 1.46e-53) (- 0.0 (/ x (tan B))) (- (/ 1.0 (sin B)) (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.4e-45) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 1.46e-53) {
		tmp = 0.0 - (x / tan(B));
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.4d-45)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 1.46d-53) then
        tmp = 0.0d0 - (x / tan(b))
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.4e-45) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 1.46e-53) {
		tmp = 0.0 - (x / Math.tan(B));
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.4e-45:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 1.46e-53:
		tmp = 0.0 - (x / math.tan(B))
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.4e-45)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 1.46e-53)
		tmp = Float64(0.0 - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.4e-45)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 1.46e-53)
		tmp = 0.0 - (x / tan(B));
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.4e-45], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.46e-53], N[(0.0 - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.4000000000000001e-45

   1. Initial program 59.5%

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

   3. Taylor expanded in B around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\color{blue}{\left(\frac{x}{B}\right)}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{sin.f64}\left(B\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 36.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{F}, \mathsf{sin.f64}\left(B\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]

   5. Simplified 36.0%

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

   6. Taylor expanded in F around -inf

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

      1. distribute-lft-in N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. neg-mul-1 N/A

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

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right), \color{blue}{\left(\frac{x}{B}\right)}\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\mathsf{neg}\left(1\right)}{\sin B}\right), \left(\frac{\color{blue}{x}}{B}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{-1}{\sin B}\right), \left(\frac{x}{B}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \sin B\right), \left(\frac{\color{blue}{x}}{B}\right)\right) \]

      9. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{sin.f64}\left(B\right)\right), \left(\frac{x}{B}\right)\right) \]

      10. /-lowering-/.f64 70.1%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \color{blue}{B}\right)\right) \]

   8. Simplified 70.1%

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

   if -1.4000000000000001e-45 < F < 1.45999999999999989e-53

   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 N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + {F}^{2}}}\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. *-lowering-*.f64 99.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 99.8%

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

   8. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right) \]

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \cos B\right), \color{blue}{\left(\mathsf{neg}\left(\sin B\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \cos B\right), \left(\mathsf{neg}\left(\color{blue}{\sin B}\right)\right)\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \left(\mathsf{neg}\left(\sin B\right)\right)\right) \]

      6. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \mathsf{neg.f64}\left(\sin B\right)\right) \]

      7. sin-lowering-sin.f64 79.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \mathsf{neg.f64}\left(\mathsf{sin.f64}\left(B\right)\right)\right) \]

   10. Simplified 79.6%

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

       1. distribute-frac-neg2 N/A

          \[\leadsto \mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right) \]

       2. clear-num N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\sin B}{x \cdot \cos B}}\right) \]

       3. associate-/l/ N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\frac{\sin B}{\cos B}}{x}}\right) \]

       4. tan-quot N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\tan B}{x}}\right) \]

       5. clear-num N/A

          \[\leadsto \mathsf{neg}\left(\frac{x}{\tan B}\right) \]

       6. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{x}{\tan B}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, \tan B\right)\right) \]

       8. tan-lowering-tan.f64 79.8%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   12. Applied egg-rr 79.8%

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

   if 1.45999999999999989e-53 < F

   1. Initial program 65.2%

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

   3. Taylor expanded in B around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\color{blue}{\left(\frac{x}{B}\right)}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{sin.f64}\left(B\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 44.2%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{F}, \mathsf{sin.f64}\left(B\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]

   5. Simplified 44.2%

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

   6. Taylor expanded in F around inf

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{1}{\sin B}\right), \color{blue}{\left(\frac{x}{B}\right)}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \sin B\right), \left(\frac{\color{blue}{x}}{B}\right)\right) \]

      3. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \left(\frac{x}{B}\right)\right) \]

      4. /-lowering-/.f64 71.0%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \color{blue}{B}\right)\right) \]

   8. Simplified 71.0%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.4 \cdot 10^{-45}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.46 \cdot 10^{-53}:\\ \;\;\;\;0 - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 63.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.4 \cdot 10^{-45}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 5.7 \cdot 10^{+88}:\\ \;\;\;\;0 - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.4e-45)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 5.7e+88) (- 0.0 (/ x (tan B))) (/ 1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.4e-45) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 5.7e+88) {
		tmp = 0.0 - (x / tan(B));
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.4d-45)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 5.7d+88) then
        tmp = 0.0d0 - (x / tan(b))
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.4e-45) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 5.7e+88) {
		tmp = 0.0 - (x / Math.tan(B));
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.4e-45:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 5.7e+88:
		tmp = 0.0 - (x / math.tan(B))
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.4e-45)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 5.7e+88)
		tmp = Float64(0.0 - Float64(x / tan(B)));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.4e-45)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 5.7e+88)
		tmp = 0.0 - (x / tan(B));
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.4e-45], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.7e+88], N[(0.0 - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.4000000000000001e-45

   1. Initial program 59.5%

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

   3. Taylor expanded in B around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\color{blue}{\left(\frac{x}{B}\right)}\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(F, \mathsf{sin.f64}\left(B\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 36.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, B\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{F}, \mathsf{sin.f64}\left(B\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), 2\right), \mathsf{*.f64}\left(2, x\right)\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, 2\right)\right)\right)\right)\right) \]

   5. Simplified 36.0%

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

   6. Taylor expanded in F around -inf

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

      1. distribute-lft-in N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. neg-mul-1 N/A

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

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right), \color{blue}{\left(\frac{x}{B}\right)}\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\mathsf{neg}\left(1\right)}{\sin B}\right), \left(\frac{\color{blue}{x}}{B}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{-1}{\sin B}\right), \left(\frac{x}{B}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \sin B\right), \left(\frac{\color{blue}{x}}{B}\right)\right) \]

      9. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{sin.f64}\left(B\right)\right), \left(\frac{x}{B}\right)\right) \]

      10. /-lowering-/.f64 70.1%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \color{blue}{B}\right)\right) \]

   8. Simplified 70.1%

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

   if -1.4000000000000001e-45 < F < 5.70000000000000021e88

   1. Initial program 98.7%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \color{blue}{\left(x \cdot \frac{1}{\tan B}\right)}\right) \]

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)}, \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + {F}^{2}}}\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

      6. *-lowering-*.f64 99.7%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(B\right)\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   7. Simplified 99.7%

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

   8. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right) \]

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \cos B\right), \color{blue}{\left(\mathsf{neg}\left(\sin B\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \cos B\right), \left(\mathsf{neg}\left(\color{blue}{\sin B}\right)\right)\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \left(\mathsf{neg}\left(\sin B\right)\right)\right) \]

      6. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \mathsf{neg.f64}\left(\sin B\right)\right) \]

      7. sin-lowering-sin.f64 72.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(B\right)\right), \mathsf{neg.f64}\left(\mathsf{sin.f64}\left(B\right)\right)\right) \]

   10. Simplified 72.8%

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

       1. distribute-frac-neg2 N/A

          \[\leadsto \mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right) \]

       2. clear-num N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\sin B}{x \cdot \cos B}}\right) \]

       3. associate-/l/ N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\frac{\sin B}{\cos B}}{x}}\right) \]

       4. tan-quot N/A

          \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{\tan B}{x}}\right) \]

       5. clear-num N/A

          \[\leadsto \mathsf{neg}\left(\frac{x}{\tan B}\right) \]

       6. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{x}{\tan B}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, \tan B\right)\right) \]

       8. tan-lowering-tan.f64 73.0%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(B\right)\right)\right) \]

   12. Applied egg-rr 73.0%

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

   if 5.70000000000000021e88 < F

   1. Initial program 46.4%

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

   3. Taylor expanded in F around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \color{blue}{\left(\frac{1}{\sin B}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\sin B}\right)\right) \]

      2. sin-lowering-sin.f64 99.7%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right)\right) \]

   5. Simplified 99.7%

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

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\sin B}\right) \]

      2. sin-lowering-sin.f64 56.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right) \]

   8. Simplified 56.0%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.4 \cdot 10^{-45}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 5.7 \cdot 10^{+88}:\\ \;\;\;\;0 - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 43.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.3 \cdot 10^{-52}:\\ \;\;\;\;\frac{\left(-1 + \frac{1 + \left(x \cdot 2\right) \cdot 0.5}{F \cdot F}\right) - x}{B}\\ \mathbf{elif}\;F \leq 2.6 \cdot 10^{-22}:\\ \;\;\;\;0 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.3e-52)
   (/ (- (+ -1.0 (/ (+ 1.0 (* (* x 2.0) 0.5)) (* F F))) x) B)
   (if (<= F 2.6e-22) (- 0.0 (/ x B)) (/ 1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.3e-52) {
		tmp = ((-1.0 + ((1.0 + ((x * 2.0) * 0.5)) / (F * F))) - x) / B;
	} else if (F <= 2.6e-22) {
		tmp = 0.0 - (x / B);
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-2.3d-52)) then
        tmp = (((-1.0d0) + ((1.0d0 + ((x * 2.0d0) * 0.5d0)) / (f * f))) - x) / b
    else if (f <= 2.6d-22) then
        tmp = 0.0d0 - (x / b)
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.3e-52) {
		tmp = ((-1.0 + ((1.0 + ((x * 2.0) * 0.5)) / (F * F))) - x) / B;
	} else if (F <= 2.6e-22) {
		tmp = 0.0 - (x / B);
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.3e-52:
		tmp = ((-1.0 + ((1.0 + ((x * 2.0) * 0.5)) / (F * F))) - x) / B
	elif F <= 2.6e-22:
		tmp = 0.0 - (x / B)
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.3e-52)
		tmp = Float64(Float64(Float64(-1.0 + Float64(Float64(1.0 + Float64(Float64(x * 2.0) * 0.5)) / Float64(F * F))) - x) / B);
	elseif (F <= 2.6e-22)
		tmp = Float64(0.0 - Float64(x / B));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.3e-52)
		tmp = ((-1.0 + ((1.0 + ((x * 2.0) * 0.5)) / (F * F))) - x) / B;
	elseif (F <= 2.6e-22)
		tmp = 0.0 - (x / B);
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.3e-52], N[(N[(N[(-1.0 + N[(N[(1.0 + N[(N[(x * 2.0), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 2.6e-22], N[(0.0 - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.29999999999999994e-52

   1. Initial program 60.1%

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

   3. Taylor expanded in B around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x\right), \color{blue}{B}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right), x\right), B\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right), x\right), B\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right), x\right), B\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2} + 2 \cdot x\right)\right)\right)\right)\right), x\right), B\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left({F}^{2}\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left(F \cdot F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      11. *-lowering-*.f64 44.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   5. Simplified 44.3%

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

   6. Taylor expanded in F around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}} - 1\right)}, x\right), B\right) \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}} + \left(\mathsf{neg}\left(1\right)\right)\right), x\right), B\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}} + -1\right), x\right), B\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right), -1\right), x\right), B\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{2} \cdot \left(2 + 2 \cdot x\right)}{{F}^{2}}\right), -1\right), x\right), B\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(2 + 2 \cdot x\right)\right), \left({F}^{2}\right)\right), -1\right), x\right), B\right) \]

      6. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2} \cdot 2 + \frac{1}{2} \cdot \left(2 \cdot x\right)\right), \left({F}^{2}\right)\right), -1\right), x\right), B\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(1 + \frac{1}{2} \cdot \left(2 \cdot x\right)\right), \left({F}^{2}\right)\right), -1\right), x\right), B\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \left(2 \cdot x\right)\right)\right), \left({F}^{2}\right)\right), -1\right), x\right), B\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(2 \cdot x\right)\right)\right), \left({F}^{2}\right)\right), -1\right), x\right), B\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(x \cdot 2\right)\right)\right), \left({F}^{2}\right)\right), -1\right), x\right), B\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, 2\right)\right)\right), \left({F}^{2}\right)\right), -1\right), x\right), B\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, 2\right)\right)\right), \left(F \cdot F\right)\right), -1\right), x\right), B\right) \]

      13. *-lowering-*.f64 52.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, 2\right)\right)\right), \mathsf{*.f64}\left(F, F\right)\right), -1\right), x\right), B\right) \]

   8. Simplified 52.4%

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

   if -2.29999999999999994e-52 < F < 2.6e-22

   1. Initial program 99.4%

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

   3. Taylor expanded in B around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x\right), \color{blue}{B}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right), x\right), B\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right), x\right), B\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right), x\right), B\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2} + 2 \cdot x\right)\right)\right)\right)\right), x\right), B\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left({F}^{2}\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left(F \cdot F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      11. *-lowering-*.f64 56.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   5. Simplified 56.7%

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

   6. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{x}{B}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{x}{\color{blue}{\mathsf{neg}\left(B\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(B\right)\right)}\right) \]

      4. neg-lowering-neg.f64 44.5%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{neg.f64}\left(B\right)\right) \]

   8. Simplified 44.5%

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

   if 2.6e-22 < F

   1. Initial program 63.9%

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

   3. Taylor expanded in F around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \color{blue}{\left(\frac{1}{\sin B}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\sin B}\right)\right) \]

      2. sin-lowering-sin.f64 95.1%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right)\right) \]

   5. Simplified 95.1%

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

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\sin B}\right) \]

      2. sin-lowering-sin.f64 50.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right) \]

   8. Simplified 50.0%

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

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.3 \cdot 10^{-52}:\\ \;\;\;\;\frac{\left(-1 + \frac{1 + \left(x \cdot 2\right) \cdot 0.5}{F \cdot F}\right) - x}{B}\\ \mathbf{elif}\;F \leq 2.6 \cdot 10^{-22}:\\ \;\;\;\;0 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 42.9% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.3 \cdot 10^{-52}:\\ \;\;\;\;\frac{\left(-1 + \frac{1 + \left(x \cdot 2\right) \cdot 0.5}{F \cdot F}\right) - x}{B}\\ \mathbf{elif}\;F \leq 2.45 \cdot 10^{-64}:\\ \;\;\;\;0 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{-1 + \left(x \cdot 2\right) \cdot -0.5}{F \cdot F}\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.3e-52)
   (/ (- (+ -1.0 (/ (+ 1.0 (* (* x 2.0) 0.5)) (* F F))) x) B)
   (if (<= F 2.45e-64)
     (- 0.0 (/ x B))
     (/ (- (+ 1.0 (/ (+ -1.0 (* (* x 2.0) -0.5)) (* F F))) x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.3e-52) {
		tmp = ((-1.0 + ((1.0 + ((x * 2.0) * 0.5)) / (F * F))) - x) / B;
	} else if (F <= 2.45e-64) {
		tmp = 0.0 - (x / B);
	} else {
		tmp = ((1.0 + ((-1.0 + ((x * 2.0) * -0.5)) / (F * F))) - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-2.3d-52)) then
        tmp = (((-1.0d0) + ((1.0d0 + ((x * 2.0d0) * 0.5d0)) / (f * f))) - x) / b
    else if (f <= 2.45d-64) then
        tmp = 0.0d0 - (x / b)
    else
        tmp = ((1.0d0 + (((-1.0d0) + ((x * 2.0d0) * (-0.5d0))) / (f * f))) - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.3e-52) {
		tmp = ((-1.0 + ((1.0 + ((x * 2.0) * 0.5)) / (F * F))) - x) / B;
	} else if (F <= 2.45e-64) {
		tmp = 0.0 - (x / B);
	} else {
		tmp = ((1.0 + ((-1.0 + ((x * 2.0) * -0.5)) / (F * F))) - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.3e-52:
		tmp = ((-1.0 + ((1.0 + ((x * 2.0) * 0.5)) / (F * F))) - x) / B
	elif F <= 2.45e-64:
		tmp = 0.0 - (x / B)
	else:
		tmp = ((1.0 + ((-1.0 + ((x * 2.0) * -0.5)) / (F * F))) - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.3e-52)
		tmp = Float64(Float64(Float64(-1.0 + Float64(Float64(1.0 + Float64(Float64(x * 2.0) * 0.5)) / Float64(F * F))) - x) / B);
	elseif (F <= 2.45e-64)
		tmp = Float64(0.0 - Float64(x / B));
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(-1.0 + Float64(Float64(x * 2.0) * -0.5)) / Float64(F * F))) - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.3e-52)
		tmp = ((-1.0 + ((1.0 + ((x * 2.0) * 0.5)) / (F * F))) - x) / B;
	elseif (F <= 2.45e-64)
		tmp = 0.0 - (x / B);
	else
		tmp = ((1.0 + ((-1.0 + ((x * 2.0) * -0.5)) / (F * F))) - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.3e-52], N[(N[(N[(-1.0 + N[(N[(1.0 + N[(N[(x * 2.0), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 2.45e-64], N[(0.0 - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + N[(N[(-1.0 + N[(N[(x * 2.0), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.29999999999999994e-52

   1. Initial program 60.1%

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

   3. Taylor expanded in B around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x\right), \color{blue}{B}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right), x\right), B\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right), x\right), B\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right), x\right), B\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2} + 2 \cdot x\right)\right)\right)\right)\right), x\right), B\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left({F}^{2}\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left(F \cdot F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      11. *-lowering-*.f64 44.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   5. Simplified 44.3%

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

   6. Taylor expanded in F around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}} - 1\right)}, x\right), B\right) \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}} + \left(\mathsf{neg}\left(1\right)\right)\right), x\right), B\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}} + -1\right), x\right), B\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right), -1\right), x\right), B\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{2} \cdot \left(2 + 2 \cdot x\right)}{{F}^{2}}\right), -1\right), x\right), B\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(2 + 2 \cdot x\right)\right), \left({F}^{2}\right)\right), -1\right), x\right), B\right) \]

      6. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2} \cdot 2 + \frac{1}{2} \cdot \left(2 \cdot x\right)\right), \left({F}^{2}\right)\right), -1\right), x\right), B\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(1 + \frac{1}{2} \cdot \left(2 \cdot x\right)\right), \left({F}^{2}\right)\right), -1\right), x\right), B\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \left(2 \cdot x\right)\right)\right), \left({F}^{2}\right)\right), -1\right), x\right), B\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(2 \cdot x\right)\right)\right), \left({F}^{2}\right)\right), -1\right), x\right), B\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(x \cdot 2\right)\right)\right), \left({F}^{2}\right)\right), -1\right), x\right), B\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, 2\right)\right)\right), \left({F}^{2}\right)\right), -1\right), x\right), B\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, 2\right)\right)\right), \left(F \cdot F\right)\right), -1\right), x\right), B\right) \]

      13. *-lowering-*.f64 52.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, 2\right)\right)\right), \mathsf{*.f64}\left(F, F\right)\right), -1\right), x\right), B\right) \]

   8. Simplified 52.4%

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

   if -2.29999999999999994e-52 < F < 2.4500000000000001e-64

   1. Initial program 99.4%

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

   3. Taylor expanded in B around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x\right), \color{blue}{B}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right), x\right), B\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right), x\right), B\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right), x\right), B\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2} + 2 \cdot x\right)\right)\right)\right)\right), x\right), B\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left({F}^{2}\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left(F \cdot F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      11. *-lowering-*.f64 57.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   5. Simplified 57.9%

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

   6. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{x}{B}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{x}{\color{blue}{\mathsf{neg}\left(B\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(B\right)\right)}\right) \]

      4. neg-lowering-neg.f64 45.2%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{neg.f64}\left(B\right)\right) \]

   8. Simplified 45.2%

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

   if 2.4500000000000001e-64 < F

   1. Initial program 65.6%

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

   3. Taylor expanded in B around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x\right), \color{blue}{B}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right), x\right), B\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right), x\right), B\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right), x\right), B\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2} + 2 \cdot x\right)\right)\right)\right)\right), x\right), B\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left({F}^{2}\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left(F \cdot F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      11. *-lowering-*.f64 37.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   5. Simplified 37.6%

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

   6. Taylor expanded in F around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\left(1 + \frac{-1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right)}, x\right), B\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right)\right), x\right), B\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{2} \cdot \left(2 + 2 \cdot x\right)}{{F}^{2}}\right)\right), x\right), B\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot \left(2 + 2 \cdot x\right)\right), \left({F}^{2}\right)\right)\right), x\right), B\right) \]

      4. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot 2 + \frac{-1}{2} \cdot \left(2 \cdot x\right)\right), \left({F}^{2}\right)\right)\right), x\right), B\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(-1 + \frac{-1}{2} \cdot \left(2 \cdot x\right)\right), \left({F}^{2}\right)\right)\right), x\right), B\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(-1, \left(\frac{-1}{2} \cdot \left(2 \cdot x\right)\right)\right), \left({F}^{2}\right)\right)\right), x\right), B\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(2 \cdot x\right)\right)\right), \left({F}^{2}\right)\right)\right), x\right), B\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(x \cdot 2\right)\right)\right), \left({F}^{2}\right)\right)\right), x\right), B\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, 2\right)\right)\right), \left({F}^{2}\right)\right)\right), x\right), B\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, 2\right)\right)\right), \left(F \cdot F\right)\right)\right), x\right), B\right) \]

      11. *-lowering-*.f64 41.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, 2\right)\right)\right), \mathsf{*.f64}\left(F, F\right)\right)\right), x\right), B\right) \]

   8. Simplified 41.1%

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1 + -0.5 \cdot \left(x \cdot 2\right)}{F \cdot F}\right)} - x}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.3 \cdot 10^{-52}:\\ \;\;\;\;\frac{\left(-1 + \frac{1 + \left(x \cdot 2\right) \cdot 0.5}{F \cdot F}\right) - x}{B}\\ \mathbf{elif}\;F \leq 2.45 \cdot 10^{-64}:\\ \;\;\;\;0 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{-1 + \left(x \cdot 2\right) \cdot -0.5}{F \cdot F}\right) - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 43.3% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -5.5 \cdot 10^{-88}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.45 \cdot 10^{-64}:\\ \;\;\;\;0 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{-1 + \left(x \cdot 2\right) \cdot -0.5}{F \cdot F}\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -5.5e-88)
   (/ (- -1.0 x) B)
   (if (<= F 2.45e-64)
     (- 0.0 (/ x B))
     (/ (- (+ 1.0 (/ (+ -1.0 (* (* x 2.0) -0.5)) (* F F))) x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -5.5e-88) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 2.45e-64) {
		tmp = 0.0 - (x / B);
	} else {
		tmp = ((1.0 + ((-1.0 + ((x * 2.0) * -0.5)) / (F * F))) - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-5.5d-88)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 2.45d-64) then
        tmp = 0.0d0 - (x / b)
    else
        tmp = ((1.0d0 + (((-1.0d0) + ((x * 2.0d0) * (-0.5d0))) / (f * f))) - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -5.5e-88) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 2.45e-64) {
		tmp = 0.0 - (x / B);
	} else {
		tmp = ((1.0 + ((-1.0 + ((x * 2.0) * -0.5)) / (F * F))) - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -5.5e-88:
		tmp = (-1.0 - x) / B
	elif F <= 2.45e-64:
		tmp = 0.0 - (x / B)
	else:
		tmp = ((1.0 + ((-1.0 + ((x * 2.0) * -0.5)) / (F * F))) - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -5.5e-88)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 2.45e-64)
		tmp = Float64(0.0 - Float64(x / B));
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(-1.0 + Float64(Float64(x * 2.0) * -0.5)) / Float64(F * F))) - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -5.5e-88)
		tmp = (-1.0 - x) / B;
	elseif (F <= 2.45e-64)
		tmp = 0.0 - (x / B);
	else
		tmp = ((1.0 + ((-1.0 + ((x * 2.0) * -0.5)) / (F * F))) - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -5.5e-88], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 2.45e-64], N[(0.0 - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + N[(N[(-1.0 + N[(N[(x * 2.0), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.49999999999999971e-88

   1. Initial program 64.0%

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

   3. Taylor expanded in B around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x\right), \color{blue}{B}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right), x\right), B\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right), x\right), B\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right), x\right), B\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2} + 2 \cdot x\right)\right)\right)\right)\right), x\right), B\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left({F}^{2}\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left(F \cdot F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      11. *-lowering-*.f64 46.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   5. Simplified 46.2%

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

   6. Taylor expanded in F around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(1 + x\right)\right)}{B} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(1 + x\right)\right)\right), \color{blue}{B}\right) \]

      4. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(x\right)\right)\right), B\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 + \left(\mathsf{neg}\left(x\right)\right)\right), B\right) \]

      6. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 - x\right), B\right) \]

      7. --lowering--.f64 49.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, x\right), B\right) \]

   8. Simplified 49.6%

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

   if -5.49999999999999971e-88 < F < 2.4500000000000001e-64

   1. Initial program 99.5%

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

   3. Taylor expanded in B around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x\right), \color{blue}{B}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right), x\right), B\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right), x\right), B\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right), x\right), B\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2} + 2 \cdot x\right)\right)\right)\right)\right), x\right), B\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left({F}^{2}\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left(F \cdot F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      11. *-lowering-*.f64 57.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   5. Simplified 57.4%

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

   6. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{x}{B}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{x}{\color{blue}{\mathsf{neg}\left(B\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(B\right)\right)}\right) \]

      4. neg-lowering-neg.f64 46.6%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{neg.f64}\left(B\right)\right) \]

   8. Simplified 46.6%

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

   if 2.4500000000000001e-64 < F

   1. Initial program 65.6%

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

   3. Taylor expanded in B around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x\right), \color{blue}{B}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right), x\right), B\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right), x\right), B\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right), x\right), B\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2} + 2 \cdot x\right)\right)\right)\right)\right), x\right), B\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left({F}^{2}\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left(F \cdot F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      11. *-lowering-*.f64 37.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   5. Simplified 37.6%

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

   6. Taylor expanded in F around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\left(1 + \frac{-1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right)}, x\right), B\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right)\right), x\right), B\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{2} \cdot \left(2 + 2 \cdot x\right)}{{F}^{2}}\right)\right), x\right), B\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot \left(2 + 2 \cdot x\right)\right), \left({F}^{2}\right)\right)\right), x\right), B\right) \]

      4. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot 2 + \frac{-1}{2} \cdot \left(2 \cdot x\right)\right), \left({F}^{2}\right)\right)\right), x\right), B\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(-1 + \frac{-1}{2} \cdot \left(2 \cdot x\right)\right), \left({F}^{2}\right)\right)\right), x\right), B\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(-1, \left(\frac{-1}{2} \cdot \left(2 \cdot x\right)\right)\right), \left({F}^{2}\right)\right)\right), x\right), B\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(2 \cdot x\right)\right)\right), \left({F}^{2}\right)\right)\right), x\right), B\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{-1}{2}, \left(x \cdot 2\right)\right)\right), \left({F}^{2}\right)\right)\right), x\right), B\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, 2\right)\right)\right), \left({F}^{2}\right)\right)\right), x\right), B\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, 2\right)\right)\right), \left(F \cdot F\right)\right)\right), x\right), B\right) \]

      11. *-lowering-*.f64 41.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, 2\right)\right)\right), \mathsf{*.f64}\left(F, F\right)\right)\right), x\right), B\right) \]

   8. Simplified 41.1%

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1 + -0.5 \cdot \left(x \cdot 2\right)}{F \cdot F}\right)} - x}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5.5 \cdot 10^{-88}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.45 \cdot 10^{-64}:\\ \;\;\;\;0 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{-1 + \left(x \cdot 2\right) \cdot -0.5}{F \cdot F}\right) - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 43.8% accurate, 21.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{-87}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 3.2 \cdot 10^{-22}:\\ \;\;\;\;0 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1e-87)
   (/ (- -1.0 x) B)
   (if (<= F 3.2e-22) (- 0.0 (/ x B)) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1e-87) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 3.2e-22) {
		tmp = 0.0 - (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 <= (-1d-87)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 3.2d-22) then
        tmp = 0.0d0 - (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 <= -1e-87) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 3.2e-22) {
		tmp = 0.0 - (x / B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1e-87:
		tmp = (-1.0 - x) / B
	elif F <= 3.2e-22:
		tmp = 0.0 - (x / B)
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1e-87)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 3.2e-22)
		tmp = Float64(0.0 - 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 <= -1e-87)
		tmp = (-1.0 - x) / B;
	elseif (F <= 3.2e-22)
		tmp = 0.0 - (x / B);
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1e-87], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 3.2e-22], N[(0.0 - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.00000000000000002e-87

   1. Initial program 64.0%

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

   3. Taylor expanded in B around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x\right), \color{blue}{B}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right), x\right), B\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right), x\right), B\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right), x\right), B\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2} + 2 \cdot x\right)\right)\right)\right)\right), x\right), B\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left({F}^{2}\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left(F \cdot F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      11. *-lowering-*.f64 46.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   5. Simplified 46.2%

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

   6. Taylor expanded in F around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(1 + x\right)\right)}{B} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(1 + x\right)\right)\right), \color{blue}{B}\right) \]

      4. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(x\right)\right)\right), B\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 + \left(\mathsf{neg}\left(x\right)\right)\right), B\right) \]

      6. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 - x\right), B\right) \]

      7. --lowering--.f64 49.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, x\right), B\right) \]

   8. Simplified 49.6%

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

   if -1.00000000000000002e-87 < F < 3.19999999999999987e-22

   1. Initial program 99.5%

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

   3. Taylor expanded in B around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x\right), \color{blue}{B}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right), x\right), B\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right), x\right), B\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right), x\right), B\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2} + 2 \cdot x\right)\right)\right)\right)\right), x\right), B\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left({F}^{2}\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left(F \cdot F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      11. *-lowering-*.f64 55.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   5. Simplified 55.6%

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

   6. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{x}{B}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{x}{\color{blue}{\mathsf{neg}\left(B\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(B\right)\right)}\right) \]

      4. neg-lowering-neg.f64 45.4%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{neg.f64}\left(B\right)\right) \]

   8. Simplified 45.4%

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

   if 3.19999999999999987e-22 < F

   1. Initial program 63.5%

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

   3. Taylor expanded in B around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x\right), \color{blue}{B}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right), x\right), B\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right), x\right), B\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right), x\right), B\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2} + 2 \cdot x\right)\right)\right)\right)\right), x\right), B\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left({F}^{2}\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left(F \cdot F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      11. *-lowering-*.f64 38.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   5. Simplified 38.5%

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

   6. Taylor expanded in F around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 - x\right), \color{blue}{B}\right) \]

      2. --lowering--.f64 41.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, x\right), B\right) \]

   8. Simplified 41.7%

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

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{-87}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 3.2 \cdot 10^{-22}:\\ \;\;\;\;0 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 36.6% accurate, 32.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -6 \cdot 10^{-90}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -6e-90) (/ (- -1.0 x) B) (- 0.0 (/ x B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -6e-90) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = 0.0 - (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 <= (-6d-90)) then
        tmp = ((-1.0d0) - x) / b
    else
        tmp = 0.0d0 - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -6e-90) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = 0.0 - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -6e-90:
		tmp = (-1.0 - x) / B
	else:
		tmp = 0.0 - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -6e-90)
		tmp = Float64(Float64(-1.0 - x) / B);
	else
		tmp = Float64(0.0 - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -6e-90)
		tmp = (-1.0 - x) / B;
	else
		tmp = 0.0 - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -6e-90], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[(0.0 - N[(x / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -6.00000000000000041e-90

   1. Initial program 64.0%

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

   3. Taylor expanded in B around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x\right), \color{blue}{B}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right), x\right), B\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right), x\right), B\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right), x\right), B\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2} + 2 \cdot x\right)\right)\right)\right)\right), x\right), B\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left({F}^{2}\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left(F \cdot F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      11. *-lowering-*.f64 46.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   5. Simplified 46.2%

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

   6. Taylor expanded in F around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(1 + x\right)\right)}{B} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(1 + x\right)\right)\right), \color{blue}{B}\right) \]

      4. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(x\right)\right)\right), B\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 + \left(\mathsf{neg}\left(x\right)\right)\right), B\right) \]

      6. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 - x\right), B\right) \]

      7. --lowering--.f64 49.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(-1, x\right), B\right) \]

   8. Simplified 49.6%

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

   if -6.00000000000000041e-90 < F

   1. Initial program 83.5%

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

   3. Taylor expanded in B around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x\right), \color{blue}{B}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right), x\right), B\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right), x\right), B\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right), x\right), B\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2} + 2 \cdot x\right)\right)\right)\right)\right), x\right), B\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left({F}^{2}\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left(F \cdot F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      11. *-lowering-*.f64 48.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   5. Simplified 48.1%

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

   6. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{x}{B}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{x}{\color{blue}{\mathsf{neg}\left(B\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(B\right)\right)}\right) \]

      4. neg-lowering-neg.f64 36.8%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{neg.f64}\left(B\right)\right) \]

   8. Simplified 36.8%

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

4. Final simplification 40.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6 \cdot 10^{-90}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 17.1% accurate, 40.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 3.1 \cdot 10^{-79}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (if (<= F 3.1e-79) (/ -1.0 B) (/ 1.0 B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= 3.1e-79) {
		tmp = -1.0 / 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 <= 3.1d-79) then
        tmp = (-1.0d0) / 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 <= 3.1e-79) {
		tmp = -1.0 / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= 3.1e-79:
		tmp = -1.0 / B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= 3.1e-79)
		tmp = Float64(-1.0 / B);
	else
		tmp = Float64(1.0 / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= 3.1e-79)
		tmp = -1.0 / B;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, 3.1e-79], N[(-1.0 / B), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 3.0999999999999999e-79

   1. Initial program 82.8%

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

   3. Taylor expanded in B around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x\right), \color{blue}{B}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right), x\right), B\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right), x\right), B\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right), x\right), B\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2} + 2 \cdot x\right)\right)\right)\right)\right), x\right), B\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left({F}^{2}\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left(F \cdot F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

      11. *-lowering-*.f64 52.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   5. Simplified 52.5%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \color{blue}{\left(\frac{F}{B}\right)}\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \left(\frac{\color{blue}{F}}{B}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \left(\frac{F}{B}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \left(\frac{F}{B}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \left(\frac{F}{B}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \left(\frac{F}{B}\right)\right) \]

      8. /-lowering-/.f64 14.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{/.f64}\left(F, \color{blue}{B}\right)\right) \]

   8. Simplified 14.6%

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

   9. Taylor expanded in F around -inf

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

       1. /-lowering-/.f64 14.5%

          \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{B}\right) \]

   11. Simplified 14.5%

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

   if 3.0999999999999999e-79 < F

   1. Initial program 66.8%

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

   3. Taylor expanded in F around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \color{blue}{\left(\frac{1}{\sin B}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\sin B}\right)\right) \]

      2. sin-lowering-sin.f64 90.1%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right)\right) \]

   5. Simplified 90.1%

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

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\sin B}\right) \]

      2. sin-lowering-sin.f64 46.3%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right) \]

   8. Simplified 46.3%

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

   9. Taylor expanded in B around 0

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

       1. /-lowering-/.f64 17.7%

          \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{B}\right) \]

   11. Simplified 17.7%

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

Alternative 25: 29.9% accurate, 64.8× speedup

Math

\[\begin{array}{l} \\ 0 - \frac{x}{B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (- 0.0 (/ x B)))

C

double code(double F, double B, double x) {
	return 0.0 - (x / B);
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = 0.0d0 - (x / b)
end function

Java

public static double code(double F, double B, double x) {
	return 0.0 - (x / B);
}

Python

def code(F, B, x):
	return 0.0 - (x / B)

Julia

function code(F, B, x)
	return Float64(0.0 - Float64(x / B))
end

MATLAB

function tmp = code(F, B, x)
	tmp = 0.0 - (x / B);
end

Wolfram

code[F_, B_, x_] := N[(0.0 - N[(x / B), $MachinePrecision]), $MachinePrecision]

Derivation

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. Add Preprocessing

3. Taylor expanded in B around 0

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x\right), \color{blue}{B}\right) \]

   2. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right), x\right), B\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right), x\right), B\right) \]

   4. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right), x\right), B\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2} + 2 \cdot x\right)\right)\right)\right)\right), x\right), B\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left({F}^{2}\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left(F \cdot F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   11. *-lowering-*.f64 47.5%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

5. Simplified 47.5%

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

6. Taylor expanded in F around 0

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

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{x}{B}\right) \]

   2. distribute-neg-frac2 N/A

      \[\leadsto \frac{x}{\color{blue}{\mathsf{neg}\left(B\right)}} \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(B\right)\right)}\right) \]

   4. neg-lowering-neg.f64 33.5%

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{neg.f64}\left(B\right)\right) \]

8. Simplified 33.5%

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

9. Final simplification 33.5%

   \[\leadsto 0 - \frac{x}{B} \]
10. Add Preprocessing

Alternative 26: 10.1% 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 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. Add Preprocessing

3. Taylor expanded in B around 0

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x\right), \color{blue}{B}\right) \]

   2. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right), x\right), B\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right), x\right), B\right) \]

   4. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\left(\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}\right)\right)\right), x\right), B\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(2 \cdot x + {F}^{2}\right)\right)\right)\right)\right), x\right), B\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2} + 2 \cdot x\right)\right)\right)\right)\right), x\right), B\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left({F}^{2}\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\left(F \cdot F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \left(2 \cdot x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

   11. *-lowering-*.f64 47.5%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{+.f64}\left(\mathsf{*.f64}\left(F, F\right), \mathsf{*.f64}\left(2, x\right)\right)\right)\right)\right)\right), x\right), B\right) \]

5. Simplified 47.5%

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

6. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{2 + {F}^{2}}}\right), \color{blue}{\left(\frac{F}{B}\right)}\right) \]

   3. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{2 + {F}^{2}}\right)\right), \left(\frac{\color{blue}{F}}{B}\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 + {F}^{2}\right)\right)\right), \left(\frac{F}{B}\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({F}^{2}\right)\right)\right)\right), \left(\frac{F}{B}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left(F \cdot F\right)\right)\right)\right), \left(\frac{F}{B}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \left(\frac{F}{B}\right)\right) \]

   8. /-lowering-/.f64 14.3%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(F, F\right)\right)\right)\right), \mathsf{/.f64}\left(F, \color{blue}{B}\right)\right) \]

8. Simplified 14.3%

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

9. Taylor expanded in F around -inf

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

    1. /-lowering-/.f64 10.6%

       \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{B}\right) \]

11. Simplified 10.6%

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

Alternative 27: 2.8% accurate, 108.0× speedup

Math

\[\begin{array}{l} \\ B \cdot 0.16666666666666666 \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (* B 0.16666666666666666))

C

double code(double F, double B, double x) {
	return B * 0.16666666666666666;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = b * 0.16666666666666666d0
end function

Java

public static double code(double F, double B, double x) {
	return B * 0.16666666666666666;
}

Python

def code(F, B, x):
	return B * 0.16666666666666666

Julia

function code(F, B, x)
	return Float64(B * 0.16666666666666666)
end

MATLAB

function tmp = code(F, B, x)
	tmp = B * 0.16666666666666666;
end

Wolfram

code[F_, B_, x_] := N[(B * 0.16666666666666666), $MachinePrecision]

Derivation

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. Add Preprocessing

3. Taylor expanded in F around inf

   \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \color{blue}{\left(\frac{1}{\sin B}\right)}\right) \]
4. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\sin B}\right)\right) \]

   2. sin-lowering-sin.f64 57.0%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{sin.f64}\left(B\right)\right)\right) \]

5. Simplified 57.0%

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

6. Taylor expanded in B around 0

   \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \color{blue}{\left(\frac{1 + \frac{1}{6} \cdot {B}^{2}}{B}\right)}\right) \]
7. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(\left(1 + \frac{1}{6} \cdot {B}^{2}\right), \color{blue}{B}\right)\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{6} \cdot {B}^{2}\right)\right), B\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left({B}^{2} \cdot \frac{1}{6}\right)\right), B\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({B}^{2}\right), \frac{1}{6}\right)\right), B\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(B \cdot B\right), \frac{1}{6}\right)\right), B\right)\right) \]

   6. *-lowering-*.f64 35.7%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{tan.f64}\left(B\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \frac{1}{6}\right)\right), B\right)\right) \]

8. Simplified 35.7%

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

9. Taylor expanded in B around inf

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

    1. *-commutative N/A

       \[\leadsto B \cdot \color{blue}{\frac{1}{6}} \]

    2. *-lowering-*.f64 2.9%

       \[\leadsto \mathsf{*.f64}\left(B, \color{blue}{\frac{1}{6}}\right) \]

11. Simplified 2.9%

    \[\leadsto \color{blue}{B \cdot 0.16666666666666666} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024138 
(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))))))