VandenBroeck and Keller, Equation (23)

Percentage Accurate: 77.3% → 99.6%

Time: 17.9s

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: 77.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1 \cdot 10^{+66}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 14.5:\\ \;\;\;\;\frac{1}{\frac{\sin B}{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1e+66)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 14.5)
       (-
        (/ 1.0 (/ (sin B) (* F (pow (+ (* F F) (+ 2.0 (* x 2.0))) -0.5))))
        t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1e+66) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 14.5) {
		tmp = (1.0 / (sin(B) / (F * pow(((F * F) + (2.0 + (x * 2.0))), -0.5)))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1d+66)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 14.5d0) then
        tmp = (1.0d0 / (sin(b) / (f * (((f * f) + (2.0d0 + (x * 2.0d0))) ** (-0.5d0))))) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1e+66) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 14.5) {
		tmp = (1.0 / (Math.sin(B) / (F * Math.pow(((F * F) + (2.0 + (x * 2.0))), -0.5)))) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1e+66:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 14.5:
		tmp = (1.0 / (math.sin(B) / (F * math.pow(((F * F) + (2.0 + (x * 2.0))), -0.5)))) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1e+66)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 14.5)
		tmp = Float64(Float64(1.0 / Float64(sin(B) / Float64(F * (Float64(Float64(F * F) + Float64(2.0 + Float64(x * 2.0))) ^ -0.5)))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1e+66)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 14.5)
		tmp = (1.0 / (sin(B) / (F * (((F * F) + (2.0 + (x * 2.0))) ^ -0.5)))) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1e+66], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 14.5], N[(N[(1.0 / N[(N[Sin[B], $MachinePrecision] / N[(F * N[Power[N[(N[(F * F), $MachinePrecision] + N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -9.99999999999999945e65

   1. Initial program 54.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 72.9%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   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 99.9%

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

   7. Simplified 99.9%

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

   if -9.99999999999999945e65 < F < 14.5

   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 99.5%

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

      1. div-inv N/A

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

      2. associate-*r/ N/A

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

      3. clear-num N/A

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

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

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

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

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

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

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

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

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

   6. Applied egg-rr 99.6%

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

   if 14.5 < F

   1. Initial program 67.0%

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

      1. +-commutative 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 77.0%

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

   5. Taylor expanded in 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 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{+66}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 14.5:\\ \;\;\;\;\frac{1}{\frac{\sin B}{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.7 \cdot 10^{+34}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 14.5:\\ \;\;\;\;\frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\frac{\sin B}{F}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.7e+34)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 14.5)
       (- (/ (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 <= -1.7e+34) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 14.5) {
		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 <= (-1.7d+34)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 14.5d0) 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 <= -1.7e+34) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 14.5) {
		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 <= -1.7e+34:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 14.5:
		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 <= -1.7e+34)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 14.5)
		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 <= -1.7e+34)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 14.5)
		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, -1.7e+34], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 14.5], 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 < -1.7e34

   1. Initial program 56.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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 99.9%

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

   7. Simplified 99.9%

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

   if -1.7e34 < F < 14.5

   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 99.5%

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

   if 14.5 < F

   1. Initial program 67.0%

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

      1. +-commutative 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 77.0%

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

   5. Taylor expanded in 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 3: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1 \cdot 10^{+66}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 14.5:\\ \;\;\;\;\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1e+66)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 14.5)
       (- (/ (* F (pow (+ (* F F) (+ 2.0 (* x 2.0))) -0.5)) (sin B)) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1e+66) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 14.5) {
		tmp = ((F * pow(((F * F) + (2.0 + (x * 2.0))), -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 <= (-1d+66)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 14.5d0) then
        tmp = ((f * (((f * f) + (2.0d0 + (x * 2.0d0))) ** (-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 <= -1e+66) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 14.5) {
		tmp = ((F * Math.pow(((F * F) + (2.0 + (x * 2.0))), -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 <= -1e+66:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 14.5:
		tmp = ((F * math.pow(((F * F) + (2.0 + (x * 2.0))), -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 <= -1e+66)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 14.5)
		tmp = Float64(Float64(Float64(F * (Float64(Float64(F * F) + Float64(2.0 + Float64(x * 2.0))) ^ -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 <= -1e+66)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 14.5)
		tmp = ((F * (((F * F) + (2.0 + (x * 2.0))) ^ -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, -1e+66], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 14.5], 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] - 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 < -9.99999999999999945e65

   1. Initial program 54.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 72.9%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   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 99.9%

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

   7. Simplified 99.9%

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

   if -9.99999999999999945e65 < F < 14.5

   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 99.5%

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

   if 14.5 < F

   1. Initial program 67.0%

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

      1. +-commutative 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 77.0%

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

   5. Taylor expanded in 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 4: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2 \cdot 10^{+19}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 14.5:\\ \;\;\;\;\frac{1}{\frac{\sin B \cdot \sqrt{F \cdot F + 2}}{F}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -2e+19)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 14.5)
       (- (/ 1.0 (/ (* (sin B) (sqrt (+ (* F F) 2.0))) 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+19) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 14.5) {
		tmp = (1.0 / ((sin(B) * sqrt(((F * F) + 2.0))) / 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+19)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 14.5d0) then
        tmp = (1.0d0 / ((sin(b) * sqrt(((f * f) + 2.0d0))) / 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+19) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 14.5) {
		tmp = (1.0 / ((Math.sin(B) * Math.sqrt(((F * F) + 2.0))) / 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+19:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 14.5:
		tmp = (1.0 / ((math.sin(B) * math.sqrt(((F * F) + 2.0))) / 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+19)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 14.5)
		tmp = Float64(Float64(1.0 / Float64(Float64(sin(B) * sqrt(Float64(Float64(F * F) + 2.0))) / 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+19)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 14.5)
		tmp = (1.0 / ((sin(B) * sqrt(((F * F) + 2.0))) / 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+19], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 14.5], N[(N[(1.0 / N[(N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision]), $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 < -2e19

   1. Initial program 60.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 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 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 -2e19 < F < 14.5

   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 99.5%

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

      1. div-inv N/A

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

      2. associate-*r/ N/A

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

      3. clear-num N/A

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

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

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

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

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

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

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

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

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in x around 0

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

      1. associate-*l/ N/A

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

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

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

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

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

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

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 99.1%

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

   9. Simplified 99.1%

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

   if 14.5 < F

   1. Initial program 67.0%

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

      1. +-commutative 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 77.0%

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

   5. Taylor expanded in 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 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{+19}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 14.5:\\ \;\;\;\;\frac{1}{\frac{\sin B \cdot \sqrt{F \cdot F + 2}}{F}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1.86:\\ \;\;\;\;\frac{{\left(2 + x \cdot 2\right)}^{-0.5}}{\frac{\sin B}{F}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.4)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.86)
       (- (/ (pow (+ 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 <= -1.4) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.86) {
		tmp = (pow((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 <= (-1.4d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.86d0) then
        tmp = (((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 <= -1.4) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.86) {
		tmp = (Math.pow((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 <= -1.4:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.86:
		tmp = (math.pow((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 <= -1.4)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.86)
		tmp = Float64(Float64((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 <= -1.4)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.86)
		tmp = (((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, -1.4], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.86], N[(N[(N[Power[N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[(N[Sin[B], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.3999999999999999

   1. Initial program 61.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 77.7%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   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 98.3%

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

   7. Simplified 98.3%

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

   if -1.3999999999999999 < F < 1.8600000000000001

   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 99.5%

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

   5. Taylor expanded in F around 0

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

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

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

      2. *-lowering-*.f64 98.9%

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

   7. Simplified 98.9%

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

   if 1.8600000000000001 < F

   1. Initial program 67.0%

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

      1. +-commutative 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 77.0%

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

   5. Taylor expanded in 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 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.86:\\ \;\;\;\;\frac{{\left(2 + x \cdot 2\right)}^{-0.5}}{\frac{\sin B}{F}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -9.5 \cdot 10^{-53}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 5.6 \cdot 10^{-50}:\\ \;\;\;\;\frac{x \cdot \cos B}{0 - \sin B}\\ \mathbf{elif}\;F \leq 14.5:\\ \;\;\;\;\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -9.5e-53)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 5.6e-50)
       (/ (* x (cos B)) (- 0.0 (sin B)))
       (if (<= F 14.5)
         (- (/ (* F (pow (+ (* F F) (+ 2.0 (* x 2.0))) -0.5)) (sin B)) (/ x B))
         (- (/ 1.0 (sin B)) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -9.5e-53) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 5.6e-50) {
		tmp = (x * cos(B)) / (0.0 - sin(B));
	} else if (F <= 14.5) {
		tmp = ((F * pow(((F * F) + (2.0 + (x * 2.0))), -0.5)) / sin(B)) - (x / B);
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-9.5d-53)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 5.6d-50) then
        tmp = (x * cos(b)) / (0.0d0 - sin(b))
    else if (f <= 14.5d0) then
        tmp = ((f * (((f * f) + (2.0d0 + (x * 2.0d0))) ** (-0.5d0))) / sin(b)) - (x / b)
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -9.5e-53) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 5.6e-50) {
		tmp = (x * Math.cos(B)) / (0.0 - Math.sin(B));
	} else if (F <= 14.5) {
		tmp = ((F * Math.pow(((F * F) + (2.0 + (x * 2.0))), -0.5)) / Math.sin(B)) - (x / B);
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -9.5e-53:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 5.6e-50:
		tmp = (x * math.cos(B)) / (0.0 - math.sin(B))
	elif F <= 14.5:
		tmp = ((F * math.pow(((F * F) + (2.0 + (x * 2.0))), -0.5)) / math.sin(B)) - (x / B)
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -9.5e-53)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 5.6e-50)
		tmp = Float64(Float64(x * cos(B)) / Float64(0.0 - sin(B)));
	elseif (F <= 14.5)
		tmp = Float64(Float64(Float64(F * (Float64(Float64(F * F) + Float64(2.0 + Float64(x * 2.0))) ^ -0.5)) / sin(B)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -9.5e-53)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 5.6e-50)
		tmp = (x * cos(B)) / (0.0 - sin(B));
	elseif (F <= 14.5)
		tmp = ((F * (((F * F) + (2.0 + (x * 2.0))) ^ -0.5)) / sin(B)) - (x / B);
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -9.5e-53], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 5.6e-50], N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[(0.0 - N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 14.5], 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], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -9.5000000000000008e-53

   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. 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 79.2%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   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 95.1%

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

   7. Simplified 95.1%

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

   if -9.5000000000000008e-53 < F < 5.5999999999999996e-50

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      6. sin-lowering-sin.f64 77.0%

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

   5. Simplified 77.0%

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

   if 5.5999999999999996e-50 < F < 14.5

   1. Initial program 98.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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 98.7%

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

   5. Taylor expanded in 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 89.0%

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

   7. Simplified 89.0%

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

   if 14.5 < F

   1. Initial program 67.0%

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

      1. +-commutative 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 77.0%

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

   5. Taylor expanded in 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 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -9.5 \cdot 10^{-53}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 5.6 \cdot 10^{-50}:\\ \;\;\;\;\frac{x \cdot \cos B}{0 - \sin B}\\ \mathbf{elif}\;F \leq 14.5:\\ \;\;\;\;\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -175:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 14.2:\\ \;\;\;\;\sqrt{\frac{1}{2 + \left(F \cdot F + x \cdot 2\right)}} \cdot \frac{F}{B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -175.0)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 14.2)
       (- (* (sqrt (/ 1.0 (+ 2.0 (+ (* F F) (* x 2.0))))) (/ F B)) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -175.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 14.2) {
		tmp = (sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0))))) * (F / B)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-175.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 14.2d0) then
        tmp = (sqrt((1.0d0 / (2.0d0 + ((f * f) + (x * 2.0d0))))) * (f / b)) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -175.0) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 14.2) {
		tmp = (Math.sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0))))) * (F / B)) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -175.0:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 14.2:
		tmp = (math.sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0))))) * (F / B)) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -175.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 14.2)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(Float64(F * F) + Float64(x * 2.0))))) * Float64(F / B)) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -175.0)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 14.2)
		tmp = (sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0))))) * (F / B)) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -175.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 14.2], 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 / 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 < -175

   1. Initial program 60.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 77.4%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   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 99.1%

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

   7. Simplified 99.1%

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

   if -175 < F < 14.199999999999999

   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 99.5%

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

   5. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

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

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

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

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

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

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

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

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

      6. +-commutative N/A

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

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

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

      8. unpow2 N/A

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

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

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

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

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

      11. /-lowering-/.f64 82.5%

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

   7. Simplified 82.5%

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

   if 14.199999999999999 < F

   1. Initial program 67.0%

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

      1. +-commutative 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 77.0%

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

   5. Taylor expanded in 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 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -175:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 14.2:\\ \;\;\;\;\sqrt{\frac{1}{2 + \left(F \cdot F + x \cdot 2\right)}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -230:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 14.5:\\ \;\;\;\;\frac{{\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\frac{B}{F}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -230.0)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 14.5)
       (- (/ (pow (+ (* F F) (+ 2.0 (* x 2.0))) -0.5) (/ 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 <= -230.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 14.5) {
		tmp = (pow(((F * F) + (2.0 + (x * 2.0))), -0.5) / (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 <= (-230.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 14.5d0) then
        tmp = ((((f * f) + (2.0d0 + (x * 2.0d0))) ** (-0.5d0)) / (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 <= -230.0) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 14.5) {
		tmp = (Math.pow(((F * F) + (2.0 + (x * 2.0))), -0.5) / (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 <= -230.0:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 14.5:
		tmp = (math.pow(((F * F) + (2.0 + (x * 2.0))), -0.5) / (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 <= -230.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 14.5)
		tmp = Float64(Float64((Float64(Float64(F * F) + Float64(2.0 + Float64(x * 2.0))) ^ -0.5) / Float64(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 <= -230.0)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 14.5)
		tmp = ((((F * F) + (2.0 + (x * 2.0))) ^ -0.5) / (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, -230.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 14.5], N[(N[(N[Power[N[(N[(F * F), $MachinePrecision] + N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[(B / 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 < -230

   1. Initial program 60.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 77.4%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   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 99.1%

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

   7. Simplified 99.1%

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

   if -230 < F < 14.5

   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 99.5%

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

   5. Taylor expanded in B around 0

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

      1. /-lowering-/.f64 82.4%

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

   7. Simplified 82.4%

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

   if 14.5 < F

   1. Initial program 67.0%

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

      1. +-commutative 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 77.0%

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

   5. Taylor expanded in 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 9: 84.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -7.8 \cdot 10^{-52}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{-45}:\\ \;\;\;\;\frac{x \cdot \cos B}{0 - \sin B}\\ \mathbf{elif}\;F \leq 13.2:\\ \;\;\;\;\frac{1}{\frac{\sin B}{F} \cdot \sqrt{F \cdot F + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -7.8e-52)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 5.2e-45)
       (/ (* x (cos B)) (- 0.0 (sin B)))
       (if (<= F 13.2)
         (/ 1.0 (* (/ (sin B) F) (sqrt (+ (* F F) 2.0))))
         (- (/ 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 <= -7.8e-52) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 5.2e-45) {
		tmp = (x * cos(B)) / (0.0 - sin(B));
	} else if (F <= 13.2) {
		tmp = 1.0 / ((sin(B) / F) * sqrt(((F * F) + 2.0)));
	} 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 <= (-7.8d-52)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 5.2d-45) then
        tmp = (x * cos(b)) / (0.0d0 - sin(b))
    else if (f <= 13.2d0) then
        tmp = 1.0d0 / ((sin(b) / f) * sqrt(((f * f) + 2.0d0)))
    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 <= -7.8e-52) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 5.2e-45) {
		tmp = (x * Math.cos(B)) / (0.0 - Math.sin(B));
	} else if (F <= 13.2) {
		tmp = 1.0 / ((Math.sin(B) / F) * Math.sqrt(((F * F) + 2.0)));
	} 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 <= -7.8e-52:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 5.2e-45:
		tmp = (x * math.cos(B)) / (0.0 - math.sin(B))
	elif F <= 13.2:
		tmp = 1.0 / ((math.sin(B) / F) * math.sqrt(((F * F) + 2.0)))
	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 <= -7.8e-52)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 5.2e-45)
		tmp = Float64(Float64(x * cos(B)) / Float64(0.0 - sin(B)));
	elseif (F <= 13.2)
		tmp = Float64(1.0 / Float64(Float64(sin(B) / F) * sqrt(Float64(Float64(F * F) + 2.0))));
	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 <= -7.8e-52)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 5.2e-45)
		tmp = (x * cos(B)) / (0.0 - sin(B));
	elseif (F <= 13.2)
		tmp = 1.0 / ((sin(B) / F) * sqrt(((F * F) + 2.0)));
	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, -7.8e-52], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 5.2e-45], N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[(0.0 - N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 13.2], N[(1.0 / N[(N[(N[Sin[B], $MachinePrecision] / F), $MachinePrecision] * N[Sqrt[N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -7.80000000000000036e-52

   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. 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 79.2%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   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 95.1%

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

   7. Simplified 95.1%

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

   if -7.80000000000000036e-52 < F < 5.19999999999999973e-45

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      6. sin-lowering-sin.f64 77.0%

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

   5. Simplified 77.0%

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

   if 5.19999999999999973e-45 < F < 13.199999999999999

   1. Initial program 98.8%

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

   3. Taylor expanded in 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 82.3%

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

   5. Simplified 82.3%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} \]
   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. div-inv N/A

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

      5. metadata-eval N/A

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

      6. sqrt-div N/A

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

      7. clear-num N/A

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

      8. /-rgt-identity N/A

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

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

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

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

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

      11. pow1/2 N/A

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

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

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

      13. +-commutative N/A

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

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

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

      15. *-lowering-*.f64 82.5%

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

   7. Applied egg-rr 82.5%

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

      1. clear-num N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

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

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

      8. unpow1/2 N/A

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

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

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

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

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

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

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

      12. /-lowering-/.f64 N/A

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

      13. sin-lowering-sin.f64 82.7%

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

   9. Applied egg-rr 82.7%

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

   if 13.199999999999999 < F

   1. Initial program 67.0%

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

      1. +-commutative 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 77.0%

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

   5. Taylor expanded in 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 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7.8 \cdot 10^{-52}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{-45}:\\ \;\;\;\;\frac{x \cdot \cos B}{0 - \sin B}\\ \mathbf{elif}\;F \leq 13.2:\\ \;\;\;\;\frac{1}{\frac{\sin B}{F} \cdot \sqrt{F \cdot F + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 85.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2.8 \cdot 10^{-52}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1.25 \cdot 10^{-47}:\\ \;\;\;\;\frac{x \cdot \cos B}{0 - \sin B}\\ \mathbf{elif}\;F \leq 6.9:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{F \cdot F + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -2.8e-52)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.25e-47)
       (/ (* x (cos B)) (- 0.0 (sin B)))
       (if (<= F 6.9)
         (/ F (* (sin B) (sqrt (+ (* F F) 2.0))))
         (- (/ 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 <= -2.8e-52) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.25e-47) {
		tmp = (x * cos(B)) / (0.0 - sin(B));
	} else if (F <= 6.9) {
		tmp = F / (sin(B) * sqrt(((F * F) + 2.0)));
	} 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 <= (-2.8d-52)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.25d-47) then
        tmp = (x * cos(b)) / (0.0d0 - sin(b))
    else if (f <= 6.9d0) then
        tmp = f / (sin(b) * sqrt(((f * f) + 2.0d0)))
    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 <= -2.8e-52) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.25e-47) {
		tmp = (x * Math.cos(B)) / (0.0 - Math.sin(B));
	} else if (F <= 6.9) {
		tmp = F / (Math.sin(B) * Math.sqrt(((F * F) + 2.0)));
	} 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 <= -2.8e-52:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.25e-47:
		tmp = (x * math.cos(B)) / (0.0 - math.sin(B))
	elif F <= 6.9:
		tmp = F / (math.sin(B) * math.sqrt(((F * F) + 2.0)))
	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 <= -2.8e-52)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.25e-47)
		tmp = Float64(Float64(x * cos(B)) / Float64(0.0 - sin(B)));
	elseif (F <= 6.9)
		tmp = Float64(F / Float64(sin(B) * sqrt(Float64(Float64(F * F) + 2.0))));
	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 <= -2.8e-52)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.25e-47)
		tmp = (x * cos(B)) / (0.0 - sin(B));
	elseif (F <= 6.9)
		tmp = F / (sin(B) * sqrt(((F * F) + 2.0)));
	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, -2.8e-52], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.25e-47], N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[(0.0 - N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 6.9], N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.79999999999999995e-52

   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. 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 79.2%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   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 95.1%

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

   7. Simplified 95.1%

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

   if -2.79999999999999995e-52 < F < 1.25000000000000003e-47

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      6. sin-lowering-sin.f64 77.0%

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

   5. Simplified 77.0%

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

   if 1.25000000000000003e-47 < F < 6.9000000000000004

   1. Initial program 98.8%

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

   3. Taylor expanded in 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 82.3%

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

   5. Simplified 82.3%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} \]
   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. div-inv N/A

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

      5. metadata-eval N/A

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

      6. sqrt-div N/A

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

      7. clear-num N/A

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

      8. /-rgt-identity N/A

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

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

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

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

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

      11. pow1/2 N/A

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

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

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

      13. +-commutative N/A

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

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

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

      15. *-lowering-*.f64 82.5%

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

   7. Applied egg-rr 82.5%

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

      1. *-commutative N/A

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

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

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

      3. unpow1/2 N/A

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

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

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

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

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

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

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

      7. sin-lowering-sin.f64 82.5%

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

   9. Applied egg-rr 82.5%

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

   if 6.9000000000000004 < F

   1. Initial program 67.0%

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

      1. +-commutative 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 77.0%

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

   5. Taylor expanded in 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 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.8 \cdot 10^{-52}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.25 \cdot 10^{-47}:\\ \;\;\;\;\frac{x \cdot \cos B}{0 - \sin B}\\ \mathbf{elif}\;F \leq 6.9:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{F \cdot F + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 84.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -7.8 \cdot 10^{-52}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 2.2 \cdot 10^{-49}:\\ \;\;\;\;\frac{x \cdot \cos B}{0 - \sin B}\\ \mathbf{elif}\;F \leq 0.047:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -7.8e-52)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 2.2e-49)
       (/ (* x (cos B)) (- 0.0 (sin B)))
       (if (<= F 0.047)
         (/ F (* (sin B) (sqrt 2.0)))
         (- (/ 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 <= -7.8e-52) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 2.2e-49) {
		tmp = (x * cos(B)) / (0.0 - sin(B));
	} else if (F <= 0.047) {
		tmp = F / (sin(B) * sqrt(2.0));
	} 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 <= (-7.8d-52)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 2.2d-49) then
        tmp = (x * cos(b)) / (0.0d0 - sin(b))
    else if (f <= 0.047d0) then
        tmp = f / (sin(b) * sqrt(2.0d0))
    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 <= -7.8e-52) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 2.2e-49) {
		tmp = (x * Math.cos(B)) / (0.0 - Math.sin(B));
	} else if (F <= 0.047) {
		tmp = F / (Math.sin(B) * Math.sqrt(2.0));
	} 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 <= -7.8e-52:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 2.2e-49:
		tmp = (x * math.cos(B)) / (0.0 - math.sin(B))
	elif F <= 0.047:
		tmp = F / (math.sin(B) * math.sqrt(2.0))
	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 <= -7.8e-52)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 2.2e-49)
		tmp = Float64(Float64(x * cos(B)) / Float64(0.0 - sin(B)));
	elseif (F <= 0.047)
		tmp = Float64(F / Float64(sin(B) * sqrt(2.0)));
	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 <= -7.8e-52)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 2.2e-49)
		tmp = (x * cos(B)) / (0.0 - sin(B));
	elseif (F <= 0.047)
		tmp = F / (sin(B) * sqrt(2.0));
	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, -7.8e-52], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 2.2e-49], N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[(0.0 - N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.047], N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -7.80000000000000036e-52

   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. 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 79.2%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   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 95.1%

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

   7. Simplified 95.1%

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

   if -7.80000000000000036e-52 < F < 2.1999999999999999e-49

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      6. sin-lowering-sin.f64 77.0%

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

   5. Simplified 77.0%

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

   if 2.1999999999999999e-49 < F < 0.047

   1. Initial program 98.8%

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

   3. Taylor expanded in 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 82.3%

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

   5. Simplified 82.3%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} \]
   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. div-inv N/A

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

      5. metadata-eval N/A

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

      6. sqrt-div N/A

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

      7. clear-num N/A

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

      8. /-rgt-identity N/A

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

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

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

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

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

      11. pow1/2 N/A

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

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

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

      13. +-commutative N/A

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

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

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

      15. *-lowering-*.f64 82.5%

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

   7. Applied egg-rr 82.5%

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

   8. Taylor expanded in F around 0

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

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

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

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

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

      3. sqrt-lowering-sqrt.f64 75.6%

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

   10. Simplified 75.6%

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

   if 0.047 < F

   1. Initial program 67.0%

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

      1. +-commutative 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 77.0%

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

   5. Taylor expanded in 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 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7.8 \cdot 10^{-52}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.2 \cdot 10^{-49}:\\ \;\;\;\;\frac{x \cdot \cos B}{0 - \sin B}\\ \mathbf{elif}\;F \leq 0.047:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 76.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -8.4 \cdot 10^{-69}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 3.05 \cdot 10^{-92}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{F \cdot F + \left(2 + x \cdot 2\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -8.4e-69)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 3.05e-92)
       (/ (- (/ F (sqrt (+ (* F F) (+ 2.0 (* x 2.0))))) x) B)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -8.4e-69) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 3.05e-92) {
		tmp = ((F / sqrt(((F * F) + (2.0 + (x * 2.0))))) - x) / B;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-8.4d-69)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 3.05d-92) then
        tmp = ((f / sqrt(((f * f) + (2.0d0 + (x * 2.0d0))))) - x) / b
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -8.4e-69) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 3.05e-92) {
		tmp = ((F / Math.sqrt(((F * F) + (2.0 + (x * 2.0))))) - x) / B;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -8.4e-69:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 3.05e-92:
		tmp = ((F / math.sqrt(((F * F) + (2.0 + (x * 2.0))))) - x) / B
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -8.4e-69)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 3.05e-92)
		tmp = Float64(Float64(Float64(F / sqrt(Float64(Float64(F * F) + Float64(2.0 + Float64(x * 2.0))))) - x) / B);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -8.4e-69)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 3.05e-92)
		tmp = ((F / sqrt(((F * F) + (2.0 + (x * 2.0))))) - x) / B;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -8.4e-69], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 3.05e-92], 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[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -8.3999999999999999e-69

   1. Initial program 64.8%

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

      1. +-commutative 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 79.6%

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

   5. Taylor expanded in 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 95.3%

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

   7. Simplified 95.3%

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

   if -8.3999999999999999e-69 < F < 3.04999999999999994e-92

   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 51.9%

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

   5. Simplified 51.9%

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

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

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

   7. Applied egg-rr 51.9%

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

   if 3.04999999999999994e-92 < F

   1. Initial program 73.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 81.6%

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

   5. Taylor expanded in 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 87.1%

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

   7. Simplified 87.1%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -8.4 \cdot 10^{-69}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 3.05 \cdot 10^{-92}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{F \cdot F + \left(2 + x \cdot 2\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 48.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.095:\\ \;\;\;\;\frac{\frac{F}{\sqrt{F \cdot F + \left(2 + x \cdot 2\right)}} - x}{B}\\ \mathbf{elif}\;B \leq 1.75 \cdot 10^{+112}:\\ \;\;\;\;\frac{1}{\sin B} \cdot \left(F \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 0.095)
   (/ (- (/ F (sqrt (+ (* F F) (+ 2.0 (* x 2.0))))) x) B)
   (if (<= B 1.75e+112)
     (* (/ 1.0 (sin B)) (* F (sqrt 0.5)))
     (/ -1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 0.095) {
		tmp = ((F / sqrt(((F * F) + (2.0 + (x * 2.0))))) - x) / B;
	} else if (B <= 1.75e+112) {
		tmp = (1.0 / sin(B)) * (F * sqrt(0.5));
	} 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 (b <= 0.095d0) then
        tmp = ((f / sqrt(((f * f) + (2.0d0 + (x * 2.0d0))))) - x) / b
    else if (b <= 1.75d+112) then
        tmp = (1.0d0 / sin(b)) * (f * sqrt(0.5d0))
    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 (B <= 0.095) {
		tmp = ((F / Math.sqrt(((F * F) + (2.0 + (x * 2.0))))) - x) / B;
	} else if (B <= 1.75e+112) {
		tmp = (1.0 / Math.sin(B)) * (F * Math.sqrt(0.5));
	} else {
		tmp = -1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if B <= 0.095:
		tmp = ((F / math.sqrt(((F * F) + (2.0 + (x * 2.0))))) - x) / B
	elif B <= 1.75e+112:
		tmp = (1.0 / math.sin(B)) * (F * math.sqrt(0.5))
	else:
		tmp = -1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 0.095)
		tmp = Float64(Float64(Float64(F / sqrt(Float64(Float64(F * F) + Float64(2.0 + Float64(x * 2.0))))) - x) / B);
	elseif (B <= 1.75e+112)
		tmp = Float64(Float64(1.0 / sin(B)) * Float64(F * sqrt(0.5)));
	else
		tmp = Float64(-1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (B <= 0.095)
		tmp = ((F / sqrt(((F * F) + (2.0 + (x * 2.0))))) - x) / B;
	elseif (B <= 1.75e+112)
		tmp = (1.0 / sin(B)) * (F * sqrt(0.5));
	else
		tmp = -1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 0.095], 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], If[LessEqual[B, 1.75e+112], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 0.095000000000000001

   1. Initial program 74.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 54.4%

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

   5. Simplified 54.4%

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

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

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

   7. Applied egg-rr 54.5%

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

   if 0.095000000000000001 < B < 1.74999999999999998e112

   1. Initial program 94.9%

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

   3. Taylor expanded in 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 35.4%

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

   5. Simplified 35.4%

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

   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 29.0%

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

   8. Simplified 29.0%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

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

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

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

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

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

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

      7. sqrt-lowering-sqrt.f64 29.2%

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

   10. Applied egg-rr 29.2%

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

   if 1.74999999999999998e112 < B

   1. Initial program 92.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 39.9%

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

   5. Simplified 39.9%

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

   6. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   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 28.1%

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

   8. Simplified 28.1%

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

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 0.095:\\ \;\;\;\;\frac{\frac{F}{\sqrt{F \cdot F + \left(2 + x \cdot 2\right)}} - x}{B}\\ \mathbf{elif}\;B \leq 1.75 \cdot 10^{+112}:\\ \;\;\;\;\frac{1}{\sin B} \cdot \left(F \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 48.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.1:\\ \;\;\;\;\frac{\frac{F}{\sqrt{F \cdot F + \left(2 + x \cdot 2\right)}} - x}{B}\\ \mathbf{elif}\;B \leq 7.2 \cdot 10^{+111}:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 0.1)
   (/ (- (/ F (sqrt (+ (* F F) (+ 2.0 (* x 2.0))))) x) B)
   (if (<= B 7.2e+111) (/ F (* (sin B) (sqrt 2.0))) (/ -1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 0.1) {
		tmp = ((F / sqrt(((F * F) + (2.0 + (x * 2.0))))) - x) / B;
	} else if (B <= 7.2e+111) {
		tmp = F / (sin(B) * sqrt(2.0));
	} 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) :: tmp
    if (b <= 0.1d0) then
        tmp = ((f / sqrt(((f * f) + (2.0d0 + (x * 2.0d0))))) - x) / b
    else if (b <= 7.2d+111) then
        tmp = f / (sin(b) * sqrt(2.0d0))
    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 (B <= 0.1) {
		tmp = ((F / Math.sqrt(((F * F) + (2.0 + (x * 2.0))))) - x) / B;
	} else if (B <= 7.2e+111) {
		tmp = F / (Math.sin(B) * Math.sqrt(2.0));
	} else {
		tmp = -1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if B <= 0.1:
		tmp = ((F / math.sqrt(((F * F) + (2.0 + (x * 2.0))))) - x) / B
	elif B <= 7.2e+111:
		tmp = F / (math.sin(B) * math.sqrt(2.0))
	else:
		tmp = -1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 0.1)
		tmp = Float64(Float64(Float64(F / sqrt(Float64(Float64(F * F) + Float64(2.0 + Float64(x * 2.0))))) - x) / B);
	elseif (B <= 7.2e+111)
		tmp = Float64(F / Float64(sin(B) * sqrt(2.0)));
	else
		tmp = Float64(-1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (B <= 0.1)
		tmp = ((F / sqrt(((F * F) + (2.0 + (x * 2.0))))) - x) / B;
	elseif (B <= 7.2e+111)
		tmp = F / (sin(B) * sqrt(2.0));
	else
		tmp = -1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 0.1], 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], If[LessEqual[B, 7.2e+111], N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 0.10000000000000001

   1. Initial program 74.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 54.4%

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

   5. Simplified 54.4%

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

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

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

   7. Applied egg-rr 54.5%

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

   if 0.10000000000000001 < B < 7.2000000000000004e111

   1. Initial program 94.9%

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

   3. Taylor expanded in 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 35.4%

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

   5. Simplified 35.4%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} \]
   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. div-inv N/A

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

      5. metadata-eval N/A

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

      6. sqrt-div N/A

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

      7. clear-num N/A

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

      8. /-rgt-identity N/A

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

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

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

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

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

      11. pow1/2 N/A

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

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

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

      13. +-commutative N/A

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

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

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

      15. *-lowering-*.f64 35.4%

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

   7. Applied egg-rr 35.4%

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

   8. Taylor expanded in F around 0

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

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

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

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

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

      3. sqrt-lowering-sqrt.f64 29.0%

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

   10. Simplified 29.0%

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

   if 7.2000000000000004e111 < B

   1. Initial program 92.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 39.9%

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

   5. Simplified 39.9%

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

   6. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   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 28.1%

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

   8. Simplified 28.1%

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

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 0.1:\\ \;\;\;\;\frac{\frac{F}{\sqrt{F \cdot F + \left(2 + x \cdot 2\right)}} - x}{B}\\ \mathbf{elif}\;B \leq 7.2 \cdot 10^{+111}:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 48.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.1:\\ \;\;\;\;\frac{\frac{F}{\sqrt{F \cdot F + \left(2 + x \cdot 2\right)}} - x}{B}\\ \mathbf{elif}\;B \leq 7.2 \cdot 10^{+111}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 0.1)
   (/ (- (/ F (sqrt (+ (* F F) (+ 2.0 (* x 2.0))))) x) B)
   (if (<= B 7.2e+111) (* F (/ (sqrt 0.5) (sin B))) (/ -1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 0.1) {
		tmp = ((F / sqrt(((F * F) + (2.0 + (x * 2.0))))) - x) / B;
	} else if (B <= 7.2e+111) {
		tmp = F * (sqrt(0.5) / sin(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 (b <= 0.1d0) then
        tmp = ((f / sqrt(((f * f) + (2.0d0 + (x * 2.0d0))))) - x) / b
    else if (b <= 7.2d+111) then
        tmp = f * (sqrt(0.5d0) / sin(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 (B <= 0.1) {
		tmp = ((F / Math.sqrt(((F * F) + (2.0 + (x * 2.0))))) - x) / B;
	} else if (B <= 7.2e+111) {
		tmp = F * (Math.sqrt(0.5) / Math.sin(B));
	} else {
		tmp = -1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if B <= 0.1:
		tmp = ((F / math.sqrt(((F * F) + (2.0 + (x * 2.0))))) - x) / B
	elif B <= 7.2e+111:
		tmp = F * (math.sqrt(0.5) / math.sin(B))
	else:
		tmp = -1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 0.1)
		tmp = Float64(Float64(Float64(F / sqrt(Float64(Float64(F * F) + Float64(2.0 + Float64(x * 2.0))))) - x) / B);
	elseif (B <= 7.2e+111)
		tmp = Float64(F * Float64(sqrt(0.5) / sin(B)));
	else
		tmp = Float64(-1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (B <= 0.1)
		tmp = ((F / sqrt(((F * F) + (2.0 + (x * 2.0))))) - x) / B;
	elseif (B <= 7.2e+111)
		tmp = F * (sqrt(0.5) / sin(B));
	else
		tmp = -1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 0.1], 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], If[LessEqual[B, 7.2e+111], N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 0.10000000000000001

   1. Initial program 74.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 54.4%

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

   5. Simplified 54.4%

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

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

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

   7. Applied egg-rr 54.5%

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

   if 0.10000000000000001 < B < 7.2000000000000004e111

   1. Initial program 94.9%

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

   3. Taylor expanded in 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 35.4%

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

   5. Simplified 35.4%

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

   6. Taylor expanded in F around 0

      \[\leadsto \mathsf{*.f64}\left(F, \color{blue}{\left(\frac{\sqrt{\frac{1}{2}}}{\sin B}\right)}\right) \]
   7. Step-by-step derivation

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

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

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

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

      3. sin-lowering-sin.f64 29.1%

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

   8. Simplified 29.1%

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

   if 7.2000000000000004e111 < B

   1. Initial program 92.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 39.9%

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

   5. Simplified 39.9%

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

   6. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   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 28.1%

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

   8. Simplified 28.1%

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

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 0.1:\\ \;\;\;\;\frac{\frac{F}{\sqrt{F \cdot F + \left(2 + x \cdot 2\right)}} - x}{B}\\ \mathbf{elif}\;B \leq 7.2 \cdot 10^{+111}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 58.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.102:\\ \;\;\;\;\frac{\left(\sqrt{\frac{1}{2 + \left(F \cdot F + x \cdot 2\right)}} \cdot \left(F + 0.16666666666666666 \cdot \left(F \cdot \left(B \cdot B\right)\right)\right) + B \cdot \left(B \cdot \left(x \cdot 0.3333333333333333\right)\right)\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 0.102)
   (/
    (-
     (+
      (*
       (sqrt (/ 1.0 (+ 2.0 (+ (* F F) (* x 2.0)))))
       (+ F (* 0.16666666666666666 (* F (* B B)))))
      (* B (* B (* x 0.3333333333333333))))
     x)
    B)
   (- (/ -1.0 (sin B)) (/ x (tan B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 0.102) {
		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 = (-1.0 / sin(B)) - (x / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (b <= 0.102d0) 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 = ((-1.0d0) / sin(b)) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (B <= 0.102) {
		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 = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if B <= 0.102:
		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 = (-1.0 / math.sin(B)) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 0.102)
		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(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (B <= 0.102)
		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 = (-1.0 / sin(B)) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 0.102], 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[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 0.101999999999999993

   1. Initial program 74.1%

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

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \frac{1}{3} \cdot x\right)\right) - x}{B}} \]
   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 53.8%

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

   if 0.101999999999999993 < B

   1. Initial program 93.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 93.1%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(F \cdot F + \left(2 + x \cdot 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   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 59.2%

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

   7. Simplified 59.2%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 0.102:\\ \;\;\;\;\frac{\left(\sqrt{\frac{1}{2 + \left(F \cdot F + x \cdot 2\right)}} \cdot \left(F + 0.16666666666666666 \cdot \left(F \cdot \left(B \cdot B\right)\right)\right) + B \cdot \left(B \cdot \left(x \cdot 0.3333333333333333\right)\right)\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 48.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 56:\\ \;\;\;\;\frac{\left(\sqrt{\frac{1}{2 + \left(F \cdot F + x \cdot 2\right)}} \cdot \left(F + 0.16666666666666666 \cdot \left(F \cdot \left(B \cdot B\right)\right)\right) + B \cdot \left(B \cdot \left(x \cdot 0.3333333333333333\right)\right)\right) - x}{B}\\ \mathbf{elif}\;B \leq 1.08 \cdot 10^{+130}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 56.0)
   (/
    (-
     (+
      (*
       (sqrt (/ 1.0 (+ 2.0 (+ (* F F) (* x 2.0)))))
       (+ F (* 0.16666666666666666 (* F (* B B)))))
      (* B (* B (* x 0.3333333333333333))))
     x)
    B)
   (if (<= B 1.08e+130) (/ 1.0 (sin B)) (/ -1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 56.0) {
		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 if (B <= 1.08e+130) {
		tmp = 1.0 / sin(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 (b <= 56.0d0) 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 if (b <= 1.08d+130) then
        tmp = 1.0d0 / sin(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 (B <= 56.0) {
		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 if (B <= 1.08e+130) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = -1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if B <= 56.0:
		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
	elif B <= 1.08e+130:
		tmp = 1.0 / math.sin(B)
	else:
		tmp = -1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 56.0)
		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);
	elseif (B <= 1.08e+130)
		tmp = Float64(1.0 / sin(B));
	else
		tmp = Float64(-1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (B <= 56.0)
		tmp = (((sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0))))) * (F + (0.16666666666666666 * (F * (B * B))))) + (B * (B * (x * 0.3333333333333333)))) - x) / B;
	elseif (B <= 1.08e+130)
		tmp = 1.0 / sin(B);
	else
		tmp = -1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 56.0], 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], If[LessEqual[B, 1.08e+130], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 56

   1. Initial program 74.2%

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

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \frac{1}{3} \cdot x\right)\right) - x}{B}} \]
   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 53.6%

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

   if 56 < B < 1.08e130

   1. Initial program 95.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 42.5%

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

   5. Simplified 42.5%

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

   6. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
   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 20.3%

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

   8. Simplified 20.3%

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

   if 1.08e130 < B

   1. Initial program 91.7%

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

   3. Taylor expanded in 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 36.6%

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

   5. Simplified 36.6%

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

   6. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   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 29.6%

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

   8. Simplified 29.6%

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

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 56:\\ \;\;\;\;\frac{\left(\sqrt{\frac{1}{2 + \left(F \cdot F + x \cdot 2\right)}} \cdot \left(F + 0.16666666666666666 \cdot \left(F \cdot \left(B \cdot B\right)\right)\right) + B \cdot \left(B \cdot \left(x \cdot 0.3333333333333333\right)\right)\right) - x}{B}\\ \mathbf{elif}\;B \leq 1.08 \cdot 10^{+130}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 45.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -4.6 \cdot 10^{+103}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq -4.6 \cdot 10^{-8}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.1 \cdot 10^{-52}:\\ \;\;\;\;0 - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.45:\\ \;\;\;\;\frac{F}{B \cdot \sqrt{F \cdot F + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4.6e+103)
   (/ (- -1.0 x) B)
   (if (<= F -4.6e-8)
     (/ -1.0 (sin B))
     (if (<= F 1.1e-52)
       (- 0.0 (/ x B))
       (if (<= F 1.45) (/ F (* B (sqrt (+ (* F F) 2.0)))) (/ (- 1.0 x) B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.6e+103) {
		tmp = (-1.0 - x) / B;
	} else if (F <= -4.6e-8) {
		tmp = -1.0 / sin(B);
	} else if (F <= 1.1e-52) {
		tmp = 0.0 - (x / B);
	} else if (F <= 1.45) {
		tmp = F / (B * sqrt(((F * F) + 2.0)));
	} 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 <= (-4.6d+103)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= (-4.6d-8)) then
        tmp = (-1.0d0) / sin(b)
    else if (f <= 1.1d-52) then
        tmp = 0.0d0 - (x / b)
    else if (f <= 1.45d0) then
        tmp = f / (b * sqrt(((f * f) + 2.0d0)))
    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 <= -4.6e+103) {
		tmp = (-1.0 - x) / B;
	} else if (F <= -4.6e-8) {
		tmp = -1.0 / Math.sin(B);
	} else if (F <= 1.1e-52) {
		tmp = 0.0 - (x / B);
	} else if (F <= 1.45) {
		tmp = F / (B * Math.sqrt(((F * F) + 2.0)));
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -4.6e+103:
		tmp = (-1.0 - x) / B
	elif F <= -4.6e-8:
		tmp = -1.0 / math.sin(B)
	elif F <= 1.1e-52:
		tmp = 0.0 - (x / B)
	elif F <= 1.45:
		tmp = F / (B * math.sqrt(((F * F) + 2.0)))
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -4.6e+103)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= -4.6e-8)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 1.1e-52)
		tmp = Float64(0.0 - Float64(x / B));
	elseif (F <= 1.45)
		tmp = Float64(F / Float64(B * sqrt(Float64(Float64(F * F) + 2.0))));
	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 <= -4.6e+103)
		tmp = (-1.0 - x) / B;
	elseif (F <= -4.6e-8)
		tmp = -1.0 / sin(B);
	elseif (F <= 1.1e-52)
		tmp = 0.0 - (x / B);
	elseif (F <= 1.45)
		tmp = F / (B * sqrt(((F * F) + 2.0)));
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -4.6e+103], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, -4.6e-8], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.1e-52], N[(0.0 - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.45], N[(F / N[(B * N[Sqrt[N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -4.60000000000000017e103

   1. Initial program 48.7%

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

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   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 39.4%

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

   5. Simplified 39.4%

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

   6. Taylor expanded in F around -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 55.1%

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

   8. Simplified 55.1%

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

   if -4.60000000000000017e103 < F < -4.6000000000000002e-8

   1. Initial program 92.3%

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

   3. Taylor expanded in 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 69.7%

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

   5. Simplified 69.7%

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

   6. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   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 60.8%

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

   8. Simplified 60.8%

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

   if -4.6000000000000002e-8 < F < 1.10000000000000005e-52

   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 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. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

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

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

      4. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(0 - x\right), B\right) \]

      5. --lowering--.f64 39.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, x\right), B\right) \]

   8. Simplified 39.4%

      \[\leadsto \color{blue}{\frac{0 - x}{B}} \]

   if 1.10000000000000005e-52 < F < 1.44999999999999996

   1. Initial program 98.9%

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

   3. Taylor expanded in 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 75.0%

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

   5. Simplified 75.0%

      \[\leadsto \color{blue}{F \cdot \frac{\sqrt{\frac{1}{2 + F \cdot F}}}{\sin B}} \]
   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. div-inv N/A

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

      5. metadata-eval N/A

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

      6. sqrt-div N/A

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

      7. clear-num N/A

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

      8. /-rgt-identity N/A

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

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

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

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

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

      11. pow1/2 N/A

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

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

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

      13. +-commutative N/A

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

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

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

      15. *-lowering-*.f64 75.2%

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

   7. Applied egg-rr 75.2%

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

   8. Taylor expanded in B around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 49.0%

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

   10. Simplified 49.0%

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

   if 1.44999999999999996 < F

   1. Initial program 67.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 39.5%

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

   5. Simplified 39.5%

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

   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 51.8%

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

   8. Simplified 51.8%

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

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.6 \cdot 10^{+103}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq -4.6 \cdot 10^{-8}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.1 \cdot 10^{-52}:\\ \;\;\;\;0 - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.45:\\ \;\;\;\;\frac{F}{B \cdot \sqrt{F \cdot F + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 45.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.3 \cdot 10^{+103}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq -3.4 \cdot 10^{-8}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 9 \cdot 10^{-53}:\\ \;\;\;\;0 - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.196:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.3e+103)
   (/ (- -1.0 x) B)
   (if (<= F -3.4e-8)
     (/ -1.0 (sin B))
     (if (<= F 9e-53)
       (- 0.0 (/ x B))
       (if (<= F 0.196) (/ (* F (sqrt 0.5)) B) (/ (- 1.0 x) B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.3e+103) {
		tmp = (-1.0 - x) / B;
	} else if (F <= -3.4e-8) {
		tmp = -1.0 / sin(B);
	} else if (F <= 9e-53) {
		tmp = 0.0 - (x / B);
	} else if (F <= 0.196) {
		tmp = (F * sqrt(0.5)) / 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 <= (-3.3d+103)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= (-3.4d-8)) then
        tmp = (-1.0d0) / sin(b)
    else if (f <= 9d-53) then
        tmp = 0.0d0 - (x / b)
    else if (f <= 0.196d0) then
        tmp = (f * sqrt(0.5d0)) / 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 <= -3.3e+103) {
		tmp = (-1.0 - x) / B;
	} else if (F <= -3.4e-8) {
		tmp = -1.0 / Math.sin(B);
	} else if (F <= 9e-53) {
		tmp = 0.0 - (x / B);
	} else if (F <= 0.196) {
		tmp = (F * Math.sqrt(0.5)) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.3e+103:
		tmp = (-1.0 - x) / B
	elif F <= -3.4e-8:
		tmp = -1.0 / math.sin(B)
	elif F <= 9e-53:
		tmp = 0.0 - (x / B)
	elif F <= 0.196:
		tmp = (F * math.sqrt(0.5)) / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.3e+103)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= -3.4e-8)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 9e-53)
		tmp = Float64(0.0 - Float64(x / B));
	elseif (F <= 0.196)
		tmp = Float64(Float64(F * sqrt(0.5)) / 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 <= -3.3e+103)
		tmp = (-1.0 - x) / B;
	elseif (F <= -3.4e-8)
		tmp = -1.0 / sin(B);
	elseif (F <= 9e-53)
		tmp = 0.0 - (x / B);
	elseif (F <= 0.196)
		tmp = (F * sqrt(0.5)) / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.3e+103], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, -3.4e-8], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 9e-53], N[(0.0 - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.196], N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -3.30000000000000009e103

   1. Initial program 48.7%

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

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   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 39.4%

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

   5. Simplified 39.4%

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

   6. Taylor expanded in F around -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 55.1%

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

   8. Simplified 55.1%

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

   if -3.30000000000000009e103 < F < -3.4e-8

   1. Initial program 92.3%

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

   3. Taylor expanded in 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 69.7%

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

   5. Simplified 69.7%

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

   6. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   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 60.8%

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

   8. Simplified 60.8%

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

   if -3.4e-8 < F < 8.9999999999999997e-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. 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. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

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

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

      4. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(0 - x\right), B\right) \]

      5. --lowering--.f64 39.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, x\right), B\right) \]

   8. Simplified 39.4%

      \[\leadsto \color{blue}{\frac{0 - x}{B}} \]

   if 8.9999999999999997e-53 < F < 0.19600000000000001

   1. Initial program 98.9%

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

   3. Taylor expanded in 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 75.0%

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

   5. Simplified 75.0%

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

   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 68.6%

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

   8. Simplified 68.6%

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

   9. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right), \color{blue}{B}\right) \]
   10. Step-by-step derivation

   11. Simplified 43.2%

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

   if 0.19600000000000001 < F

   1. Initial program 67.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 39.5%

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

   5. Simplified 39.5%

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

   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 51.8%

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

   8. Simplified 51.8%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.3 \cdot 10^{+103}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq -3.4 \cdot 10^{-8}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 9 \cdot 10^{-53}:\\ \;\;\;\;0 - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.196:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 48.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.435:\\ \;\;\;\;\frac{\frac{F}{\sqrt{F \cdot F + \left(2 + x \cdot 2\right)}} - x}{B}\\ \mathbf{elif}\;B \leq 8.2 \cdot 10^{+129}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 0.435)
   (/ (- (/ F (sqrt (+ (* F F) (+ 2.0 (* x 2.0))))) x) B)
   (if (<= B 8.2e+129) (/ 1.0 (sin B)) (/ -1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 0.435) {
		tmp = ((F / sqrt(((F * F) + (2.0 + (x * 2.0))))) - x) / B;
	} else if (B <= 8.2e+129) {
		tmp = 1.0 / sin(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 (b <= 0.435d0) then
        tmp = ((f / sqrt(((f * f) + (2.0d0 + (x * 2.0d0))))) - x) / b
    else if (b <= 8.2d+129) then
        tmp = 1.0d0 / sin(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 (B <= 0.435) {
		tmp = ((F / Math.sqrt(((F * F) + (2.0 + (x * 2.0))))) - x) / B;
	} else if (B <= 8.2e+129) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = -1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if B <= 0.435:
		tmp = ((F / math.sqrt(((F * F) + (2.0 + (x * 2.0))))) - x) / B
	elif B <= 8.2e+129:
		tmp = 1.0 / math.sin(B)
	else:
		tmp = -1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 0.435)
		tmp = Float64(Float64(Float64(F / sqrt(Float64(Float64(F * F) + Float64(2.0 + Float64(x * 2.0))))) - x) / B);
	elseif (B <= 8.2e+129)
		tmp = Float64(1.0 / sin(B));
	else
		tmp = Float64(-1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (B <= 0.435)
		tmp = ((F / sqrt(((F * F) + (2.0 + (x * 2.0))))) - x) / B;
	elseif (B <= 8.2e+129)
		tmp = 1.0 / sin(B);
	else
		tmp = -1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 0.435], 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], If[LessEqual[B, 8.2e+129], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 0.434999999999999998

   1. Initial program 74.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 54.4%

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

   5. Simplified 54.4%

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

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

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

   7. Applied egg-rr 54.5%

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

   if 0.434999999999999998 < B < 8.2000000000000005e129

   1. Initial program 95.3%

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

   3. Taylor expanded in 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 40.8%

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

   5. Simplified 40.8%

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

   6. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
   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 19.5%

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

   8. Simplified 19.5%

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

   if 8.2000000000000005e129 < B

   1. Initial program 91.7%

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

   3. Taylor expanded in 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 36.6%

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

   5. Simplified 36.6%

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

   6. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   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 29.6%

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

   8. Simplified 29.6%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 0.435:\\ \;\;\;\;\frac{\frac{F}{\sqrt{F \cdot F + \left(2 + x \cdot 2\right)}} - x}{B}\\ \mathbf{elif}\;B \leq 8.2 \cdot 10^{+129}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 48.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 1.02:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{F \cdot F + 2}} - x}{B}\\ \mathbf{elif}\;B \leq 5.8 \cdot 10^{+131}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 1.02)
   (/ (- (* F (sqrt (/ 1.0 (+ (* F F) 2.0)))) x) B)
   (if (<= B 5.8e+131) (/ 1.0 (sin B)) (/ -1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 1.02) {
		tmp = ((F * sqrt((1.0 / ((F * F) + 2.0)))) - x) / B;
	} else if (B <= 5.8e+131) {
		tmp = 1.0 / sin(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 (b <= 1.02d0) then
        tmp = ((f * sqrt((1.0d0 / ((f * f) + 2.0d0)))) - x) / b
    else if (b <= 5.8d+131) then
        tmp = 1.0d0 / sin(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 (B <= 1.02) {
		tmp = ((F * Math.sqrt((1.0 / ((F * F) + 2.0)))) - x) / B;
	} else if (B <= 5.8e+131) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = -1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if B <= 1.02:
		tmp = ((F * math.sqrt((1.0 / ((F * F) + 2.0)))) - x) / B
	elif B <= 5.8e+131:
		tmp = 1.0 / math.sin(B)
	else:
		tmp = -1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 1.02)
		tmp = Float64(Float64(Float64(F * sqrt(Float64(1.0 / Float64(Float64(F * F) + 2.0)))) - x) / B);
	elseif (B <= 5.8e+131)
		tmp = Float64(1.0 / sin(B));
	else
		tmp = Float64(-1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (B <= 1.02)
		tmp = ((F * sqrt((1.0 / ((F * F) + 2.0)))) - x) / B;
	elseif (B <= 5.8e+131)
		tmp = 1.0 / sin(B);
	else
		tmp = -1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 1.02], N[(N[(N[(F * N[Sqrt[N[(1.0 / N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[B, 5.8e+131], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.02

   1. Initial program 74.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 54.4%

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

   5. Simplified 54.4%

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

   6. Taylor expanded in x around 0

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

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

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

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

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

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

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 54.4%

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

   8. Simplified 54.4%

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

   if 1.02 < B < 5.8000000000000002e131

   1. Initial program 95.3%

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

   3. Taylor expanded in 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 40.8%

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

   5. Simplified 40.8%

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

   6. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
   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 19.5%

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

   8. Simplified 19.5%

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

   if 5.8000000000000002e131 < B

   1. Initial program 91.7%

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

   3. Taylor expanded in 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 36.6%

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

   5. Simplified 36.6%

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

   6. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   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 29.6%

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

   8. Simplified 29.6%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.02:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{F \cdot F + 2}} - x}{B}\\ \mathbf{elif}\;B \leq 5.8 \cdot 10^{+131}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 44.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -4.6 \cdot 10^{+103}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq -3.4 \cdot 10^{-8}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 8.6 \cdot 10^{-38}:\\ \;\;\;\;0 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4.6e+103)
   (/ (- -1.0 x) B)
   (if (<= F -3.4e-8)
     (/ -1.0 (sin B))
     (if (<= F 8.6e-38) (- 0.0 (/ x B)) (/ (- 1.0 x) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.6e+103) {
		tmp = (-1.0 - x) / B;
	} else if (F <= -3.4e-8) {
		tmp = -1.0 / sin(B);
	} else if (F <= 8.6e-38) {
		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 <= (-4.6d+103)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= (-3.4d-8)) then
        tmp = (-1.0d0) / sin(b)
    else if (f <= 8.6d-38) 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 <= -4.6e+103) {
		tmp = (-1.0 - x) / B;
	} else if (F <= -3.4e-8) {
		tmp = -1.0 / Math.sin(B);
	} else if (F <= 8.6e-38) {
		tmp = 0.0 - (x / B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -4.6e+103:
		tmp = (-1.0 - x) / B
	elif F <= -3.4e-8:
		tmp = -1.0 / math.sin(B)
	elif F <= 8.6e-38:
		tmp = 0.0 - (x / B)
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -4.6e+103)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= -3.4e-8)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 8.6e-38)
		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 <= -4.6e+103)
		tmp = (-1.0 - x) / B;
	elseif (F <= -3.4e-8)
		tmp = -1.0 / sin(B);
	elseif (F <= 8.6e-38)
		tmp = 0.0 - (x / B);
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -4.6e+103], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, -3.4e-8], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 8.6e-38], N[(0.0 - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -4.60000000000000017e103

   1. Initial program 48.7%

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

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   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 39.4%

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

   5. Simplified 39.4%

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

   6. Taylor expanded in F around -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 55.1%

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

   8. Simplified 55.1%

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

   if -4.60000000000000017e103 < F < -3.4e-8

   1. Initial program 92.3%

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

   3. Taylor expanded in 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 69.7%

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

   5. Simplified 69.7%

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

   6. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   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 60.8%

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

   8. Simplified 60.8%

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

   if -3.4e-8 < F < 8.6000000000000004e-38

   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 48.7%

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

   5. Simplified 48.7%

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

   6. Taylor expanded in F around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

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

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

      4. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(0 - x\right), B\right) \]

      5. --lowering--.f64 38.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, x\right), B\right) \]

   8. Simplified 38.3%

      \[\leadsto \color{blue}{\frac{0 - x}{B}} \]

   if 8.6000000000000004e-38 < F

   1. Initial program 70.2%

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

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   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 40.6%

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

   5. Simplified 40.6%

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

   6. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
   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 47.3%

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

   8. Simplified 47.3%

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

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.6 \cdot 10^{+103}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq -3.4 \cdot 10^{-8}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 8.6 \cdot 10^{-38}:\\ \;\;\;\;0 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 44.3% accurate, 16.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -7.2:\\ \;\;\;\;\frac{\left(-1 + \frac{x + 1}{F} \cdot \frac{1}{F}\right) - x}{B}\\ \mathbf{elif}\;F \leq 1.95 \cdot 10^{-39}:\\ \;\;\;\;0 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -7.2)
   (/ (- (+ -1.0 (* (/ (+ x 1.0) F) (/ 1.0 F))) x) B)
   (if (<= F 1.95e-39) (- 0.0 (/ x B)) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -7.2) {
		tmp = ((-1.0 + (((x + 1.0) / F) * (1.0 / F))) - x) / B;
	} else if (F <= 1.95e-39) {
		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 <= (-7.2d0)) then
        tmp = (((-1.0d0) + (((x + 1.0d0) / f) * (1.0d0 / f))) - x) / b
    else if (f <= 1.95d-39) 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 <= -7.2) {
		tmp = ((-1.0 + (((x + 1.0) / F) * (1.0 / F))) - x) / B;
	} else if (F <= 1.95e-39) {
		tmp = 0.0 - (x / B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -7.2:
		tmp = ((-1.0 + (((x + 1.0) / F) * (1.0 / F))) - x) / B
	elif F <= 1.95e-39:
		tmp = 0.0 - (x / B)
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -7.2)
		tmp = Float64(Float64(Float64(-1.0 + Float64(Float64(Float64(x + 1.0) / F) * Float64(1.0 / F))) - x) / B);
	elseif (F <= 1.95e-39)
		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 <= -7.2)
		tmp = ((-1.0 + (((x + 1.0) / F) * (1.0 / F))) - x) / B;
	elseif (F <= 1.95e-39)
		tmp = 0.0 - (x / B);
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -7.2], N[(N[(N[(-1.0 + N[(N[(N[(x + 1.0), $MachinePrecision] / F), $MachinePrecision] * N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.95e-39], 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 < -7.20000000000000018

   1. Initial program 61.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 37.5%

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

   5. Simplified 37.5%

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

   6. Taylor expanded in F around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}} - 1\right)}, x\right), B\right) \]
   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. *-lowering-*.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 47.8%

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

   8. Simplified 47.8%

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

      1. associate-/r* N/A

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

      2. div-inv N/A

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

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

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

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

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

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

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. /-lowering-/.f64 47.8%

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

   10. Applied egg-rr 47.8%

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

   if -7.20000000000000018 < F < 1.95000000000000015e-39

   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 48.3%

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

   5. Simplified 48.3%

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

   6. Taylor expanded in F around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

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

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

      4. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(0 - x\right), B\right) \]

      5. --lowering--.f64 38.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, x\right), B\right) \]

   8. Simplified 38.0%

      \[\leadsto \color{blue}{\frac{0 - x}{B}} \]

   if 1.95000000000000015e-39 < F

   1. Initial program 70.2%

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

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   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 40.6%

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

   5. Simplified 40.6%

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

   6. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
   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 47.3%

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

   8. Simplified 47.3%

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

4. Final simplification 43.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7.2:\\ \;\;\;\;\frac{\left(-1 + \frac{x + 1}{F} \cdot \frac{1}{F}\right) - x}{B}\\ \mathbf{elif}\;F \leq 1.95 \cdot 10^{-39}:\\ \;\;\;\;0 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 44.2% accurate, 21.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.08 \cdot 10^{-69}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.05 \cdot 10^{-38}:\\ \;\;\;\;0 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.08e-69)
   (/ (- -1.0 x) B)
   (if (<= F 1.05e-38) (- 0.0 (/ x B)) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.08e-69) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.05e-38) {
		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 <= (-1.08d-69)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 1.05d-38) 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 <= -1.08e-69) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.05e-38) {
		tmp = 0.0 - (x / B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.08e-69:
		tmp = (-1.0 - x) / B
	elif F <= 1.05e-38:
		tmp = 0.0 - (x / B)
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.08e-69)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 1.05e-38)
		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 <= -1.08e-69)
		tmp = (-1.0 - x) / B;
	elseif (F <= 1.05e-38)
		tmp = 0.0 - (x / B);
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.08e-69], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.05e-38], 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 < -1.0800000000000001e-69

   1. Initial program 64.8%

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

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   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 45.8%

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

   8. Simplified 45.8%

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

   if -1.0800000000000001e-69 < F < 1.05000000000000006e-38

   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 49.1%

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

   5. Simplified 49.1%

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

   6. Taylor expanded in F around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

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

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

      4. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(0 - x\right), B\right) \]

      5. --lowering--.f64 39.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, x\right), B\right) \]

   8. Simplified 39.0%

      \[\leadsto \color{blue}{\frac{0 - x}{B}} \]

   if 1.05000000000000006e-38 < F

   1. Initial program 70.2%

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

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   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 40.6%

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

   5. Simplified 40.6%

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

   6. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
   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 47.3%

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

   8. Simplified 47.3%

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

4. Final simplification 43.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.08 \cdot 10^{-69}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.05 \cdot 10^{-38}:\\ \;\;\;\;0 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 37.0% accurate, 32.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.6 \cdot 10^{-52}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.6e-52) (/ (- -1.0 x) B) (- 0.0 (/ x B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.6e-52) {
		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 <= (-1.6d-52)) 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 <= -1.6e-52) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = 0.0 - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.6e-52:
		tmp = (-1.0 - x) / B
	else:
		tmp = 0.0 - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.6e-52)
		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 <= -1.6e-52)
		tmp = (-1.0 - x) / B;
	else
		tmp = 0.0 - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.6e-52], 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 < -1.60000000000000005e-52

   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 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 37.3%

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

   5. Simplified 37.3%

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

   6. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   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 45.7%

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

   8. Simplified 45.7%

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

   if -1.60000000000000005e-52 < F

   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. 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 45.2%

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

   5. Simplified 45.2%

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

   6. Taylor expanded in F around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

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

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

      4. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(0 - x\right), B\right) \]

      5. --lowering--.f64 33.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, x\right), B\right) \]

   8. Simplified 33.7%

      \[\leadsto \color{blue}{\frac{0 - x}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.6 \cdot 10^{-52}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 29.8% accurate, 64.8× speedup

Math

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

FPCore

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

C

double code(double F, double B, double x) {
	return (-1.0 - 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 = ((-1.0d0) - x) / b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.5%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
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 42.5%

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

5. Simplified 42.5%

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

6. Taylor expanded in F around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
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 30.2%

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

8. Simplified 30.2%

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

Alternative 27: 10.4% accurate, 108.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.5%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
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 42.5%

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

5. Simplified 42.5%

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

6. Taylor expanded in F around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
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 30.2%

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

8. Simplified 30.2%

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

9. Taylor expanded in x around 0

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

    1. /-lowering-/.f64 10.0%

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

11. Simplified 10.0%

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

Reproduce

herbie shell --seed 2024144 
(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))))))