VandenBroeck and Keller, Equation (23)

Percentage Accurate: 77.3% → 99.2%

Time: 22.7s

Alternatives: 33

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: 99.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3.8 \cdot 10^{+144}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 3.5 \cdot 10^{+161}:\\ \;\;\;\;F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -3.8e+144)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 3.5e+161)
       (- (* F (/ (pow (fma x 2.0 (fma F F 2.0)) -0.5) (sin B))) t_0)
       (+ (* x (/ -1.0 (tan B))) (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -3.8e+144) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 3.5e+161) {
		tmp = (F * (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) / sin(B))) - t_0;
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -3.8e+144)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 3.5e+161)
		tmp = Float64(Float64(F * Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) / sin(B))) - t_0);
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -3.8e+144], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 3.5e+161], N[(N[(F * N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.80000000000000026e144

   1. Initial program 32.1%

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

   2. Taylor expanded in F around -inf 99.7%

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

      1. expm1-log1p-u 46.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}\right)\right)} \]

      2. expm1-udef 46.2%

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

      3. div-inv 46.2%

         \[\leadsto e^{\mathsf{log1p}\left(\left(-\color{blue}{\frac{x}{\tan B}}\right) + \frac{-1}{\sin B}\right)} - 1 \]

      4. neg-mul-1 46.2%

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

      5. fma-def 46.2%

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

   4. Applied egg-rr 46.2%

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

      1. expm1-def 46.2%

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

      2. expm1-log1p 99.8%

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

      3. rem-log-exp 39.5%

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

      4. fma-udef 39.5%

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

      5. neg-mul-1 39.5%

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

      6. prod-exp 33.1%

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

      7. *-commutative 33.1%

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

      8. prod-exp 39.5%

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

      9. rem-log-exp 99.8%

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

      10. unsub-neg 99.8%

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

   6. Simplified 99.8%

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

   if -3.80000000000000026e144 < F < 3.49999999999999988e161

   1. Initial program 93.7%

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

      1. +-commutative 93.7%

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

      2. unsub-neg 93.7%

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

      3. associate-*l/ 99.5%

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

      4. associate-*r/ 99.6%

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

      5. *-commutative 99.6%

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

   3. Simplified 99.7%

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

   if 3.49999999999999988e161 < F

   1. Initial program 20.2%

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

   2. Taylor expanded in F around inf 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.8 \cdot 10^{+144}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 3.5 \cdot 10^{+161}:\\ \;\;\;\;F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -2.4 \cdot 10^{+32}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.35:\\ \;\;\;\;t_0 + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{1}{\frac{\sin B}{\frac{F}{F + 0.5 \cdot \frac{2 + x \cdot 2}{F}}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B)))))
   (if (<= F -2.4e+32)
     (- (/ -1.0 (sin B)) (/ x (tan B)))
     (if (<= F 1.35)
       (+ t_0 (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)))
       (+
        t_0
        (/ 1.0 (/ (sin B) (/ F (+ F (* 0.5 (/ (+ 2.0 (* x 2.0)) F)))))))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double tmp;
	if (F <= -2.4e+32) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 1.35) {
		tmp = t_0 + ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5));
	} else {
		tmp = t_0 + (1.0 / (sin(B) / (F / (F + (0.5 * ((2.0 + (x * 2.0)) / F))))));
	}
	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 * ((-1.0d0) / tan(b))
    if (f <= (-2.4d+32)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 1.35d0) then
        tmp = t_0 + ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)))
    else
        tmp = t_0 + (1.0d0 / (sin(b) / (f / (f + (0.5d0 * ((2.0d0 + (x * 2.0d0)) / f))))))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / Math.tan(B));
	double tmp;
	if (F <= -2.4e+32) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 1.35) {
		tmp = t_0 + ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5));
	} else {
		tmp = t_0 + (1.0 / (Math.sin(B) / (F / (F + (0.5 * ((2.0 + (x * 2.0)) / F))))));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * (-1.0 / math.tan(B))
	tmp = 0
	if F <= -2.4e+32:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 1.35:
		tmp = t_0 + ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5))
	else:
		tmp = t_0 + (1.0 / (math.sin(B) / (F / (F + (0.5 * ((2.0 + (x * 2.0)) / F))))))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	tmp = 0.0
	if (F <= -2.4e+32)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 1.35)
		tmp = Float64(t_0 + Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)));
	else
		tmp = Float64(t_0 + Float64(1.0 / Float64(sin(B) / Float64(F / Float64(F + Float64(0.5 * Float64(Float64(2.0 + Float64(x * 2.0)) / F)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x * (-1.0 / tan(B));
	tmp = 0.0;
	if (F <= -2.4e+32)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 1.35)
		tmp = t_0 + ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5));
	else
		tmp = t_0 + (1.0 / (sin(B) / (F / (F + (0.5 * ((2.0 + (x * 2.0)) / F))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.4e+32], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.35], N[(t$95$0 + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(1.0 / N[(N[Sin[B], $MachinePrecision] / N[(F / N[(F + N[(0.5 * N[(N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.39999999999999991e32

   1. Initial program 51.9%

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

   2. Taylor expanded in F around -inf 99.7%

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

      1. expm1-log1p-u 45.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}\right)\right)} \]

      2. expm1-udef 45.1%

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

      3. div-inv 45.1%

         \[\leadsto e^{\mathsf{log1p}\left(\left(-\color{blue}{\frac{x}{\tan B}}\right) + \frac{-1}{\sin B}\right)} - 1 \]

      4. neg-mul-1 45.1%

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

      5. fma-def 45.1%

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

   4. Applied egg-rr 45.1%

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

      1. expm1-def 45.1%

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

      2. expm1-log1p 99.7%

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

      3. rem-log-exp 44.4%

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

      4. fma-udef 44.4%

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

      5. neg-mul-1 44.4%

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

      6. prod-exp 39.8%

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

      7. *-commutative 39.8%

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

      8. prod-exp 44.4%

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

      9. rem-log-exp 99.7%

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

      10. unsub-neg 99.7%

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

   6. Simplified 99.7%

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

   if -2.39999999999999991e32 < F < 1.3500000000000001

   1. Initial program 99.6%

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

   if 1.3500000000000001 < F

   1. Initial program 47.4%

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

      1. associate-*l/ 67.9%

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

      2. +-commutative 67.9%

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

      3. *-commutative 67.9%

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

      4. fma-udef 67.9%

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

      5. fma-def 67.9%

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

      6. metadata-eval 67.9%

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

      7. metadata-eval 67.9%

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

      8. associate-/l* 67.9%

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

      9. fma-def 67.9%

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

      10. fma-udef 67.9%

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

      11. *-commutative 67.9%

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

      12. fma-def 67.9%

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

      13. fma-def 67.9%

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

   3. Applied egg-rr 67.9%

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

      1. clear-num 67.9%

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

      2. inv-pow 67.9%

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

      3. div-inv 67.9%

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

      4. pow-flip 67.9%

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

      5. metadata-eval 67.9%

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

      6. pow1/2 67.9%

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

   5. Applied egg-rr 67.9%

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

      1. unpow-1 67.9%

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

      2. associate-/l* 67.9%

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

   7. Simplified 67.9%

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

   8. Taylor expanded in F around inf 99.7%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\frac{\sin B}{\frac{F}{\color{blue}{F + 0.5 \cdot \frac{2 \cdot x + 2}{F}}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.4 \cdot 10^{+32}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.35:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\frac{\sin B}{\frac{F}{F + 0.5 \cdot \frac{2 + x \cdot 2}{F}}}}\\ \end{array} \]

Alternative 3: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.4 \cdot 10^{+32}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.35:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\frac{\sin B}{\frac{F}{F + 0.5 \cdot \frac{2 + x \cdot 2}{F}}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.4e+32)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 1.35)
     (+
      (/ -1.0 (/ (tan B) x))
      (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)))
     (+
      (* x (/ -1.0 (tan B)))
      (/ 1.0 (/ (sin B) (/ F (+ F (* 0.5 (/ (+ 2.0 (* x 2.0)) F))))))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.4e+32) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 1.35) {
		tmp = (-1.0 / (tan(B) / x)) + ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5));
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / (sin(B) / (F / (F + (0.5 * ((2.0 + (x * 2.0)) / F))))));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-2.4d+32)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 1.35d0) then
        tmp = ((-1.0d0) / (tan(b) / x)) + ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)))
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / (sin(b) / (f / (f + (0.5d0 * ((2.0d0 + (x * 2.0d0)) / f))))))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.4e+32) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 1.35) {
		tmp = (-1.0 / (Math.tan(B) / x)) + ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5));
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / (Math.sin(B) / (F / (F + (0.5 * ((2.0 + (x * 2.0)) / F))))));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.4e+32:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 1.35:
		tmp = (-1.0 / (math.tan(B) / x)) + ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5))
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / (math.sin(B) / (F / (F + (0.5 * ((2.0 + (x * 2.0)) / F))))))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.4e+32)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 1.35)
		tmp = Float64(Float64(-1.0 / Float64(tan(B) / x)) + Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / Float64(sin(B) / Float64(F / Float64(F + Float64(0.5 * Float64(Float64(2.0 + Float64(x * 2.0)) / F)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.4e+32)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 1.35)
		tmp = (-1.0 / (tan(B) / x)) + ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5));
	else
		tmp = (x * (-1.0 / tan(B))) + (1.0 / (sin(B) / (F / (F + (0.5 * ((2.0 + (x * 2.0)) / F))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.4e+32], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.35], N[(N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sin[B], $MachinePrecision] / N[(F / N[(F + N[(0.5 * N[(N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.39999999999999991e32

   1. Initial program 51.9%

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

   2. Taylor expanded in F around -inf 99.7%

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

      1. expm1-log1p-u 45.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}\right)\right)} \]

      2. expm1-udef 45.1%

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

      3. div-inv 45.1%

         \[\leadsto e^{\mathsf{log1p}\left(\left(-\color{blue}{\frac{x}{\tan B}}\right) + \frac{-1}{\sin B}\right)} - 1 \]

      4. neg-mul-1 45.1%

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

      5. fma-def 45.1%

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

   4. Applied egg-rr 45.1%

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

      1. expm1-def 45.1%

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

      2. expm1-log1p 99.7%

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

      3. rem-log-exp 44.4%

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

      4. fma-udef 44.4%

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

      5. neg-mul-1 44.4%

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

      6. prod-exp 39.8%

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

      7. *-commutative 39.8%

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

      8. prod-exp 44.4%

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

      9. rem-log-exp 99.7%

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

      10. unsub-neg 99.7%

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

   6. Simplified 99.7%

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

   if -2.39999999999999991e32 < F < 1.3500000000000001

   1. Initial program 99.6%

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

      1. div-inv 99.7%

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

      2. clear-num 99.6%

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

   3. Applied egg-rr 99.6%

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

   if 1.3500000000000001 < F

   1. Initial program 47.4%

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

      1. associate-*l/ 67.9%

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

      2. +-commutative 67.9%

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

      3. *-commutative 67.9%

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

      4. fma-udef 67.9%

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

      5. fma-def 67.9%

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

      6. metadata-eval 67.9%

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

      7. metadata-eval 67.9%

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

      8. associate-/l* 67.9%

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

      9. fma-def 67.9%

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

      10. fma-udef 67.9%

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

      11. *-commutative 67.9%

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

      12. fma-def 67.9%

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

      13. fma-def 67.9%

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

   3. Applied egg-rr 67.9%

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

      1. clear-num 67.9%

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

      2. inv-pow 67.9%

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

      3. div-inv 67.9%

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

      4. pow-flip 67.9%

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

      5. metadata-eval 67.9%

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

      6. pow1/2 67.9%

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

   5. Applied egg-rr 67.9%

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

      1. unpow-1 67.9%

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

      2. associate-/l* 67.9%

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

   7. Simplified 67.9%

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

   8. Taylor expanded in F around inf 99.7%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\frac{\sin B}{\frac{F}{\color{blue}{F + 0.5 \cdot \frac{2 \cdot x + 2}{F}}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.4 \cdot 10^{+32}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.35:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\frac{\sin B}{\frac{F}{F + 0.5 \cdot \frac{2 + x \cdot 2}{F}}}}\\ \end{array} \]

Alternative 4: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ t_1 := 2 + x \cdot 2\\ \mathbf{if}\;F \leq -0.92:\\ \;\;\;\;t_0 + \frac{1}{\frac{\sin B}{\frac{F}{\frac{-1 - x}{F} - F}}}\\ \mathbf{elif}\;F \leq 0.95:\\ \;\;\;\;F \cdot \left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{t_1}}\right) - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{1}{\frac{\sin B}{\frac{F}{F + 0.5 \cdot \frac{t_1}{F}}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B)))) (t_1 (+ 2.0 (* x 2.0))))
   (if (<= F -0.92)
     (+ t_0 (/ 1.0 (/ (sin B) (/ F (- (/ (- -1.0 x) F) F)))))
     (if (<= F 0.95)
       (- (* F (* (/ 1.0 (sin B)) (sqrt (/ 1.0 t_1)))) (/ x (tan B)))
       (+ t_0 (/ 1.0 (/ (sin B) (/ F (+ F (* 0.5 (/ t_1 F)))))))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double t_1 = 2.0 + (x * 2.0);
	double tmp;
	if (F <= -0.92) {
		tmp = t_0 + (1.0 / (sin(B) / (F / (((-1.0 - x) / F) - F))));
	} else if (F <= 0.95) {
		tmp = (F * ((1.0 / sin(B)) * sqrt((1.0 / t_1)))) - (x / tan(B));
	} else {
		tmp = t_0 + (1.0 / (sin(B) / (F / (F + (0.5 * (t_1 / F))))));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x * ((-1.0d0) / tan(b))
    t_1 = 2.0d0 + (x * 2.0d0)
    if (f <= (-0.92d0)) then
        tmp = t_0 + (1.0d0 / (sin(b) / (f / ((((-1.0d0) - x) / f) - f))))
    else if (f <= 0.95d0) then
        tmp = (f * ((1.0d0 / sin(b)) * sqrt((1.0d0 / t_1)))) - (x / tan(b))
    else
        tmp = t_0 + (1.0d0 / (sin(b) / (f / (f + (0.5d0 * (t_1 / f))))))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / Math.tan(B));
	double t_1 = 2.0 + (x * 2.0);
	double tmp;
	if (F <= -0.92) {
		tmp = t_0 + (1.0 / (Math.sin(B) / (F / (((-1.0 - x) / F) - F))));
	} else if (F <= 0.95) {
		tmp = (F * ((1.0 / Math.sin(B)) * Math.sqrt((1.0 / t_1)))) - (x / Math.tan(B));
	} else {
		tmp = t_0 + (1.0 / (Math.sin(B) / (F / (F + (0.5 * (t_1 / F))))));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * (-1.0 / math.tan(B))
	t_1 = 2.0 + (x * 2.0)
	tmp = 0
	if F <= -0.92:
		tmp = t_0 + (1.0 / (math.sin(B) / (F / (((-1.0 - x) / F) - F))))
	elif F <= 0.95:
		tmp = (F * ((1.0 / math.sin(B)) * math.sqrt((1.0 / t_1)))) - (x / math.tan(B))
	else:
		tmp = t_0 + (1.0 / (math.sin(B) / (F / (F + (0.5 * (t_1 / F))))))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	t_1 = Float64(2.0 + Float64(x * 2.0))
	tmp = 0.0
	if (F <= -0.92)
		tmp = Float64(t_0 + Float64(1.0 / Float64(sin(B) / Float64(F / Float64(Float64(Float64(-1.0 - x) / F) - F)))));
	elseif (F <= 0.95)
		tmp = Float64(Float64(F * Float64(Float64(1.0 / sin(B)) * sqrt(Float64(1.0 / t_1)))) - Float64(x / tan(B)));
	else
		tmp = Float64(t_0 + Float64(1.0 / Float64(sin(B) / Float64(F / Float64(F + Float64(0.5 * Float64(t_1 / F)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x * (-1.0 / tan(B));
	t_1 = 2.0 + (x * 2.0);
	tmp = 0.0;
	if (F <= -0.92)
		tmp = t_0 + (1.0 / (sin(B) / (F / (((-1.0 - x) / F) - F))));
	elseif (F <= 0.95)
		tmp = (F * ((1.0 / sin(B)) * sqrt((1.0 / t_1)))) - (x / tan(B));
	else
		tmp = t_0 + (1.0 / (sin(B) / (F / (F + (0.5 * (t_1 / F))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.92], N[(t$95$0 + N[(1.0 / N[(N[Sin[B], $MachinePrecision] / N[(F / N[(N[(N[(-1.0 - x), $MachinePrecision] / F), $MachinePrecision] - F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.95], N[(N[(F * N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(1.0 / N[(N[Sin[B], $MachinePrecision] / N[(F / N[(F + N[(0.5 * N[(t$95$1 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.92000000000000004

   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. associate-*l/ 68.7%

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

      2. +-commutative 68.7%

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

      3. *-commutative 68.7%

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

      4. fma-udef 68.7%

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

      5. fma-def 68.7%

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

      6. metadata-eval 68.7%

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

      7. metadata-eval 68.7%

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

      8. associate-/l* 68.6%

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

      9. fma-def 68.6%

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

      10. fma-udef 68.6%

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

      11. *-commutative 68.6%

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

      12. fma-def 68.6%

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

      13. fma-def 68.6%

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

   3. Applied egg-rr 68.6%

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

      1. clear-num 68.6%

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

      2. inv-pow 68.6%

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

      3. div-inv 68.6%

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

      4. pow-flip 68.6%

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

      5. metadata-eval 68.6%

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

      6. pow1/2 68.6%

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

   5. Applied egg-rr 68.6%

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

      1. unpow-1 68.6%

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

      2. associate-/l* 68.7%

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

   7. Simplified 68.7%

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

   8. Taylor expanded in F around -inf 99.0%

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

      1. neg-mul-1 99.0%

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

      2. unsub-neg 99.0%

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

      3. associate-*r/ 99.0%

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

      4. +-commutative 99.0%

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

      5. distribute-lft-in 99.0%

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

      6. metadata-eval 99.0%

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

      7. metadata-eval 99.0%

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

      8. associate-*r* 99.0%

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

      9. metadata-eval 99.0%

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

      10. distribute-lft-in 99.0%

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

      11. mul-1-neg 99.0%

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

   10. Simplified 99.0%

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

   if -0.92000000000000004 < F < 0.94999999999999996

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. unsub-neg 99.5%

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

      3. associate-*l/ 99.5%

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

      4. associate-*r/ 99.5%

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

      5. *-commutative 99.5%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in F around 0 99.6%

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

   if 0.94999999999999996 < F

   1. Initial program 47.4%

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

      1. associate-*l/ 67.9%

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

      2. +-commutative 67.9%

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

      3. *-commutative 67.9%

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

      4. fma-udef 67.9%

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

      5. fma-def 67.9%

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

      6. metadata-eval 67.9%

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

      7. metadata-eval 67.9%

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

      8. associate-/l* 67.9%

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

      9. fma-def 67.9%

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

      10. fma-udef 67.9%

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

      11. *-commutative 67.9%

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

      12. fma-def 67.9%

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

      13. fma-def 67.9%

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

   3. Applied egg-rr 67.9%

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

      1. clear-num 67.9%

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

      2. inv-pow 67.9%

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

      3. div-inv 67.9%

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

      4. pow-flip 67.9%

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

      5. metadata-eval 67.9%

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

      6. pow1/2 67.9%

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

   5. Applied egg-rr 67.9%

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

      1. unpow-1 67.9%

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

      2. associate-/l* 67.9%

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

   7. Simplified 67.9%

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

   8. Taylor expanded in F around inf 99.7%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\frac{\sin B}{\frac{F}{\color{blue}{F + 0.5 \cdot \frac{2 \cdot x + 2}{F}}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.92:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\frac{\sin B}{\frac{F}{\frac{-1 - x}{F} - F}}}\\ \mathbf{elif}\;F \leq 0.95:\\ \;\;\;\;F \cdot \left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\right) - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\frac{\sin B}{\frac{F}{F + 0.5 \cdot \frac{2 + x \cdot 2}{F}}}}\\ \end{array} \]

Alternative 5: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ t_1 := 2 + x \cdot 2\\ \mathbf{if}\;F \leq -1.05:\\ \;\;\;\;t_0 + \frac{1}{\frac{\sin B}{\frac{F}{\frac{-1 - x}{F} - F}}}\\ \mathbf{elif}\;F \leq 0.98:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{t_1}}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{1}{\frac{\sin B}{\frac{F}{F + 0.5 \cdot \frac{t_1}{F}}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B)))) (t_1 (+ 2.0 (* x 2.0))))
   (if (<= F -1.05)
     (+ t_0 (/ 1.0 (/ (sin B) (/ F (- (/ (- -1.0 x) F) F)))))
     (if (<= F 0.98)
       (- (* F (/ (sqrt (/ 1.0 t_1)) (sin B))) (/ x (tan B)))
       (+ t_0 (/ 1.0 (/ (sin B) (/ F (+ F (* 0.5 (/ t_1 F)))))))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double t_1 = 2.0 + (x * 2.0);
	double tmp;
	if (F <= -1.05) {
		tmp = t_0 + (1.0 / (sin(B) / (F / (((-1.0 - x) / F) - F))));
	} else if (F <= 0.98) {
		tmp = (F * (sqrt((1.0 / t_1)) / sin(B))) - (x / tan(B));
	} else {
		tmp = t_0 + (1.0 / (sin(B) / (F / (F + (0.5 * (t_1 / F))))));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x * ((-1.0d0) / tan(b))
    t_1 = 2.0d0 + (x * 2.0d0)
    if (f <= (-1.05d0)) then
        tmp = t_0 + (1.0d0 / (sin(b) / (f / ((((-1.0d0) - x) / f) - f))))
    else if (f <= 0.98d0) then
        tmp = (f * (sqrt((1.0d0 / t_1)) / sin(b))) - (x / tan(b))
    else
        tmp = t_0 + (1.0d0 / (sin(b) / (f / (f + (0.5d0 * (t_1 / f))))))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / Math.tan(B));
	double t_1 = 2.0 + (x * 2.0);
	double tmp;
	if (F <= -1.05) {
		tmp = t_0 + (1.0 / (Math.sin(B) / (F / (((-1.0 - x) / F) - F))));
	} else if (F <= 0.98) {
		tmp = (F * (Math.sqrt((1.0 / t_1)) / Math.sin(B))) - (x / Math.tan(B));
	} else {
		tmp = t_0 + (1.0 / (Math.sin(B) / (F / (F + (0.5 * (t_1 / F))))));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * (-1.0 / math.tan(B))
	t_1 = 2.0 + (x * 2.0)
	tmp = 0
	if F <= -1.05:
		tmp = t_0 + (1.0 / (math.sin(B) / (F / (((-1.0 - x) / F) - F))))
	elif F <= 0.98:
		tmp = (F * (math.sqrt((1.0 / t_1)) / math.sin(B))) - (x / math.tan(B))
	else:
		tmp = t_0 + (1.0 / (math.sin(B) / (F / (F + (0.5 * (t_1 / F))))))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	t_1 = Float64(2.0 + Float64(x * 2.0))
	tmp = 0.0
	if (F <= -1.05)
		tmp = Float64(t_0 + Float64(1.0 / Float64(sin(B) / Float64(F / Float64(Float64(Float64(-1.0 - x) / F) - F)))));
	elseif (F <= 0.98)
		tmp = Float64(Float64(F * Float64(sqrt(Float64(1.0 / t_1)) / sin(B))) - Float64(x / tan(B)));
	else
		tmp = Float64(t_0 + Float64(1.0 / Float64(sin(B) / Float64(F / Float64(F + Float64(0.5 * Float64(t_1 / F)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x * (-1.0 / tan(B));
	t_1 = 2.0 + (x * 2.0);
	tmp = 0.0;
	if (F <= -1.05)
		tmp = t_0 + (1.0 / (sin(B) / (F / (((-1.0 - x) / F) - F))));
	elseif (F <= 0.98)
		tmp = (F * (sqrt((1.0 / t_1)) / sin(B))) - (x / tan(B));
	else
		tmp = t_0 + (1.0 / (sin(B) / (F / (F + (0.5 * (t_1 / F))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.05], N[(t$95$0 + N[(1.0 / N[(N[Sin[B], $MachinePrecision] / N[(F / N[(N[(N[(-1.0 - x), $MachinePrecision] / F), $MachinePrecision] - F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.98], N[(N[(F * N[(N[Sqrt[N[(1.0 / t$95$1), $MachinePrecision]], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(1.0 / N[(N[Sin[B], $MachinePrecision] / N[(F / N[(F + N[(0.5 * N[(t$95$1 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.05000000000000004

   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. associate-*l/ 68.7%

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

      2. +-commutative 68.7%

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

      3. *-commutative 68.7%

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

      4. fma-udef 68.7%

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

      5. fma-def 68.7%

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

      6. metadata-eval 68.7%

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

      7. metadata-eval 68.7%

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

      8. associate-/l* 68.6%

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

      9. fma-def 68.6%

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

      10. fma-udef 68.6%

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

      11. *-commutative 68.6%

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

      12. fma-def 68.6%

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

      13. fma-def 68.6%

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

   3. Applied egg-rr 68.6%

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

      1. clear-num 68.6%

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

      2. inv-pow 68.6%

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

      3. div-inv 68.6%

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

      4. pow-flip 68.6%

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

      5. metadata-eval 68.6%

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

      6. pow1/2 68.6%

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

   5. Applied egg-rr 68.6%

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

      1. unpow-1 68.6%

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

      2. associate-/l* 68.7%

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

   7. Simplified 68.7%

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

   8. Taylor expanded in F around -inf 99.0%

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

      1. neg-mul-1 99.0%

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

      2. unsub-neg 99.0%

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

      3. associate-*r/ 99.0%

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

      4. +-commutative 99.0%

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

      5. distribute-lft-in 99.0%

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

      6. metadata-eval 99.0%

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

      7. metadata-eval 99.0%

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

      8. associate-*r* 99.0%

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

      9. metadata-eval 99.0%

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

      10. distribute-lft-in 99.0%

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

      11. mul-1-neg 99.0%

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

   10. Simplified 99.0%

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

   if -1.05000000000000004 < F < 0.97999999999999998

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. unsub-neg 99.5%

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

      3. associate-*l/ 99.5%

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

      4. associate-*r/ 99.5%

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

      5. *-commutative 99.5%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in F around 0 99.6%

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

   if 0.97999999999999998 < F

   1. Initial program 47.4%

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

      1. associate-*l/ 67.9%

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

      2. +-commutative 67.9%

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

      3. *-commutative 67.9%

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

      4. fma-udef 67.9%

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

      5. fma-def 67.9%

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

      6. metadata-eval 67.9%

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

      7. metadata-eval 67.9%

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

      8. associate-/l* 67.9%

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

      9. fma-def 67.9%

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

      10. fma-udef 67.9%

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

      11. *-commutative 67.9%

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

      12. fma-def 67.9%

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

      13. fma-def 67.9%

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

   3. Applied egg-rr 67.9%

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

      1. clear-num 67.9%

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

      2. inv-pow 67.9%

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

      3. div-inv 67.9%

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

      4. pow-flip 67.9%

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

      5. metadata-eval 67.9%

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

      6. pow1/2 67.9%

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

   5. Applied egg-rr 67.9%

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

      1. unpow-1 67.9%

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

      2. associate-/l* 67.9%

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

   7. Simplified 67.9%

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

   8. Taylor expanded in F around inf 99.7%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\frac{\sin B}{\frac{F}{\color{blue}{F + 0.5 \cdot \frac{2 \cdot x + 2}{F}}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.05:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\frac{\sin B}{\frac{F}{\frac{-1 - x}{F} - F}}}\\ \mathbf{elif}\;F \leq 0.98:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{2 + x \cdot 2}}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\frac{\sin B}{\frac{F}{F + 0.5 \cdot \frac{2 + x \cdot 2}{F}}}}\\ \end{array} \]

Alternative 6: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -0.88:\\ \;\;\;\;t_0 + \frac{1}{\frac{\sin B}{\frac{F}{\frac{-1 - x}{F} - F}}}\\ \mathbf{elif}\;F \leq 0.82:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{1}{\frac{\sin B}{\frac{F}{F + 0.5 \cdot \frac{2 + x \cdot 2}{F}}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B)))))
   (if (<= F -0.88)
     (+ t_0 (/ 1.0 (/ (sin B) (/ F (- (/ (- -1.0 x) F) F)))))
     (if (<= F 0.82)
       (- (* F (/ (sqrt 0.5) (sin B))) (/ x (tan B)))
       (+
        t_0
        (/ 1.0 (/ (sin B) (/ F (+ F (* 0.5 (/ (+ 2.0 (* x 2.0)) F)))))))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double tmp;
	if (F <= -0.88) {
		tmp = t_0 + (1.0 / (sin(B) / (F / (((-1.0 - x) / F) - F))));
	} else if (F <= 0.82) {
		tmp = (F * (sqrt(0.5) / sin(B))) - (x / tan(B));
	} else {
		tmp = t_0 + (1.0 / (sin(B) / (F / (F + (0.5 * ((2.0 + (x * 2.0)) / F))))));
	}
	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 * ((-1.0d0) / tan(b))
    if (f <= (-0.88d0)) then
        tmp = t_0 + (1.0d0 / (sin(b) / (f / ((((-1.0d0) - x) / f) - f))))
    else if (f <= 0.82d0) then
        tmp = (f * (sqrt(0.5d0) / sin(b))) - (x / tan(b))
    else
        tmp = t_0 + (1.0d0 / (sin(b) / (f / (f + (0.5d0 * ((2.0d0 + (x * 2.0d0)) / f))))))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / Math.tan(B));
	double tmp;
	if (F <= -0.88) {
		tmp = t_0 + (1.0 / (Math.sin(B) / (F / (((-1.0 - x) / F) - F))));
	} else if (F <= 0.82) {
		tmp = (F * (Math.sqrt(0.5) / Math.sin(B))) - (x / Math.tan(B));
	} else {
		tmp = t_0 + (1.0 / (Math.sin(B) / (F / (F + (0.5 * ((2.0 + (x * 2.0)) / F))))));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * (-1.0 / math.tan(B))
	tmp = 0
	if F <= -0.88:
		tmp = t_0 + (1.0 / (math.sin(B) / (F / (((-1.0 - x) / F) - F))))
	elif F <= 0.82:
		tmp = (F * (math.sqrt(0.5) / math.sin(B))) - (x / math.tan(B))
	else:
		tmp = t_0 + (1.0 / (math.sin(B) / (F / (F + (0.5 * ((2.0 + (x * 2.0)) / F))))))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	tmp = 0.0
	if (F <= -0.88)
		tmp = Float64(t_0 + Float64(1.0 / Float64(sin(B) / Float64(F / Float64(Float64(Float64(-1.0 - x) / F) - F)))));
	elseif (F <= 0.82)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / sin(B))) - Float64(x / tan(B)));
	else
		tmp = Float64(t_0 + Float64(1.0 / Float64(sin(B) / Float64(F / Float64(F + Float64(0.5 * Float64(Float64(2.0 + Float64(x * 2.0)) / F)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x * (-1.0 / tan(B));
	tmp = 0.0;
	if (F <= -0.88)
		tmp = t_0 + (1.0 / (sin(B) / (F / (((-1.0 - x) / F) - F))));
	elseif (F <= 0.82)
		tmp = (F * (sqrt(0.5) / sin(B))) - (x / tan(B));
	else
		tmp = t_0 + (1.0 / (sin(B) / (F / (F + (0.5 * ((2.0 + (x * 2.0)) / F))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.88], N[(t$95$0 + N[(1.0 / N[(N[Sin[B], $MachinePrecision] / N[(F / N[(N[(N[(-1.0 - x), $MachinePrecision] / F), $MachinePrecision] - F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.82], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(1.0 / N[(N[Sin[B], $MachinePrecision] / N[(F / N[(F + N[(0.5 * N[(N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.880000000000000004

   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. associate-*l/ 68.7%

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

      2. +-commutative 68.7%

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

      3. *-commutative 68.7%

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

      4. fma-udef 68.7%

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

      5. fma-def 68.7%

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

      6. metadata-eval 68.7%

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

      7. metadata-eval 68.7%

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

      8. associate-/l* 68.6%

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

      9. fma-def 68.6%

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

      10. fma-udef 68.6%

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

      11. *-commutative 68.6%

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

      12. fma-def 68.6%

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

      13. fma-def 68.6%

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

   3. Applied egg-rr 68.6%

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

      1. clear-num 68.6%

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

      2. inv-pow 68.6%

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

      3. div-inv 68.6%

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

      4. pow-flip 68.6%

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

      5. metadata-eval 68.6%

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

      6. pow1/2 68.6%

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

   5. Applied egg-rr 68.6%

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

      1. unpow-1 68.6%

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

      2. associate-/l* 68.7%

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

   7. Simplified 68.7%

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

   8. Taylor expanded in F around -inf 99.0%

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

      1. neg-mul-1 99.0%

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

      2. unsub-neg 99.0%

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

      3. associate-*r/ 99.0%

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

      4. +-commutative 99.0%

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

      5. distribute-lft-in 99.0%

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

      6. metadata-eval 99.0%

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

      7. metadata-eval 99.0%

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

      8. associate-*r* 99.0%

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

      9. metadata-eval 99.0%

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

      10. distribute-lft-in 99.0%

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

      11. mul-1-neg 99.0%

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

   10. Simplified 99.0%

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

   if -0.880000000000000004 < F < 0.819999999999999951

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. unsub-neg 99.5%

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

      3. associate-*l/ 99.5%

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

      4. associate-*r/ 99.5%

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

      5. *-commutative 99.5%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in F around 0 99.6%

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

   5. Taylor expanded in x around 0 99.1%

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

   if 0.819999999999999951 < F

   1. Initial program 47.4%

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

      1. associate-*l/ 67.9%

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

      2. +-commutative 67.9%

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

      3. *-commutative 67.9%

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

      4. fma-udef 67.9%

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

      5. fma-def 67.9%

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

      6. metadata-eval 67.9%

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

      7. metadata-eval 67.9%

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

      8. associate-/l* 67.9%

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

      9. fma-def 67.9%

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

      10. fma-udef 67.9%

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

      11. *-commutative 67.9%

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

      12. fma-def 67.9%

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

      13. fma-def 67.9%

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

   3. Applied egg-rr 67.9%

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

      1. clear-num 67.9%

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

      2. inv-pow 67.9%

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

      3. div-inv 67.9%

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

      4. pow-flip 67.9%

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

      5. metadata-eval 67.9%

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

      6. pow1/2 67.9%

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

   5. Applied egg-rr 67.9%

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

      1. unpow-1 67.9%

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

      2. associate-/l* 67.9%

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

   7. Simplified 67.9%

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

   8. Taylor expanded in F around inf 99.7%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\frac{\sin B}{\frac{F}{\color{blue}{F + 0.5 \cdot \frac{2 \cdot x + 2}{F}}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.88:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\frac{\sin B}{\frac{F}{\frac{-1 - x}{F} - F}}}\\ \mathbf{elif}\;F \leq 0.82:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\frac{\sin B}{\frac{F}{F + 0.5 \cdot \frac{2 + x \cdot 2}{F}}}}\\ \end{array} \]

Alternative 7: 91.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + x \cdot 2\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2.4 \cdot 10^{+32}:\\ \;\;\;\;\frac{-1}{\sin B} - t_1\\ \mathbf{elif}\;F \leq -1.6 \cdot 10^{-29}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.2 \cdot 10^{-45}:\\ \;\;\;\;\sqrt{\frac{1}{t_0}} \cdot \frac{F}{B} - t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\frac{\sin B}{\frac{F}{F + 0.5 \cdot \frac{t_0}{F}}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (+ 2.0 (* x 2.0))) (t_1 (/ x (tan B))))
   (if (<= F -2.4e+32)
     (- (/ -1.0 (sin B)) t_1)
     (if (<= F -1.6e-29)
       (- (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)) (/ x B))
       (if (<= F 1.2e-45)
         (- (* (sqrt (/ 1.0 t_0)) (/ F B)) t_1)
         (+
          (* x (/ -1.0 (tan B)))
          (/ 1.0 (/ (sin B) (/ F (+ F (* 0.5 (/ t_0 F))))))))))))

C

double code(double F, double B, double x) {
	double t_0 = 2.0 + (x * 2.0);
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -2.4e+32) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= -1.6e-29) {
		tmp = ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B);
	} else if (F <= 1.2e-45) {
		tmp = (sqrt((1.0 / t_0)) * (F / B)) - t_1;
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / (sin(B) / (F / (F + (0.5 * (t_0 / F))))));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 + (x * 2.0d0)
    t_1 = x / tan(b)
    if (f <= (-2.4d+32)) then
        tmp = ((-1.0d0) / sin(b)) - t_1
    else if (f <= (-1.6d-29)) then
        tmp = ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0))) - (x / b)
    else if (f <= 1.2d-45) then
        tmp = (sqrt((1.0d0 / t_0)) * (f / b)) - t_1
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / (sin(b) / (f / (f + (0.5d0 * (t_0 / f))))))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = 2.0 + (x * 2.0);
	double t_1 = x / Math.tan(B);
	double tmp;
	if (F <= -2.4e+32) {
		tmp = (-1.0 / Math.sin(B)) - t_1;
	} else if (F <= -1.6e-29) {
		tmp = ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B);
	} else if (F <= 1.2e-45) {
		tmp = (Math.sqrt((1.0 / t_0)) * (F / B)) - t_1;
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / (Math.sin(B) / (F / (F + (0.5 * (t_0 / F))))));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = 2.0 + (x * 2.0)
	t_1 = x / math.tan(B)
	tmp = 0
	if F <= -2.4e+32:
		tmp = (-1.0 / math.sin(B)) - t_1
	elif F <= -1.6e-29:
		tmp = ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B)
	elif F <= 1.2e-45:
		tmp = (math.sqrt((1.0 / t_0)) * (F / B)) - t_1
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / (math.sin(B) / (F / (F + (0.5 * (t_0 / F))))))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(2.0 + Float64(x * 2.0))
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -2.4e+32)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= -1.6e-29)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)) - Float64(x / B));
	elseif (F <= 1.2e-45)
		tmp = Float64(Float64(sqrt(Float64(1.0 / t_0)) * Float64(F / B)) - t_1);
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / Float64(sin(B) / Float64(F / Float64(F + Float64(0.5 * Float64(t_0 / F)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = 2.0 + (x * 2.0);
	t_1 = x / tan(B);
	tmp = 0.0;
	if (F <= -2.4e+32)
		tmp = (-1.0 / sin(B)) - t_1;
	elseif (F <= -1.6e-29)
		tmp = ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5)) - (x / B);
	elseif (F <= 1.2e-45)
		tmp = (sqrt((1.0 / t_0)) * (F / B)) - t_1;
	else
		tmp = (x * (-1.0 / tan(B))) + (1.0 / (sin(B) / (F / (F + (0.5 * (t_0 / F))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.4e+32], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, -1.6e-29], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.2e-45], N[(N[(N[Sqrt[N[(1.0 / t$95$0), $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Sin[B], $MachinePrecision] / N[(F / N[(F + N[(0.5 * N[(t$95$0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.39999999999999991e32

   1. Initial program 51.9%

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

   2. Taylor expanded in F around -inf 99.7%

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

      1. expm1-log1p-u 45.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}\right)\right)} \]

      2. expm1-udef 45.1%

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

      3. div-inv 45.1%

         \[\leadsto e^{\mathsf{log1p}\left(\left(-\color{blue}{\frac{x}{\tan B}}\right) + \frac{-1}{\sin B}\right)} - 1 \]

      4. neg-mul-1 45.1%

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

      5. fma-def 45.1%

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

   4. Applied egg-rr 45.1%

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

      1. expm1-def 45.1%

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

      2. expm1-log1p 99.7%

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

      3. rem-log-exp 44.4%

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

      4. fma-udef 44.4%

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

      5. neg-mul-1 44.4%

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

      6. prod-exp 39.8%

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

      7. *-commutative 39.8%

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

      8. prod-exp 44.4%

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

      9. rem-log-exp 99.7%

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

      10. unsub-neg 99.7%

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

   6. Simplified 99.7%

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

   if -2.39999999999999991e32 < F < -1.6e-29

   1. Initial program 99.6%

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

   2. Taylor expanded in B around 0 96.6%

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

   if -1.6e-29 < F < 1.19999999999999995e-45

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unsub-neg 99.6%

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

      3. associate-*l/ 99.5%

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

      4. associate-*r/ 99.5%

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

      5. *-commutative 99.5%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in F around 0 99.7%

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

   5. Taylor expanded in B around 0 87.7%

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

   if 1.19999999999999995e-45 < F

   1. Initial program 54.1%

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

      1. associate-*l/ 72.0%

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

      2. +-commutative 72.0%

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

      3. *-commutative 72.0%

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

      4. fma-udef 72.0%

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

      5. fma-def 72.0%

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

      6. metadata-eval 72.0%

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

      7. metadata-eval 72.0%

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

      8. associate-/l* 72.0%

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

      9. fma-def 72.0%

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

      10. fma-udef 72.0%

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

      11. *-commutative 72.0%

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

      12. fma-def 72.0%

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

      13. fma-def 72.0%

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

   3. Applied egg-rr 72.0%

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

      1. clear-num 72.0%

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

      2. inv-pow 72.0%

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

      3. div-inv 72.0%

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

      4. pow-flip 72.0%

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

      5. metadata-eval 72.0%

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

      6. pow1/2 72.0%

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

   5. Applied egg-rr 72.0%

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

      1. unpow-1 72.0%

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

      2. associate-/l* 72.0%

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

   7. Simplified 72.0%

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

   8. Taylor expanded in F around inf 94.5%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\frac{\sin B}{\frac{F}{\color{blue}{F + 0.5 \cdot \frac{2 \cdot x + 2}{F}}}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.4 \cdot 10^{+32}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.6 \cdot 10^{-29}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.2 \cdot 10^{-45}:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\frac{\sin B}{\frac{F}{F + 0.5 \cdot \frac{2 + x \cdot 2}{F}}}}\\ \end{array} \]

Alternative 8: 92.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2.4 \cdot 10^{+32}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq -3.8 \cdot 10^{-30}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.64:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -2.4e+32)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F -3.8e-30)
       (- (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)) (/ x B))
       (if (<= F 0.64)
         (- (* (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))) (/ F B)) t_0)
         (+ (* x (/ -1.0 (tan B))) (/ 1.0 (sin B))))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -2.4e+32) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= -3.8e-30) {
		tmp = ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B);
	} else if (F <= 0.64) {
		tmp = (sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - t_0;
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-2.4d+32)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= (-3.8d-30)) then
        tmp = ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0))) - (x / b)
    else if (f <= 0.64d0) then
        tmp = (sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))) * (f / b)) - t_0
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -2.4e+32) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= -3.8e-30) {
		tmp = ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B);
	} else if (F <= 0.64) {
		tmp = (Math.sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - t_0;
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -2.4e+32:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= -3.8e-30:
		tmp = ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B)
	elif F <= 0.64:
		tmp = (math.sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - t_0
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -2.4e+32)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= -3.8e-30)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)) - Float64(x / B));
	elseif (F <= 0.64)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0)))) * Float64(F / B)) - t_0);
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -2.4e+32)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= -3.8e-30)
		tmp = ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5)) - (x / B);
	elseif (F <= 0.64)
		tmp = (sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - t_0;
	else
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.4e+32], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -3.8e-30], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.64], N[(N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.39999999999999991e32

   1. Initial program 51.9%

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

   2. Taylor expanded in F around -inf 99.7%

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

      1. expm1-log1p-u 45.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}\right)\right)} \]

      2. expm1-udef 45.1%

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

      3. div-inv 45.1%

         \[\leadsto e^{\mathsf{log1p}\left(\left(-\color{blue}{\frac{x}{\tan B}}\right) + \frac{-1}{\sin B}\right)} - 1 \]

      4. neg-mul-1 45.1%

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

      5. fma-def 45.1%

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

   4. Applied egg-rr 45.1%

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

      1. expm1-def 45.1%

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

      2. expm1-log1p 99.7%

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

      3. rem-log-exp 44.4%

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

      4. fma-udef 44.4%

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

      5. neg-mul-1 44.4%

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

      6. prod-exp 39.8%

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

      7. *-commutative 39.8%

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

      8. prod-exp 44.4%

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

      9. rem-log-exp 99.7%

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

      10. unsub-neg 99.7%

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

   6. Simplified 99.7%

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

   if -2.39999999999999991e32 < F < -3.8000000000000003e-30

   1. Initial program 99.6%

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

   2. Taylor expanded in B around 0 96.6%

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

   if -3.8000000000000003e-30 < F < 0.640000000000000013

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. unsub-neg 99.5%

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

      3. associate-*l/ 99.5%

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

      4. associate-*r/ 99.5%

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

      5. *-commutative 99.5%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in F around 0 99.7%

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

   5. Taylor expanded in B around 0 84.6%

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

   if 0.640000000000000013 < F

   1. Initial program 47.4%

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

   2. Taylor expanded in F around inf 99.6%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.4 \cdot 10^{+32}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -3.8 \cdot 10^{-30}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.64:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 9: 92.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.00052:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq -1.15 \cdot 10^{-29}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\sin B}{F}}\\ \mathbf{elif}\;F \leq 0.85:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -0.00052)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F -1.15e-29)
       (/ (sqrt 0.5) (/ (sin B) F))
       (if (<= F 0.85)
         (- (* (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))) (/ F B)) t_0)
         (+ (* x (/ -1.0 (tan B))) (/ 1.0 (sin B))))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -0.00052) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= -1.15e-29) {
		tmp = sqrt(0.5) / (sin(B) / F);
	} else if (F <= 0.85) {
		tmp = (sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - t_0;
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-0.00052d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= (-1.15d-29)) then
        tmp = sqrt(0.5d0) / (sin(b) / f)
    else if (f <= 0.85d0) then
        tmp = (sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))) * (f / b)) - t_0
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -0.00052) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= -1.15e-29) {
		tmp = Math.sqrt(0.5) / (Math.sin(B) / F);
	} else if (F <= 0.85) {
		tmp = (Math.sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - t_0;
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -0.00052:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= -1.15e-29:
		tmp = math.sqrt(0.5) / (math.sin(B) / F)
	elif F <= 0.85:
		tmp = (math.sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - t_0
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -0.00052)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= -1.15e-29)
		tmp = Float64(sqrt(0.5) / Float64(sin(B) / F));
	elseif (F <= 0.85)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0)))) * Float64(F / B)) - t_0);
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -0.00052)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= -1.15e-29)
		tmp = sqrt(0.5) / (sin(B) / F);
	elseif (F <= 0.85)
		tmp = (sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - t_0;
	else
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.00052], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -1.15e-29], N[(N[Sqrt[0.5], $MachinePrecision] / N[(N[Sin[B], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.85], N[(N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -5.19999999999999954e-4

   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. Taylor expanded in F around -inf 98.0%

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

      1. expm1-log1p-u 45.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}\right)\right)} \]

      2. expm1-udef 45.5%

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

      3. div-inv 45.5%

         \[\leadsto e^{\mathsf{log1p}\left(\left(-\color{blue}{\frac{x}{\tan B}}\right) + \frac{-1}{\sin B}\right)} - 1 \]

      4. neg-mul-1 45.5%

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

      5. fma-def 45.5%

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

   4. Applied egg-rr 45.5%

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

      1. expm1-def 45.5%

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

      2. expm1-log1p 98.0%

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

      3. rem-log-exp 45.1%

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

      4. fma-udef 45.1%

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

      5. neg-mul-1 45.1%

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

      6. prod-exp 40.9%

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

      7. *-commutative 40.9%

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

      8. prod-exp 45.1%

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

      9. rem-log-exp 98.0%

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

      10. unsub-neg 98.0%

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

   6. Simplified 98.0%

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

   if -5.19999999999999954e-4 < F < -1.14999999999999996e-29

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. unsub-neg 99.4%

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

      3. associate-*l/ 99.2%

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

      4. associate-*r/ 99.7%

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

      5. *-commutative 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in F around 0 98.1%

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

   5. Taylor expanded in x around 0 92.4%

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

      1. associate-/l* 92.8%

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

   7. Simplified 92.8%

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

   if -1.14999999999999996e-29 < F < 0.849999999999999978

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. unsub-neg 99.5%

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

      3. associate-*l/ 99.5%

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

      4. associate-*r/ 99.5%

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

      5. *-commutative 99.5%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in F around 0 99.7%

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

   5. Taylor expanded in B around 0 84.6%

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

   if 0.849999999999999978 < F

   1. Initial program 47.4%

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

   2. Taylor expanded in F around inf 99.6%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.00052:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.15 \cdot 10^{-29}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\sin B}{F}}\\ \mathbf{elif}\;F \leq 0.85:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 10: 92.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -0.035:\\ \;\;\;\;t_0 + \frac{1}{\frac{\sin B}{\frac{F}{\frac{-1 - x}{F} - F}}}\\ \mathbf{elif}\;F \leq -1.08 \cdot 10^{-29}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\sin B}{F}}\\ \mathbf{elif}\;F \leq 0.2:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B)))))
   (if (<= F -0.035)
     (+ t_0 (/ 1.0 (/ (sin B) (/ F (- (/ (- -1.0 x) F) F)))))
     (if (<= F -1.08e-29)
       (/ (sqrt 0.5) (/ (sin B) F))
       (if (<= F 0.2)
         (- (* (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))) (/ F B)) (/ x (tan B)))
         (+ t_0 (/ 1.0 (sin B))))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double tmp;
	if (F <= -0.035) {
		tmp = t_0 + (1.0 / (sin(B) / (F / (((-1.0 - x) / F) - F))));
	} else if (F <= -1.08e-29) {
		tmp = sqrt(0.5) / (sin(B) / F);
	} else if (F <= 0.2) {
		tmp = (sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - (x / tan(B));
	} else {
		tmp = t_0 + (1.0 / sin(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * ((-1.0d0) / tan(b))
    if (f <= (-0.035d0)) then
        tmp = t_0 + (1.0d0 / (sin(b) / (f / ((((-1.0d0) - x) / f) - f))))
    else if (f <= (-1.08d-29)) then
        tmp = sqrt(0.5d0) / (sin(b) / f)
    else if (f <= 0.2d0) then
        tmp = (sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))) * (f / b)) - (x / tan(b))
    else
        tmp = t_0 + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / Math.tan(B));
	double tmp;
	if (F <= -0.035) {
		tmp = t_0 + (1.0 / (Math.sin(B) / (F / (((-1.0 - x) / F) - F))));
	} else if (F <= -1.08e-29) {
		tmp = Math.sqrt(0.5) / (Math.sin(B) / F);
	} else if (F <= 0.2) {
		tmp = (Math.sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - (x / Math.tan(B));
	} else {
		tmp = t_0 + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * (-1.0 / math.tan(B))
	tmp = 0
	if F <= -0.035:
		tmp = t_0 + (1.0 / (math.sin(B) / (F / (((-1.0 - x) / F) - F))))
	elif F <= -1.08e-29:
		tmp = math.sqrt(0.5) / (math.sin(B) / F)
	elif F <= 0.2:
		tmp = (math.sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - (x / math.tan(B))
	else:
		tmp = t_0 + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	tmp = 0.0
	if (F <= -0.035)
		tmp = Float64(t_0 + Float64(1.0 / Float64(sin(B) / Float64(F / Float64(Float64(Float64(-1.0 - x) / F) - F)))));
	elseif (F <= -1.08e-29)
		tmp = Float64(sqrt(0.5) / Float64(sin(B) / F));
	elseif (F <= 0.2)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0)))) * Float64(F / B)) - Float64(x / tan(B)));
	else
		tmp = Float64(t_0 + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x * (-1.0 / tan(B));
	tmp = 0.0;
	if (F <= -0.035)
		tmp = t_0 + (1.0 / (sin(B) / (F / (((-1.0 - x) / F) - F))));
	elseif (F <= -1.08e-29)
		tmp = sqrt(0.5) / (sin(B) / F);
	elseif (F <= 0.2)
		tmp = (sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - (x / tan(B));
	else
		tmp = t_0 + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.035], N[(t$95$0 + N[(1.0 / N[(N[Sin[B], $MachinePrecision] / N[(F / N[(N[(N[(-1.0 - x), $MachinePrecision] / F), $MachinePrecision] - F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.08e-29], N[(N[Sqrt[0.5], $MachinePrecision] / N[(N[Sin[B], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.2], N[(N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.035000000000000003

   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. associate-*l/ 68.7%

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

      2. +-commutative 68.7%

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

      3. *-commutative 68.7%

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

      4. fma-udef 68.7%

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

      5. fma-def 68.7%

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

      6. metadata-eval 68.7%

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

      7. metadata-eval 68.7%

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

      8. associate-/l* 68.6%

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

      9. fma-def 68.6%

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

      10. fma-udef 68.6%

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

      11. *-commutative 68.6%

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

      12. fma-def 68.6%

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

      13. fma-def 68.6%

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

   3. Applied egg-rr 68.6%

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

      1. clear-num 68.6%

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

      2. inv-pow 68.6%

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

      3. div-inv 68.6%

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

      4. pow-flip 68.6%

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

      5. metadata-eval 68.6%

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

      6. pow1/2 68.6%

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

   5. Applied egg-rr 68.6%

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

      1. unpow-1 68.6%

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

      2. associate-/l* 68.7%

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

   7. Simplified 68.7%

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

   8. Taylor expanded in F around -inf 99.0%

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

      1. neg-mul-1 99.0%

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

      2. unsub-neg 99.0%

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

      3. associate-*r/ 99.0%

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

      4. +-commutative 99.0%

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

      5. distribute-lft-in 99.0%

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

      6. metadata-eval 99.0%

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

      7. metadata-eval 99.0%

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

      8. associate-*r* 99.0%

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

      9. metadata-eval 99.0%

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

      10. distribute-lft-in 99.0%

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

      11. mul-1-neg 99.0%

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

   10. Simplified 99.0%

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

   if -0.035000000000000003 < F < -1.07999999999999995e-29

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. unsub-neg 99.4%

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

      3. associate-*l/ 99.2%

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

      4. associate-*r/ 99.7%

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

      5. *-commutative 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in F around 0 98.1%

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

   5. Taylor expanded in x around 0 92.4%

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

      1. associate-/l* 92.8%

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

   7. Simplified 92.8%

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

   if -1.07999999999999995e-29 < F < 0.20000000000000001

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. unsub-neg 99.5%

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

      3. associate-*l/ 99.5%

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

      4. associate-*r/ 99.5%

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

      5. *-commutative 99.5%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in F around 0 99.7%

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

   5. Taylor expanded in B around 0 84.6%

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

   if 0.20000000000000001 < F

   1. Initial program 47.4%

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

   2. Taylor expanded in F around inf 99.6%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.035:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\frac{\sin B}{\frac{F}{\frac{-1 - x}{F} - F}}}\\ \mathbf{elif}\;F \leq -1.08 \cdot 10^{-29}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\sin B}{F}}\\ \mathbf{elif}\;F \leq 0.2:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 11: 84.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -0.00014:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.58 \cdot 10^{-108}:\\ \;\;\;\;t_0 \cdot \left(F \cdot \sqrt{0.5}\right)\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{-101}:\\ \;\;\;\;\frac{\left(-x\right) \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 2.4 \cdot 10^{-58}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\sin B}{F}}\\ \mathbf{elif}\;F \leq 12000000:\\ \;\;\;\;t_1 + \frac{F}{B \cdot \left(F + 0.5 \cdot \frac{2 + x \cdot 2}{F}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1 + t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))) (t_1 (* x (/ -1.0 (tan B)))))
   (if (<= F -0.00014)
     (- (/ -1.0 (sin B)) (/ x (tan B)))
     (if (<= F -1.58e-108)
       (* t_0 (* F (sqrt 0.5)))
       (if (<= F 2.3e-101)
         (/ (* (- x) (cos B)) (sin B))
         (if (<= F 2.4e-58)
           (/ (sqrt 0.5) (/ (sin B) F))
           (if (<= F 12000000.0)
             (+ t_1 (/ F (* B (+ F (* 0.5 (/ (+ 2.0 (* x 2.0)) F))))))
             (+ t_1 t_0))))))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / sin(B);
	double t_1 = x * (-1.0 / tan(B));
	double tmp;
	if (F <= -0.00014) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= -1.58e-108) {
		tmp = t_0 * (F * sqrt(0.5));
	} else if (F <= 2.3e-101) {
		tmp = (-x * cos(B)) / sin(B);
	} else if (F <= 2.4e-58) {
		tmp = sqrt(0.5) / (sin(B) / F);
	} else if (F <= 12000000.0) {
		tmp = t_1 + (F / (B * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))));
	} else {
		tmp = t_1 + t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 / sin(b)
    t_1 = x * ((-1.0d0) / tan(b))
    if (f <= (-0.00014d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= (-1.58d-108)) then
        tmp = t_0 * (f * sqrt(0.5d0))
    else if (f <= 2.3d-101) then
        tmp = (-x * cos(b)) / sin(b)
    else if (f <= 2.4d-58) then
        tmp = sqrt(0.5d0) / (sin(b) / f)
    else if (f <= 12000000.0d0) then
        tmp = t_1 + (f / (b * (f + (0.5d0 * ((2.0d0 + (x * 2.0d0)) / f)))))
    else
        tmp = t_1 + t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = 1.0 / Math.sin(B);
	double t_1 = x * (-1.0 / Math.tan(B));
	double tmp;
	if (F <= -0.00014) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= -1.58e-108) {
		tmp = t_0 * (F * Math.sqrt(0.5));
	} else if (F <= 2.3e-101) {
		tmp = (-x * Math.cos(B)) / Math.sin(B);
	} else if (F <= 2.4e-58) {
		tmp = Math.sqrt(0.5) / (Math.sin(B) / F);
	} else if (F <= 12000000.0) {
		tmp = t_1 + (F / (B * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))));
	} else {
		tmp = t_1 + t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = 1.0 / math.sin(B)
	t_1 = x * (-1.0 / math.tan(B))
	tmp = 0
	if F <= -0.00014:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= -1.58e-108:
		tmp = t_0 * (F * math.sqrt(0.5))
	elif F <= 2.3e-101:
		tmp = (-x * math.cos(B)) / math.sin(B)
	elif F <= 2.4e-58:
		tmp = math.sqrt(0.5) / (math.sin(B) / F)
	elif F <= 12000000.0:
		tmp = t_1 + (F / (B * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))))
	else:
		tmp = t_1 + t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / sin(B))
	t_1 = Float64(x * Float64(-1.0 / tan(B)))
	tmp = 0.0
	if (F <= -0.00014)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= -1.58e-108)
		tmp = Float64(t_0 * Float64(F * sqrt(0.5)));
	elseif (F <= 2.3e-101)
		tmp = Float64(Float64(Float64(-x) * cos(B)) / sin(B));
	elseif (F <= 2.4e-58)
		tmp = Float64(sqrt(0.5) / Float64(sin(B) / F));
	elseif (F <= 12000000.0)
		tmp = Float64(t_1 + Float64(F / Float64(B * Float64(F + Float64(0.5 * Float64(Float64(2.0 + Float64(x * 2.0)) / F))))));
	else
		tmp = Float64(t_1 + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = 1.0 / sin(B);
	t_1 = x * (-1.0 / tan(B));
	tmp = 0.0;
	if (F <= -0.00014)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= -1.58e-108)
		tmp = t_0 * (F * sqrt(0.5));
	elseif (F <= 2.3e-101)
		tmp = (-x * cos(B)) / sin(B);
	elseif (F <= 2.4e-58)
		tmp = sqrt(0.5) / (sin(B) / F);
	elseif (F <= 12000000.0)
		tmp = t_1 + (F / (B * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))));
	else
		tmp = t_1 + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.00014], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.58e-108], N[(t$95$0 * N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.3e-101], N[(N[((-x) * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.4e-58], N[(N[Sqrt[0.5], $MachinePrecision] / N[(N[Sin[B], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 12000000.0], N[(t$95$1 + N[(F / N[(B * N[(F + N[(0.5 * N[(N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + t$95$0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if F < -1.3999999999999999e-4

   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. Taylor expanded in F around -inf 98.0%

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

      1. expm1-log1p-u 45.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}\right)\right)} \]

      2. expm1-udef 45.5%

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

      3. div-inv 45.5%

         \[\leadsto e^{\mathsf{log1p}\left(\left(-\color{blue}{\frac{x}{\tan B}}\right) + \frac{-1}{\sin B}\right)} - 1 \]

      4. neg-mul-1 45.5%

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

      5. fma-def 45.5%

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

   4. Applied egg-rr 45.5%

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

      1. expm1-def 45.5%

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

      2. expm1-log1p 98.0%

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

      3. rem-log-exp 45.1%

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

      4. fma-udef 45.1%

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

      5. neg-mul-1 45.1%

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

      6. prod-exp 40.9%

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

      7. *-commutative 40.9%

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

      8. prod-exp 45.1%

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

      9. rem-log-exp 98.0%

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

      10. unsub-neg 98.0%

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

   6. Simplified 98.0%

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

   if -1.3999999999999999e-4 < F < -1.5799999999999999e-108

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. unsub-neg 99.5%

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

      3. associate-*l/ 99.4%

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

      4. associate-*r/ 99.7%

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

      5. *-commutative 99.7%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in F around 0 99.1%

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

   5. Taylor expanded in x around 0 70.1%

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

      1. div-inv 70.3%

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

      2. *-commutative 70.3%

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

   7. Applied egg-rr 70.3%

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

   if -1.5799999999999999e-108 < F < 2.2999999999999999e-101

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. unsub-neg 99.5%

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

      3. associate-*l/ 99.5%

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

      4. associate-*r/ 99.5%

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

      5. *-commutative 99.5%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in F around 0 99.7%

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

   5. Taylor expanded in B around 0 91.2%

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

   6. Taylor expanded in F around 0 86.0%

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

      1. associate-*r/ 86.0%

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

      2. *-commutative 86.0%

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

      3. associate-*r* 86.0%

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

      4. neg-mul-1 86.0%

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

   8. Simplified 86.0%

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

   if 2.2999999999999999e-101 < F < 2.4000000000000001e-58

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unsub-neg 99.6%

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

      3. associate-*l/ 99.6%

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

      4. associate-*r/ 99.3%

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

      5. *-commutative 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in F around 0 99.2%

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

   5. Taylor expanded in x around 0 85.7%

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

      1. associate-/l* 85.7%

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

   7. Simplified 85.7%

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

   if 2.4000000000000001e-58 < F < 1.2e7

   1. Initial program 94.8%

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

      1. associate-*l/ 99.6%

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

      2. +-commutative 99.6%

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

      3. *-commutative 99.6%

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

      4. fma-udef 99.6%

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

      5. fma-def 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. associate-/l* 99.6%

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

      9. fma-def 99.6%

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

      10. fma-udef 99.6%

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

      11. *-commutative 99.6%

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

      12. fma-def 99.6%

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

      13. fma-def 99.6%

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

   3. Applied egg-rr 99.6%

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

      1. clear-num 99.6%

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

      2. inv-pow 99.6%

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

      3. div-inv 99.6%

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

      4. pow-flip 99.6%

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

      5. metadata-eval 99.6%

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

      6. pow1/2 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. unpow-1 99.6%

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

      2. associate-/l* 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in F around inf 70.0%

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

   9. Taylor expanded in B around 0 68.6%

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

   if 1.2e7 < F

   1. Initial program 46.6%

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

   2. Taylor expanded in F around inf 99.7%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.00014:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.58 \cdot 10^{-108}:\\ \;\;\;\;\frac{1}{\sin B} \cdot \left(F \cdot \sqrt{0.5}\right)\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{-101}:\\ \;\;\;\;\frac{\left(-x\right) \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 2.4 \cdot 10^{-58}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\sin B}{F}}\\ \mathbf{elif}\;F \leq 12000000:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B \cdot \left(F + 0.5 \cdot \frac{2 + x \cdot 2}{F}\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 12: 84.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -0.012:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -4.1 \cdot 10^{-108}:\\ \;\;\;\;t_0 \cdot \left(F \cdot \sqrt{0.5}\right)\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{-102}:\\ \;\;\;\;t_1 + \frac{F}{\sin B} \cdot \frac{F}{x}\\ \mathbf{elif}\;F \leq 1.85 \cdot 10^{-58}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\sin B}{F}}\\ \mathbf{elif}\;F \leq 12000000:\\ \;\;\;\;t_1 + \frac{F}{B \cdot \left(F + 0.5 \cdot \frac{2 + x \cdot 2}{F}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1 + t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))) (t_1 (* x (/ -1.0 (tan B)))))
   (if (<= F -0.012)
     (- (/ -1.0 (sin B)) (/ x (tan B)))
     (if (<= F -4.1e-108)
       (* t_0 (* F (sqrt 0.5)))
       (if (<= F 7.5e-102)
         (+ t_1 (* (/ F (sin B)) (/ F x)))
         (if (<= F 1.85e-58)
           (/ (sqrt 0.5) (/ (sin B) F))
           (if (<= F 12000000.0)
             (+ t_1 (/ F (* B (+ F (* 0.5 (/ (+ 2.0 (* x 2.0)) F))))))
             (+ t_1 t_0))))))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / sin(B);
	double t_1 = x * (-1.0 / tan(B));
	double tmp;
	if (F <= -0.012) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= -4.1e-108) {
		tmp = t_0 * (F * sqrt(0.5));
	} else if (F <= 7.5e-102) {
		tmp = t_1 + ((F / sin(B)) * (F / x));
	} else if (F <= 1.85e-58) {
		tmp = sqrt(0.5) / (sin(B) / F);
	} else if (F <= 12000000.0) {
		tmp = t_1 + (F / (B * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))));
	} else {
		tmp = t_1 + t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 / sin(b)
    t_1 = x * ((-1.0d0) / tan(b))
    if (f <= (-0.012d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= (-4.1d-108)) then
        tmp = t_0 * (f * sqrt(0.5d0))
    else if (f <= 7.5d-102) then
        tmp = t_1 + ((f / sin(b)) * (f / x))
    else if (f <= 1.85d-58) then
        tmp = sqrt(0.5d0) / (sin(b) / f)
    else if (f <= 12000000.0d0) then
        tmp = t_1 + (f / (b * (f + (0.5d0 * ((2.0d0 + (x * 2.0d0)) / f)))))
    else
        tmp = t_1 + t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = 1.0 / Math.sin(B);
	double t_1 = x * (-1.0 / Math.tan(B));
	double tmp;
	if (F <= -0.012) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= -4.1e-108) {
		tmp = t_0 * (F * Math.sqrt(0.5));
	} else if (F <= 7.5e-102) {
		tmp = t_1 + ((F / Math.sin(B)) * (F / x));
	} else if (F <= 1.85e-58) {
		tmp = Math.sqrt(0.5) / (Math.sin(B) / F);
	} else if (F <= 12000000.0) {
		tmp = t_1 + (F / (B * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))));
	} else {
		tmp = t_1 + t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = 1.0 / math.sin(B)
	t_1 = x * (-1.0 / math.tan(B))
	tmp = 0
	if F <= -0.012:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= -4.1e-108:
		tmp = t_0 * (F * math.sqrt(0.5))
	elif F <= 7.5e-102:
		tmp = t_1 + ((F / math.sin(B)) * (F / x))
	elif F <= 1.85e-58:
		tmp = math.sqrt(0.5) / (math.sin(B) / F)
	elif F <= 12000000.0:
		tmp = t_1 + (F / (B * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))))
	else:
		tmp = t_1 + t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / sin(B))
	t_1 = Float64(x * Float64(-1.0 / tan(B)))
	tmp = 0.0
	if (F <= -0.012)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= -4.1e-108)
		tmp = Float64(t_0 * Float64(F * sqrt(0.5)));
	elseif (F <= 7.5e-102)
		tmp = Float64(t_1 + Float64(Float64(F / sin(B)) * Float64(F / x)));
	elseif (F <= 1.85e-58)
		tmp = Float64(sqrt(0.5) / Float64(sin(B) / F));
	elseif (F <= 12000000.0)
		tmp = Float64(t_1 + Float64(F / Float64(B * Float64(F + Float64(0.5 * Float64(Float64(2.0 + Float64(x * 2.0)) / F))))));
	else
		tmp = Float64(t_1 + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = 1.0 / sin(B);
	t_1 = x * (-1.0 / tan(B));
	tmp = 0.0;
	if (F <= -0.012)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= -4.1e-108)
		tmp = t_0 * (F * sqrt(0.5));
	elseif (F <= 7.5e-102)
		tmp = t_1 + ((F / sin(B)) * (F / x));
	elseif (F <= 1.85e-58)
		tmp = sqrt(0.5) / (sin(B) / F);
	elseif (F <= 12000000.0)
		tmp = t_1 + (F / (B * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))));
	else
		tmp = t_1 + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.012], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -4.1e-108], N[(t$95$0 * N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7.5e-102], N[(t$95$1 + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(F / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.85e-58], N[(N[Sqrt[0.5], $MachinePrecision] / N[(N[Sin[B], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 12000000.0], N[(t$95$1 + N[(F / N[(B * N[(F + N[(0.5 * N[(N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + t$95$0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if F < -0.012

   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. Taylor expanded in F around -inf 98.0%

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

      1. expm1-log1p-u 45.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}\right)\right)} \]

      2. expm1-udef 45.5%

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

      3. div-inv 45.5%

         \[\leadsto e^{\mathsf{log1p}\left(\left(-\color{blue}{\frac{x}{\tan B}}\right) + \frac{-1}{\sin B}\right)} - 1 \]

      4. neg-mul-1 45.5%

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

      5. fma-def 45.5%

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

   4. Applied egg-rr 45.5%

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

      1. expm1-def 45.5%

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

      2. expm1-log1p 98.0%

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

      3. rem-log-exp 45.1%

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

      4. fma-udef 45.1%

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

      5. neg-mul-1 45.1%

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

      6. prod-exp 40.9%

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

      7. *-commutative 40.9%

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

      8. prod-exp 45.1%

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

      9. rem-log-exp 98.0%

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

      10. unsub-neg 98.0%

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

   6. Simplified 98.0%

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

   if -0.012 < F < -4.10000000000000037e-108

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. unsub-neg 99.5%

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

      3. associate-*l/ 99.4%

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

      4. associate-*r/ 99.7%

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

      5. *-commutative 99.7%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in F around 0 99.1%

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

   5. Taylor expanded in x around 0 70.1%

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

      1. div-inv 70.3%

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

      2. *-commutative 70.3%

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

   7. Applied egg-rr 70.3%

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

   if -4.10000000000000037e-108 < F < 7.5000000000000008e-102

   1. Initial program 99.5%

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

      1. associate-*l/ 99.5%

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

      2. +-commutative 99.5%

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

      3. *-commutative 99.5%

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

      4. fma-udef 99.5%

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

      5. fma-def 99.5%

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

      6. metadata-eval 99.5%

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

      7. metadata-eval 99.5%

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

      8. associate-/l* 99.5%

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

      9. fma-def 99.5%

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

      10. fma-udef 99.5%

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

      11. *-commutative 99.5%

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

      12. fma-def 99.5%

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

      13. fma-def 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. clear-num 99.4%

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

      2. inv-pow 99.4%

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

      3. div-inv 99.4%

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

      4. pow-flip 99.5%

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

      5. metadata-eval 99.5%

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

      6. pow1/2 99.5%

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

   5. Applied egg-rr 99.5%

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

      1. unpow-1 99.5%

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

      2. associate-/l* 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in F around inf 85.8%

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

   9. Taylor expanded in x around inf 80.1%

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

       1. unpow2 80.1%

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

       2. times-frac 86.0%

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

   11. Simplified 86.0%

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

   if 7.5000000000000008e-102 < F < 1.8500000000000001e-58

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unsub-neg 99.6%

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

      3. associate-*l/ 99.6%

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

      4. associate-*r/ 99.3%

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

      5. *-commutative 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in F around 0 99.2%

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

   5. Taylor expanded in x around 0 85.7%

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

      1. associate-/l* 85.7%

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

   7. Simplified 85.7%

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

   if 1.8500000000000001e-58 < F < 1.2e7

   1. Initial program 94.8%

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

      1. associate-*l/ 99.6%

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

      2. +-commutative 99.6%

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

      3. *-commutative 99.6%

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

      4. fma-udef 99.6%

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

      5. fma-def 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. associate-/l* 99.6%

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

      9. fma-def 99.6%

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

      10. fma-udef 99.6%

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

      11. *-commutative 99.6%

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

      12. fma-def 99.6%

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

      13. fma-def 99.6%

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

   3. Applied egg-rr 99.6%

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

      1. clear-num 99.6%

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

      2. inv-pow 99.6%

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

      3. div-inv 99.6%

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

      4. pow-flip 99.6%

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

      5. metadata-eval 99.6%

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

      6. pow1/2 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. unpow-1 99.6%

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

      2. associate-/l* 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in F around inf 70.0%

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

   9. Taylor expanded in B around 0 68.6%

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

   if 1.2e7 < F

   1. Initial program 46.6%

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

   2. Taylor expanded in F around inf 99.7%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.012:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -4.1 \cdot 10^{-108}:\\ \;\;\;\;\frac{1}{\sin B} \cdot \left(F \cdot \sqrt{0.5}\right)\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{-102}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot \frac{F}{x}\\ \mathbf{elif}\;F \leq 1.85 \cdot 10^{-58}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\sin B}{F}}\\ \mathbf{elif}\;F \leq 12000000:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B \cdot \left(F + 0.5 \cdot \frac{2 + x \cdot 2}{F}\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 13: 70.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.039:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -3.2 \cdot 10^{-108} \lor \neg \left(F \leq 2 \cdot 10^{-107}\right) \land F \leq 1.25 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\sin B}{F}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B \cdot \left(F + 0.5 \cdot \frac{2 + x \cdot 2}{F}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.039)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (or (<= F -3.2e-108) (and (not (<= F 2e-107)) (<= F 1.25e-57)))
     (/ (sqrt 0.5) (/ (sin B) F))
     (+
      (* x (/ -1.0 (tan B)))
      (/ F (* B (+ F (* 0.5 (/ (+ 2.0 (* x 2.0)) F)))))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.039) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if ((F <= -3.2e-108) || (!(F <= 2e-107) && (F <= 1.25e-57))) {
		tmp = sqrt(0.5) / (sin(B) / F);
	} else {
		tmp = (x * (-1.0 / tan(B))) + (F / (B * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-0.039d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if ((f <= (-3.2d-108)) .or. (.not. (f <= 2d-107)) .and. (f <= 1.25d-57)) then
        tmp = sqrt(0.5d0) / (sin(b) / f)
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (f / (b * (f + (0.5d0 * ((2.0d0 + (x * 2.0d0)) / f)))))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.039) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if ((F <= -3.2e-108) || (!(F <= 2e-107) && (F <= 1.25e-57))) {
		tmp = Math.sqrt(0.5) / (Math.sin(B) / F);
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (F / (B * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -0.039:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif (F <= -3.2e-108) or (not (F <= 2e-107) and (F <= 1.25e-57)):
		tmp = math.sqrt(0.5) / (math.sin(B) / F)
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (F / (B * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.039)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif ((F <= -3.2e-108) || (!(F <= 2e-107) && (F <= 1.25e-57)))
		tmp = Float64(sqrt(0.5) / Float64(sin(B) / F));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(F / Float64(B * Float64(F + Float64(0.5 * Float64(Float64(2.0 + Float64(x * 2.0)) / F))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -0.039)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif ((F <= -3.2e-108) || (~((F <= 2e-107)) && (F <= 1.25e-57)))
		tmp = sqrt(0.5) / (sin(B) / F);
	else
		tmp = (x * (-1.0 / tan(B))) + (F / (B * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.039], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[F, -3.2e-108], And[N[Not[LessEqual[F, 2e-107]], $MachinePrecision], LessEqual[F, 1.25e-57]]], N[(N[Sqrt[0.5], $MachinePrecision] / N[(N[Sin[B], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(F / N[(B * N[(F + N[(0.5 * N[(N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.0389999999999999999

   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. Taylor expanded in F around -inf 98.0%

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

   3. Taylor expanded in B around 0 79.9%

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

   if -0.0389999999999999999 < F < -3.2e-108 or 2e-107 < F < 1.25e-57

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. unsub-neg 99.5%

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

      3. associate-*l/ 99.4%

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

      4. associate-*r/ 99.6%

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

      5. *-commutative 99.6%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in F around 0 99.1%

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

   5. Taylor expanded in x around 0 73.5%

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

      1. associate-/l* 73.5%

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

   7. Simplified 73.5%

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

   if -3.2e-108 < F < 2e-107 or 1.25e-57 < F

   1. Initial program 74.8%

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

      1. associate-*l/ 84.5%

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

      2. +-commutative 84.5%

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

      3. *-commutative 84.5%

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

      4. fma-udef 84.5%

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

      5. fma-def 84.5%

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

      6. metadata-eval 84.5%

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

      7. metadata-eval 84.5%

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

      8. associate-/l* 84.5%

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

      9. fma-def 84.5%

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

      10. fma-udef 84.5%

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

      11. *-commutative 84.5%

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

      12. fma-def 84.5%

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

      13. fma-def 84.5%

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

   3. Applied egg-rr 84.5%

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

      1. clear-num 84.5%

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

      2. inv-pow 84.5%

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

      3. div-inv 84.5%

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

      4. pow-flip 84.5%

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

      5. metadata-eval 84.5%

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

      6. pow1/2 84.5%

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

   5. Applied egg-rr 84.5%

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

      1. unpow-1 84.5%

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

      2. associate-/l* 84.5%

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

   7. Simplified 84.5%

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

   8. Taylor expanded in F around inf 90.0%

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

   9. Taylor expanded in B around 0 81.5%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{\left(F + 0.5 \cdot \frac{2 \cdot x + 2}{F}\right) \cdot B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.039:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -3.2 \cdot 10^{-108} \lor \neg \left(F \leq 2 \cdot 10^{-107}\right) \land F \leq 1.25 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\sin B}{F}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B \cdot \left(F + 0.5 \cdot \frac{2 + x \cdot 2}{F}\right)}\\ \end{array} \]

Alternative 14: 70.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{0.5}}{\frac{\sin B}{F}}\\ \mathbf{if}\;F \leq -0.082:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.02 \cdot 10^{-108}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 1.35 \cdot 10^{-101}:\\ \;\;\;\;\frac{-\cos B}{\frac{\sin B}{x}}\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{-59}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B \cdot \left(F + 0.5 \cdot \frac{2 + x \cdot 2}{F}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (sqrt 0.5) (/ (sin B) F))))
   (if (<= F -0.082)
     (- (/ -1.0 (sin B)) (/ x B))
     (if (<= F -1.02e-108)
       t_0
       (if (<= F 1.35e-101)
         (/ (- (cos B)) (/ (sin B) x))
         (if (<= F 6.2e-59)
           t_0
           (+
            (* x (/ -1.0 (tan B)))
            (/ F (* B (+ F (* 0.5 (/ (+ 2.0 (* x 2.0)) F))))))))))))

C

double code(double F, double B, double x) {
	double t_0 = sqrt(0.5) / (sin(B) / F);
	double tmp;
	if (F <= -0.082) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -1.02e-108) {
		tmp = t_0;
	} else if (F <= 1.35e-101) {
		tmp = -cos(B) / (sin(B) / x);
	} else if (F <= 6.2e-59) {
		tmp = t_0;
	} else {
		tmp = (x * (-1.0 / tan(B))) + (F / (B * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))));
	}
	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 = sqrt(0.5d0) / (sin(b) / f)
    if (f <= (-0.082d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-1.02d-108)) then
        tmp = t_0
    else if (f <= 1.35d-101) then
        tmp = -cos(b) / (sin(b) / x)
    else if (f <= 6.2d-59) then
        tmp = t_0
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (f / (b * (f + (0.5d0 * ((2.0d0 + (x * 2.0d0)) / f)))))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = Math.sqrt(0.5) / (Math.sin(B) / F);
	double tmp;
	if (F <= -0.082) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -1.02e-108) {
		tmp = t_0;
	} else if (F <= 1.35e-101) {
		tmp = -Math.cos(B) / (Math.sin(B) / x);
	} else if (F <= 6.2e-59) {
		tmp = t_0;
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (F / (B * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = math.sqrt(0.5) / (math.sin(B) / F)
	tmp = 0
	if F <= -0.082:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -1.02e-108:
		tmp = t_0
	elif F <= 1.35e-101:
		tmp = -math.cos(B) / (math.sin(B) / x)
	elif F <= 6.2e-59:
		tmp = t_0
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (F / (B * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(sqrt(0.5) / Float64(sin(B) / F))
	tmp = 0.0
	if (F <= -0.082)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -1.02e-108)
		tmp = t_0;
	elseif (F <= 1.35e-101)
		tmp = Float64(Float64(-cos(B)) / Float64(sin(B) / x));
	elseif (F <= 6.2e-59)
		tmp = t_0;
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(F / Float64(B * Float64(F + Float64(0.5 * Float64(Float64(2.0 + Float64(x * 2.0)) / F))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = sqrt(0.5) / (sin(B) / F);
	tmp = 0.0;
	if (F <= -0.082)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -1.02e-108)
		tmp = t_0;
	elseif (F <= 1.35e-101)
		tmp = -cos(B) / (sin(B) / x);
	elseif (F <= 6.2e-59)
		tmp = t_0;
	else
		tmp = (x * (-1.0 / tan(B))) + (F / (B * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[Sqrt[0.5], $MachinePrecision] / N[(N[Sin[B], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.082], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.02e-108], t$95$0, If[LessEqual[F, 1.35e-101], N[((-N[Cos[B], $MachinePrecision]) / N[(N[Sin[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 6.2e-59], t$95$0, N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(F / N[(B * N[(F + N[(0.5 * N[(N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.0820000000000000034

   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. Taylor expanded in F around -inf 98.0%

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

   3. Taylor expanded in B around 0 79.9%

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

   if -0.0820000000000000034 < F < -1.02000000000000008e-108 or 1.3500000000000001e-101 < F < 6.19999999999999998e-59

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. unsub-neg 99.5%

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

      3. associate-*l/ 99.4%

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

      4. associate-*r/ 99.6%

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

      5. *-commutative 99.6%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in F around 0 99.1%

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

   5. Taylor expanded in x around 0 73.5%

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

      1. associate-/l* 73.5%

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

   7. Simplified 73.5%

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

   if -1.02000000000000008e-108 < F < 1.3500000000000001e-101

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. unsub-neg 99.5%

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

      3. associate-*l/ 99.5%

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

      4. associate-*r/ 99.5%

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

      5. *-commutative 99.5%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in F around 0 99.7%

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

   5. Taylor expanded in B around 0 91.2%

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

   6. Taylor expanded in F around 0 86.0%

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

      1. mul-1-neg 86.0%

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

      2. associate-/l* 85.9%

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

   8. Simplified 85.9%

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

   if 6.19999999999999998e-59 < F

   1. Initial program 57.6%

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

      1. associate-*l/ 74.1%

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

      2. +-commutative 74.1%

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

      3. *-commutative 74.1%

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

      4. fma-udef 74.1%

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

      5. fma-def 74.1%

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

      6. metadata-eval 74.1%

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

      7. metadata-eval 74.1%

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

      8. associate-/l* 74.1%

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

      9. fma-def 74.1%

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

      10. fma-udef 74.1%

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

      11. *-commutative 74.1%

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

      12. fma-def 74.1%

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

      13. fma-def 74.1%

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

   3. Applied egg-rr 74.1%

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

      1. clear-num 74.1%

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

      2. inv-pow 74.1%

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

      3. div-inv 74.1%

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

      4. pow-flip 74.1%

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

      5. metadata-eval 74.1%

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

      6. pow1/2 74.1%

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

   5. Applied egg-rr 74.1%

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

      1. unpow-1 74.1%

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

      2. associate-/l* 74.1%

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

   7. Simplified 74.1%

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

   8. Taylor expanded in F around inf 92.9%

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

   9. Taylor expanded in B around 0 78.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{\left(F + 0.5 \cdot \frac{2 \cdot x + 2}{F}\right) \cdot B}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.082:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.02 \cdot 10^{-108}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\sin B}{F}}\\ \mathbf{elif}\;F \leq 1.35 \cdot 10^{-101}:\\ \;\;\;\;\frac{-\cos B}{\frac{\sin B}{x}}\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{-59}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\sin B}{F}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B \cdot \left(F + 0.5 \cdot \frac{2 + x \cdot 2}{F}\right)}\\ \end{array} \]

Alternative 15: 70.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{0.5}}{\frac{\sin B}{F}}\\ \mathbf{if}\;F \leq -0.0055:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.32 \cdot 10^{-108}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 1.22 \cdot 10^{-101}:\\ \;\;\;\;\frac{\left(-x\right) \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 1.02 \cdot 10^{-57}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B \cdot \left(F + 0.5 \cdot \frac{2 + x \cdot 2}{F}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (sqrt 0.5) (/ (sin B) F))))
   (if (<= F -0.0055)
     (- (/ -1.0 (sin B)) (/ x B))
     (if (<= F -1.32e-108)
       t_0
       (if (<= F 1.22e-101)
         (/ (* (- x) (cos B)) (sin B))
         (if (<= F 1.02e-57)
           t_0
           (+
            (* x (/ -1.0 (tan B)))
            (/ F (* B (+ F (* 0.5 (/ (+ 2.0 (* x 2.0)) F))))))))))))

C

double code(double F, double B, double x) {
	double t_0 = sqrt(0.5) / (sin(B) / F);
	double tmp;
	if (F <= -0.0055) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -1.32e-108) {
		tmp = t_0;
	} else if (F <= 1.22e-101) {
		tmp = (-x * cos(B)) / sin(B);
	} else if (F <= 1.02e-57) {
		tmp = t_0;
	} else {
		tmp = (x * (-1.0 / tan(B))) + (F / (B * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))));
	}
	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 = sqrt(0.5d0) / (sin(b) / f)
    if (f <= (-0.0055d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-1.32d-108)) then
        tmp = t_0
    else if (f <= 1.22d-101) then
        tmp = (-x * cos(b)) / sin(b)
    else if (f <= 1.02d-57) then
        tmp = t_0
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (f / (b * (f + (0.5d0 * ((2.0d0 + (x * 2.0d0)) / f)))))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = Math.sqrt(0.5) / (Math.sin(B) / F);
	double tmp;
	if (F <= -0.0055) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -1.32e-108) {
		tmp = t_0;
	} else if (F <= 1.22e-101) {
		tmp = (-x * Math.cos(B)) / Math.sin(B);
	} else if (F <= 1.02e-57) {
		tmp = t_0;
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (F / (B * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = math.sqrt(0.5) / (math.sin(B) / F)
	tmp = 0
	if F <= -0.0055:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -1.32e-108:
		tmp = t_0
	elif F <= 1.22e-101:
		tmp = (-x * math.cos(B)) / math.sin(B)
	elif F <= 1.02e-57:
		tmp = t_0
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (F / (B * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(sqrt(0.5) / Float64(sin(B) / F))
	tmp = 0.0
	if (F <= -0.0055)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -1.32e-108)
		tmp = t_0;
	elseif (F <= 1.22e-101)
		tmp = Float64(Float64(Float64(-x) * cos(B)) / sin(B));
	elseif (F <= 1.02e-57)
		tmp = t_0;
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(F / Float64(B * Float64(F + Float64(0.5 * Float64(Float64(2.0 + Float64(x * 2.0)) / F))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = sqrt(0.5) / (sin(B) / F);
	tmp = 0.0;
	if (F <= -0.0055)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -1.32e-108)
		tmp = t_0;
	elseif (F <= 1.22e-101)
		tmp = (-x * cos(B)) / sin(B);
	elseif (F <= 1.02e-57)
		tmp = t_0;
	else
		tmp = (x * (-1.0 / tan(B))) + (F / (B * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[Sqrt[0.5], $MachinePrecision] / N[(N[Sin[B], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.0055], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.32e-108], t$95$0, If[LessEqual[F, 1.22e-101], N[(N[((-x) * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.02e-57], t$95$0, N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(F / N[(B * N[(F + N[(0.5 * N[(N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.0054999999999999997

   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. Taylor expanded in F around -inf 98.0%

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

   3. Taylor expanded in B around 0 79.9%

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

   if -0.0054999999999999997 < F < -1.31999999999999996e-108 or 1.2199999999999999e-101 < F < 1.02e-57

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. unsub-neg 99.5%

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

      3. associate-*l/ 99.4%

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

      4. associate-*r/ 99.6%

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

      5. *-commutative 99.6%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in F around 0 99.1%

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

   5. Taylor expanded in x around 0 73.5%

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

      1. associate-/l* 73.5%

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

   7. Simplified 73.5%

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

   if -1.31999999999999996e-108 < F < 1.2199999999999999e-101

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. unsub-neg 99.5%

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

      3. associate-*l/ 99.5%

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

      4. associate-*r/ 99.5%

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

      5. *-commutative 99.5%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in F around 0 99.7%

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

   5. Taylor expanded in B around 0 91.2%

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

   6. Taylor expanded in F around 0 86.0%

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

      1. associate-*r/ 86.0%

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

      2. *-commutative 86.0%

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(x \cdot \cos B\right)}}{\sin B} \]

      3. associate-*r* 86.0%

         \[\leadsto \frac{\color{blue}{\left(-1 \cdot x\right) \cdot \cos B}}{\sin B} \]

      4. neg-mul-1 86.0%

         \[\leadsto \frac{\color{blue}{\left(-x\right)} \cdot \cos B}{\sin B} \]

   8. Simplified 86.0%

      \[\leadsto \color{blue}{\frac{\left(-x\right) \cdot \cos B}{\sin B}} \]

   if 1.02e-57 < F

   1. Initial program 57.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. associate-*l/ 74.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} \]

      2. +-commutative 74.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot {\color{blue}{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)}}{\sin B} \]

      3. *-commutative 74.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot {\left(\color{blue}{x \cdot 2} + \left(F \cdot F + 2\right)\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \]

      4. fma-udef 74.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot {\color{blue}{\left(\mathsf{fma}\left(x, 2, F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)}}{\sin B} \]

      5. fma-def 74.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \]

      6. metadata-eval 74.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(-\color{blue}{0.5}\right)}}{\sin B} \]

      7. metadata-eval 74.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{-0.5}}}{\sin B} \]

      8. associate-/l* 74.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}} \]

      9. fma-def 74.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F + 2}\right)\right)}^{-0.5}}} \]

      10. fma-udef 74.1%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\frac{\sin B}{{\color{blue}{\left(x \cdot 2 + \left(F \cdot F + 2\right)\right)}}^{-0.5}}} \]

      11. *-commutative 74.1%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\frac{\sin B}{{\left(\color{blue}{2 \cdot x} + \left(F \cdot F + 2\right)\right)}^{-0.5}}} \]

      12. fma-def 74.1%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\frac{\sin B}{{\color{blue}{\left(\mathsf{fma}\left(2, x, F \cdot F + 2\right)\right)}}^{-0.5}}} \]

      13. fma-def 74.1%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)\right)}^{-0.5}}} \]

   3. Applied egg-rr 74.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}} \]
   4. Step-by-step derivation

      1. clear-num 74.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\frac{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}{F}}} \]

      2. inv-pow 74.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{{\left(\frac{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}{F}\right)}^{-1}} \]

      3. div-inv 74.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + {\left(\frac{\color{blue}{\sin B \cdot \frac{1}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}}{F}\right)}^{-1} \]

      4. pow-flip 74.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + {\left(\frac{\sin B \cdot \color{blue}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(--0.5\right)}}}{F}\right)}^{-1} \]

      5. metadata-eval 74.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + {\left(\frac{\sin B \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{0.5}}}{F}\right)}^{-1} \]

      6. pow1/2 74.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + {\left(\frac{\sin B \cdot \color{blue}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}{F}\right)}^{-1} \]

   5. Applied egg-rr 74.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{{\left(\frac{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}{F}\right)}^{-1}} \]
   6. Step-by-step derivation

      1. unpow-1 74.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\frac{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}{F}}} \]

      2. associate-/l* 74.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\color{blue}{\frac{\sin B}{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}}} \]

   7. Simplified 74.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\frac{\sin B}{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}}} \]

   8. Taylor expanded in F around inf 92.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\frac{\sin B}{\frac{F}{\color{blue}{F + 0.5 \cdot \frac{2 \cdot x + 2}{F}}}}} \]

   9. Taylor expanded in B around 0 78.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{\left(F + 0.5 \cdot \frac{2 \cdot x + 2}{F}\right) \cdot B}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.0055:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.32 \cdot 10^{-108}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\sin B}{F}}\\ \mathbf{elif}\;F \leq 1.22 \cdot 10^{-101}:\\ \;\;\;\;\frac{\left(-x\right) \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 1.02 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\sin B}{F}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B \cdot \left(F + 0.5 \cdot \frac{2 + x \cdot 2}{F}\right)}\\ \end{array} \]

Alternative 16: 70.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.36:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -3.9 \cdot 10^{-108}:\\ \;\;\;\;\frac{1}{\sin B} \cdot \left(F \cdot \sqrt{0.5}\right)\\ \mathbf{elif}\;F \leq 1.35 \cdot 10^{-101}:\\ \;\;\;\;\frac{\left(-x\right) \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 5.3 \cdot 10^{-58}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\sin B}{F}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B \cdot \left(F + 0.5 \cdot \frac{2 + x \cdot 2}{F}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.36)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F -3.9e-108)
     (* (/ 1.0 (sin B)) (* F (sqrt 0.5)))
     (if (<= F 1.35e-101)
       (/ (* (- x) (cos B)) (sin B))
       (if (<= F 5.3e-58)
         (/ (sqrt 0.5) (/ (sin B) F))
         (+
          (* x (/ -1.0 (tan B)))
          (/ F (* B (+ F (* 0.5 (/ (+ 2.0 (* x 2.0)) F)))))))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.36) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -3.9e-108) {
		tmp = (1.0 / sin(B)) * (F * sqrt(0.5));
	} else if (F <= 1.35e-101) {
		tmp = (-x * cos(B)) / sin(B);
	} else if (F <= 5.3e-58) {
		tmp = sqrt(0.5) / (sin(B) / F);
	} else {
		tmp = (x * (-1.0 / tan(B))) + (F / (B * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-0.36d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-3.9d-108)) then
        tmp = (1.0d0 / sin(b)) * (f * sqrt(0.5d0))
    else if (f <= 1.35d-101) then
        tmp = (-x * cos(b)) / sin(b)
    else if (f <= 5.3d-58) then
        tmp = sqrt(0.5d0) / (sin(b) / f)
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (f / (b * (f + (0.5d0 * ((2.0d0 + (x * 2.0d0)) / f)))))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.36) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -3.9e-108) {
		tmp = (1.0 / Math.sin(B)) * (F * Math.sqrt(0.5));
	} else if (F <= 1.35e-101) {
		tmp = (-x * Math.cos(B)) / Math.sin(B);
	} else if (F <= 5.3e-58) {
		tmp = Math.sqrt(0.5) / (Math.sin(B) / F);
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (F / (B * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -0.36:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -3.9e-108:
		tmp = (1.0 / math.sin(B)) * (F * math.sqrt(0.5))
	elif F <= 1.35e-101:
		tmp = (-x * math.cos(B)) / math.sin(B)
	elif F <= 5.3e-58:
		tmp = math.sqrt(0.5) / (math.sin(B) / F)
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (F / (B * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.36)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -3.9e-108)
		tmp = Float64(Float64(1.0 / sin(B)) * Float64(F * sqrt(0.5)));
	elseif (F <= 1.35e-101)
		tmp = Float64(Float64(Float64(-x) * cos(B)) / sin(B));
	elseif (F <= 5.3e-58)
		tmp = Float64(sqrt(0.5) / Float64(sin(B) / F));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(F / Float64(B * Float64(F + Float64(0.5 * Float64(Float64(2.0 + Float64(x * 2.0)) / F))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -0.36)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -3.9e-108)
		tmp = (1.0 / sin(B)) * (F * sqrt(0.5));
	elseif (F <= 1.35e-101)
		tmp = (-x * cos(B)) / sin(B);
	elseif (F <= 5.3e-58)
		tmp = sqrt(0.5) / (sin(B) / F);
	else
		tmp = (x * (-1.0 / tan(B))) + (F / (B * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.36], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -3.9e-108], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.35e-101], N[(N[((-x) * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.3e-58], N[(N[Sqrt[0.5], $MachinePrecision] / N[(N[Sin[B], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(F / N[(B * N[(F + N[(0.5 * N[(N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -0.35999999999999999

   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. Taylor expanded in F around -inf 98.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 79.9%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{-1}{\sin B} \]

   if -0.35999999999999999 < F < -3.89999999999999995e-108

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative 99.5%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      2. unsub-neg 99.5%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      3. associate-*l/ 99.4%

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      4. associate-*r/ 99.7%

         \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      5. *-commutative 99.7%

         \[\leadsto \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \cdot F} - x \cdot \frac{1}{\tan B} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]

   4. Taylor expanded in F around 0 99.1%

      \[\leadsto F \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 \cdot x + 2}}\right)} - \frac{x}{\tan B} \]

   5. Taylor expanded in x around 0 70.1%

      \[\leadsto \color{blue}{\frac{\sqrt{0.5} \cdot F}{\sin B}} \]
   6. Step-by-step derivation

      1. div-inv 70.3%

         \[\leadsto \color{blue}{\left(\sqrt{0.5} \cdot F\right) \cdot \frac{1}{\sin B}} \]

      2. *-commutative 70.3%

         \[\leadsto \color{blue}{\left(F \cdot \sqrt{0.5}\right)} \cdot \frac{1}{\sin B} \]

   7. Applied egg-rr 70.3%

      \[\leadsto \color{blue}{\left(F \cdot \sqrt{0.5}\right) \cdot \frac{1}{\sin B}} \]

   if -3.89999999999999995e-108 < F < 1.3500000000000001e-101

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative 99.5%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      2. unsub-neg 99.5%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      3. associate-*l/ 99.5%

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      4. associate-*r/ 99.5%

         \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      5. *-commutative 99.5%

         \[\leadsto \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \cdot F} - x \cdot \frac{1}{\tan B} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]

   4. Taylor expanded in F around 0 99.7%

      \[\leadsto F \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 \cdot x + 2}}\right)} - \frac{x}{\tan B} \]

   5. Taylor expanded in B around 0 91.2%

      \[\leadsto \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 \cdot x + 2}}} - \frac{x}{\tan B} \]

   6. Taylor expanded in F around 0 86.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{\cos B \cdot x}{\sin B}} \]
   7. Step-by-step derivation

      1. associate-*r/ 86.0%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(\cos B \cdot x\right)}{\sin B}} \]

      2. *-commutative 86.0%

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(x \cdot \cos B\right)}}{\sin B} \]

      3. associate-*r* 86.0%

         \[\leadsto \frac{\color{blue}{\left(-1 \cdot x\right) \cdot \cos B}}{\sin B} \]

      4. neg-mul-1 86.0%

         \[\leadsto \frac{\color{blue}{\left(-x\right)} \cdot \cos B}{\sin B} \]

   8. Simplified 86.0%

      \[\leadsto \color{blue}{\frac{\left(-x\right) \cdot \cos B}{\sin B}} \]

   if 1.3500000000000001e-101 < F < 5.3000000000000003e-58

   1. Initial program 99.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative 99.6%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      2. unsub-neg 99.6%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      3. associate-*l/ 99.6%

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      4. associate-*r/ 99.3%

         \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      5. *-commutative 99.3%

         \[\leadsto \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \cdot F} - x \cdot \frac{1}{\tan B} \]

   3. Simplified 99.3%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]

   4. Taylor expanded in F around 0 99.2%

      \[\leadsto F \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 \cdot x + 2}}\right)} - \frac{x}{\tan B} \]

   5. Taylor expanded in x around 0 85.7%

      \[\leadsto \color{blue}{\frac{\sqrt{0.5} \cdot F}{\sin B}} \]
   6. Step-by-step derivation

      1. associate-/l* 85.7%

         \[\leadsto \color{blue}{\frac{\sqrt{0.5}}{\frac{\sin B}{F}}} \]

   7. Simplified 85.7%

      \[\leadsto \color{blue}{\frac{\sqrt{0.5}}{\frac{\sin B}{F}}} \]

   if 5.3000000000000003e-58 < F

   1. Initial program 57.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. associate-*l/ 74.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} \]

      2. +-commutative 74.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot {\color{blue}{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)}}{\sin B} \]

      3. *-commutative 74.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot {\left(\color{blue}{x \cdot 2} + \left(F \cdot F + 2\right)\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \]

      4. fma-udef 74.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot {\color{blue}{\left(\mathsf{fma}\left(x, 2, F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)}}{\sin B} \]

      5. fma-def 74.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \]

      6. metadata-eval 74.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(-\color{blue}{0.5}\right)}}{\sin B} \]

      7. metadata-eval 74.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{-0.5}}}{\sin B} \]

      8. associate-/l* 74.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}} \]

      9. fma-def 74.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F + 2}\right)\right)}^{-0.5}}} \]

      10. fma-udef 74.1%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\frac{\sin B}{{\color{blue}{\left(x \cdot 2 + \left(F \cdot F + 2\right)\right)}}^{-0.5}}} \]

      11. *-commutative 74.1%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\frac{\sin B}{{\left(\color{blue}{2 \cdot x} + \left(F \cdot F + 2\right)\right)}^{-0.5}}} \]

      12. fma-def 74.1%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\frac{\sin B}{{\color{blue}{\left(\mathsf{fma}\left(2, x, F \cdot F + 2\right)\right)}}^{-0.5}}} \]

      13. fma-def 74.1%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)\right)}^{-0.5}}} \]

   3. Applied egg-rr 74.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}} \]
   4. Step-by-step derivation

      1. clear-num 74.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\frac{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}{F}}} \]

      2. inv-pow 74.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{{\left(\frac{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}{F}\right)}^{-1}} \]

      3. div-inv 74.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + {\left(\frac{\color{blue}{\sin B \cdot \frac{1}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}}{F}\right)}^{-1} \]

      4. pow-flip 74.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + {\left(\frac{\sin B \cdot \color{blue}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(--0.5\right)}}}{F}\right)}^{-1} \]

      5. metadata-eval 74.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + {\left(\frac{\sin B \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{0.5}}}{F}\right)}^{-1} \]

      6. pow1/2 74.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + {\left(\frac{\sin B \cdot \color{blue}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}{F}\right)}^{-1} \]

   5. Applied egg-rr 74.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{{\left(\frac{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}{F}\right)}^{-1}} \]
   6. Step-by-step derivation

      1. unpow-1 74.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\frac{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}{F}}} \]

      2. associate-/l* 74.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\color{blue}{\frac{\sin B}{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}}} \]

   7. Simplified 74.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\frac{\sin B}{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}}} \]

   8. Taylor expanded in F around inf 92.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\frac{\sin B}{\frac{F}{\color{blue}{F + 0.5 \cdot \frac{2 \cdot x + 2}{F}}}}} \]

   9. Taylor expanded in B around 0 78.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{\left(F + 0.5 \cdot \frac{2 \cdot x + 2}{F}\right) \cdot B}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.36:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -3.9 \cdot 10^{-108}:\\ \;\;\;\;\frac{1}{\sin B} \cdot \left(F \cdot \sqrt{0.5}\right)\\ \mathbf{elif}\;F \leq 1.35 \cdot 10^{-101}:\\ \;\;\;\;\frac{\left(-x\right) \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 5.3 \cdot 10^{-58}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\sin B}{F}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B \cdot \left(F + 0.5 \cdot \frac{2 + x \cdot 2}{F}\right)}\\ \end{array} \]

Alternative 17: 77.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.0014:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.02 \cdot 10^{-108}:\\ \;\;\;\;\frac{1}{\sin B} \cdot \left(F \cdot \sqrt{0.5}\right)\\ \mathbf{elif}\;F \leq 1.7 \cdot 10^{-103}:\\ \;\;\;\;\frac{\left(-x\right) \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 6.9 \cdot 10^{-62}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\sin B}{F}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B \cdot \left(F + 0.5 \cdot \frac{2 + x \cdot 2}{F}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.0014)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F -1.02e-108)
     (* (/ 1.0 (sin B)) (* F (sqrt 0.5)))
     (if (<= F 1.7e-103)
       (/ (* (- x) (cos B)) (sin B))
       (if (<= F 6.9e-62)
         (/ (sqrt 0.5) (/ (sin B) F))
         (+
          (* x (/ -1.0 (tan B)))
          (/ F (* B (+ F (* 0.5 (/ (+ 2.0 (* x 2.0)) F)))))))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.0014) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= -1.02e-108) {
		tmp = (1.0 / sin(B)) * (F * sqrt(0.5));
	} else if (F <= 1.7e-103) {
		tmp = (-x * cos(B)) / sin(B);
	} else if (F <= 6.9e-62) {
		tmp = sqrt(0.5) / (sin(B) / F);
	} else {
		tmp = (x * (-1.0 / tan(B))) + (F / (B * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-0.0014d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= (-1.02d-108)) then
        tmp = (1.0d0 / sin(b)) * (f * sqrt(0.5d0))
    else if (f <= 1.7d-103) then
        tmp = (-x * cos(b)) / sin(b)
    else if (f <= 6.9d-62) then
        tmp = sqrt(0.5d0) / (sin(b) / f)
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (f / (b * (f + (0.5d0 * ((2.0d0 + (x * 2.0d0)) / f)))))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.0014) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= -1.02e-108) {
		tmp = (1.0 / Math.sin(B)) * (F * Math.sqrt(0.5));
	} else if (F <= 1.7e-103) {
		tmp = (-x * Math.cos(B)) / Math.sin(B);
	} else if (F <= 6.9e-62) {
		tmp = Math.sqrt(0.5) / (Math.sin(B) / F);
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (F / (B * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -0.0014:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= -1.02e-108:
		tmp = (1.0 / math.sin(B)) * (F * math.sqrt(0.5))
	elif F <= 1.7e-103:
		tmp = (-x * math.cos(B)) / math.sin(B)
	elif F <= 6.9e-62:
		tmp = math.sqrt(0.5) / (math.sin(B) / F)
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (F / (B * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.0014)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= -1.02e-108)
		tmp = Float64(Float64(1.0 / sin(B)) * Float64(F * sqrt(0.5)));
	elseif (F <= 1.7e-103)
		tmp = Float64(Float64(Float64(-x) * cos(B)) / sin(B));
	elseif (F <= 6.9e-62)
		tmp = Float64(sqrt(0.5) / Float64(sin(B) / F));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(F / Float64(B * Float64(F + Float64(0.5 * Float64(Float64(2.0 + Float64(x * 2.0)) / F))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -0.0014)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= -1.02e-108)
		tmp = (1.0 / sin(B)) * (F * sqrt(0.5));
	elseif (F <= 1.7e-103)
		tmp = (-x * cos(B)) / sin(B);
	elseif (F <= 6.9e-62)
		tmp = sqrt(0.5) / (sin(B) / F);
	else
		tmp = (x * (-1.0 / tan(B))) + (F / (B * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.0014], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.02e-108], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.7e-103], N[(N[((-x) * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 6.9e-62], N[(N[Sqrt[0.5], $MachinePrecision] / N[(N[Sin[B], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(F / N[(B * N[(F + N[(0.5 * N[(N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -0.00139999999999999999

   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. Taylor expanded in F around -inf 98.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. expm1-log1p-u 45.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}\right)\right)} \]

      2. expm1-udef 45.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}\right)} - 1} \]

      3. div-inv 45.5%

         \[\leadsto e^{\mathsf{log1p}\left(\left(-\color{blue}{\frac{x}{\tan B}}\right) + \frac{-1}{\sin B}\right)} - 1 \]

      4. neg-mul-1 45.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{-1 \cdot \frac{x}{\tan B}} + \frac{-1}{\sin B}\right)} - 1 \]

      5. fma-def 45.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(-1, \frac{x}{\tan B}, \frac{-1}{\sin B}\right)}\right)} - 1 \]

   4. Applied egg-rr 45.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(-1, \frac{x}{\tan B}, \frac{-1}{\sin B}\right)\right)} - 1} \]
   5. Step-by-step derivation

      1. expm1-def 45.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(-1, \frac{x}{\tan B}, \frac{-1}{\sin B}\right)\right)\right)} \]

      2. expm1-log1p 98.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \frac{x}{\tan B}, \frac{-1}{\sin B}\right)} \]

      3. rem-log-exp 45.1%

         \[\leadsto \color{blue}{\log \left(e^{\mathsf{fma}\left(-1, \frac{x}{\tan B}, \frac{-1}{\sin B}\right)}\right)} \]

      4. fma-udef 45.1%

         \[\leadsto \log \left(e^{\color{blue}{-1 \cdot \frac{x}{\tan B} + \frac{-1}{\sin B}}}\right) \]

      5. neg-mul-1 45.1%

         \[\leadsto \log \left(e^{\color{blue}{\left(-\frac{x}{\tan B}\right)} + \frac{-1}{\sin B}}\right) \]

      6. prod-exp 40.9%

         \[\leadsto \log \color{blue}{\left(e^{-\frac{x}{\tan B}} \cdot e^{\frac{-1}{\sin B}}\right)} \]

      7. *-commutative 40.9%

         \[\leadsto \log \color{blue}{\left(e^{\frac{-1}{\sin B}} \cdot e^{-\frac{x}{\tan B}}\right)} \]

      8. prod-exp 45.1%

         \[\leadsto \log \color{blue}{\left(e^{\frac{-1}{\sin B} + \left(-\frac{x}{\tan B}\right)}\right)} \]

      9. rem-log-exp 98.0%

         \[\leadsto \color{blue}{\frac{-1}{\sin B} + \left(-\frac{x}{\tan B}\right)} \]

      10. unsub-neg 98.0%

          \[\leadsto \color{blue}{\frac{-1}{\sin B} - \frac{x}{\tan B}} \]

   6. Simplified 98.0%

      \[\leadsto \color{blue}{\frac{-1}{\sin B} - \frac{x}{\tan B}} \]

   if -0.00139999999999999999 < F < -1.02000000000000008e-108

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative 99.5%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      2. unsub-neg 99.5%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      3. associate-*l/ 99.4%

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      4. associate-*r/ 99.7%

         \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      5. *-commutative 99.7%

         \[\leadsto \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \cdot F} - x \cdot \frac{1}{\tan B} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]

   4. Taylor expanded in F around 0 99.1%

      \[\leadsto F \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 \cdot x + 2}}\right)} - \frac{x}{\tan B} \]

   5. Taylor expanded in x around 0 70.1%

      \[\leadsto \color{blue}{\frac{\sqrt{0.5} \cdot F}{\sin B}} \]
   6. Step-by-step derivation

      1. div-inv 70.3%

         \[\leadsto \color{blue}{\left(\sqrt{0.5} \cdot F\right) \cdot \frac{1}{\sin B}} \]

      2. *-commutative 70.3%

         \[\leadsto \color{blue}{\left(F \cdot \sqrt{0.5}\right)} \cdot \frac{1}{\sin B} \]

   7. Applied egg-rr 70.3%

      \[\leadsto \color{blue}{\left(F \cdot \sqrt{0.5}\right) \cdot \frac{1}{\sin B}} \]

   if -1.02000000000000008e-108 < F < 1.70000000000000001e-103

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative 99.5%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      2. unsub-neg 99.5%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      3. associate-*l/ 99.5%

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      4. associate-*r/ 99.5%

         \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      5. *-commutative 99.5%

         \[\leadsto \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \cdot F} - x \cdot \frac{1}{\tan B} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]

   4. Taylor expanded in F around 0 99.7%

      \[\leadsto F \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 \cdot x + 2}}\right)} - \frac{x}{\tan B} \]

   5. Taylor expanded in B around 0 91.2%

      \[\leadsto \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 \cdot x + 2}}} - \frac{x}{\tan B} \]

   6. Taylor expanded in F around 0 86.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{\cos B \cdot x}{\sin B}} \]
   7. Step-by-step derivation

      1. associate-*r/ 86.0%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(\cos B \cdot x\right)}{\sin B}} \]

      2. *-commutative 86.0%

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(x \cdot \cos B\right)}}{\sin B} \]

      3. associate-*r* 86.0%

         \[\leadsto \frac{\color{blue}{\left(-1 \cdot x\right) \cdot \cos B}}{\sin B} \]

      4. neg-mul-1 86.0%

         \[\leadsto \frac{\color{blue}{\left(-x\right)} \cdot \cos B}{\sin B} \]

   8. Simplified 86.0%

      \[\leadsto \color{blue}{\frac{\left(-x\right) \cdot \cos B}{\sin B}} \]

   if 1.70000000000000001e-103 < F < 6.89999999999999958e-62

   1. Initial program 99.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative 99.6%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      2. unsub-neg 99.6%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      3. associate-*l/ 99.6%

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      4. associate-*r/ 99.3%

         \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      5. *-commutative 99.3%

         \[\leadsto \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \cdot F} - x \cdot \frac{1}{\tan B} \]

   3. Simplified 99.3%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]

   4. Taylor expanded in F around 0 99.2%

      \[\leadsto F \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 \cdot x + 2}}\right)} - \frac{x}{\tan B} \]

   5. Taylor expanded in x around 0 85.7%

      \[\leadsto \color{blue}{\frac{\sqrt{0.5} \cdot F}{\sin B}} \]
   6. Step-by-step derivation

      1. associate-/l* 85.7%

         \[\leadsto \color{blue}{\frac{\sqrt{0.5}}{\frac{\sin B}{F}}} \]

   7. Simplified 85.7%

      \[\leadsto \color{blue}{\frac{\sqrt{0.5}}{\frac{\sin B}{F}}} \]

   if 6.89999999999999958e-62 < F

   1. Initial program 57.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. associate-*l/ 74.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} \]

      2. +-commutative 74.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot {\color{blue}{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)}}{\sin B} \]

      3. *-commutative 74.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot {\left(\color{blue}{x \cdot 2} + \left(F \cdot F + 2\right)\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \]

      4. fma-udef 74.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot {\color{blue}{\left(\mathsf{fma}\left(x, 2, F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)}}{\sin B} \]

      5. fma-def 74.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \]

      6. metadata-eval 74.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(-\color{blue}{0.5}\right)}}{\sin B} \]

      7. metadata-eval 74.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{-0.5}}}{\sin B} \]

      8. associate-/l* 74.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}} \]

      9. fma-def 74.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F + 2}\right)\right)}^{-0.5}}} \]

      10. fma-udef 74.1%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\frac{\sin B}{{\color{blue}{\left(x \cdot 2 + \left(F \cdot F + 2\right)\right)}}^{-0.5}}} \]

      11. *-commutative 74.1%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\frac{\sin B}{{\left(\color{blue}{2 \cdot x} + \left(F \cdot F + 2\right)\right)}^{-0.5}}} \]

      12. fma-def 74.1%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\frac{\sin B}{{\color{blue}{\left(\mathsf{fma}\left(2, x, F \cdot F + 2\right)\right)}}^{-0.5}}} \]

      13. fma-def 74.1%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)\right)}^{-0.5}}} \]

   3. Applied egg-rr 74.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}} \]
   4. Step-by-step derivation

      1. clear-num 74.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\frac{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}{F}}} \]

      2. inv-pow 74.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{{\left(\frac{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}{F}\right)}^{-1}} \]

      3. div-inv 74.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + {\left(\frac{\color{blue}{\sin B \cdot \frac{1}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}}{F}\right)}^{-1} \]

      4. pow-flip 74.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + {\left(\frac{\sin B \cdot \color{blue}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(--0.5\right)}}}{F}\right)}^{-1} \]

      5. metadata-eval 74.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + {\left(\frac{\sin B \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{0.5}}}{F}\right)}^{-1} \]

      6. pow1/2 74.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + {\left(\frac{\sin B \cdot \color{blue}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}{F}\right)}^{-1} \]

   5. Applied egg-rr 74.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{{\left(\frac{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}{F}\right)}^{-1}} \]
   6. Step-by-step derivation

      1. unpow-1 74.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\frac{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}{F}}} \]

      2. associate-/l* 74.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\color{blue}{\frac{\sin B}{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}}} \]

   7. Simplified 74.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\frac{\sin B}{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}}} \]

   8. Taylor expanded in F around inf 92.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\frac{\sin B}{\frac{F}{\color{blue}{F + 0.5 \cdot \frac{2 \cdot x + 2}{F}}}}} \]

   9. Taylor expanded in B around 0 78.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{\left(F + 0.5 \cdot \frac{2 \cdot x + 2}{F}\right) \cdot B}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.0014:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.02 \cdot 10^{-108}:\\ \;\;\;\;\frac{1}{\sin B} \cdot \left(F \cdot \sqrt{0.5}\right)\\ \mathbf{elif}\;F \leq 1.7 \cdot 10^{-103}:\\ \;\;\;\;\frac{\left(-x\right) \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 6.9 \cdot 10^{-62}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\sin B}{F}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B \cdot \left(F + 0.5 \cdot \frac{2 + x \cdot 2}{F}\right)}\\ \end{array} \]

Alternative 18: 63.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ t_1 := \sqrt{\frac{1}{2 + x \cdot 2}}\\ \mathbf{if}\;F \leq -0.235:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -4.5 \cdot 10^{-145}:\\ \;\;\;\;\left(0.3333333333333333 \cdot \left(B \cdot x\right) + t_1 \cdot \frac{F}{B}\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.65 \cdot 10^{-307}:\\ \;\;\;\;t_0 + \frac{F}{B} \cdot \frac{-1}{F}\\ \mathbf{elif}\;F \leq 2.35 \cdot 10^{-138}:\\ \;\;\;\;\frac{F \cdot t_1 - x}{B}\\ \mathbf{elif}\;F \leq 0.0235:\\ \;\;\;\;t_0 + \frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B)))) (t_1 (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))))
   (if (<= F -0.235)
     (- (/ -1.0 (sin B)) (/ x B))
     (if (<= F -4.5e-145)
       (- (+ (* 0.3333333333333333 (* B x)) (* t_1 (/ F B))) (/ x B))
       (if (<= F -1.65e-307)
         (+ t_0 (* (/ F B) (/ -1.0 F)))
         (if (<= F 2.35e-138)
           (/ (- (* F t_1) x) B)
           (if (<= F 0.0235)
             (+ t_0 (/ -1.0 B))
             (- (/ 1.0 (sin B)) (/ x B)))))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double t_1 = sqrt((1.0 / (2.0 + (x * 2.0))));
	double tmp;
	if (F <= -0.235) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -4.5e-145) {
		tmp = ((0.3333333333333333 * (B * x)) + (t_1 * (F / B))) - (x / B);
	} else if (F <= -1.65e-307) {
		tmp = t_0 + ((F / B) * (-1.0 / F));
	} else if (F <= 2.35e-138) {
		tmp = ((F * t_1) - x) / B;
	} else if (F <= 0.0235) {
		tmp = t_0 + (-1.0 / B);
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x * ((-1.0d0) / tan(b))
    t_1 = sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))
    if (f <= (-0.235d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-4.5d-145)) then
        tmp = ((0.3333333333333333d0 * (b * x)) + (t_1 * (f / b))) - (x / b)
    else if (f <= (-1.65d-307)) then
        tmp = t_0 + ((f / b) * ((-1.0d0) / f))
    else if (f <= 2.35d-138) then
        tmp = ((f * t_1) - x) / b
    else if (f <= 0.0235d0) then
        tmp = t_0 + ((-1.0d0) / b)
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / Math.tan(B));
	double t_1 = Math.sqrt((1.0 / (2.0 + (x * 2.0))));
	double tmp;
	if (F <= -0.235) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -4.5e-145) {
		tmp = ((0.3333333333333333 * (B * x)) + (t_1 * (F / B))) - (x / B);
	} else if (F <= -1.65e-307) {
		tmp = t_0 + ((F / B) * (-1.0 / F));
	} else if (F <= 2.35e-138) {
		tmp = ((F * t_1) - x) / B;
	} else if (F <= 0.0235) {
		tmp = t_0 + (-1.0 / B);
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * (-1.0 / math.tan(B))
	t_1 = math.sqrt((1.0 / (2.0 + (x * 2.0))))
	tmp = 0
	if F <= -0.235:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -4.5e-145:
		tmp = ((0.3333333333333333 * (B * x)) + (t_1 * (F / B))) - (x / B)
	elif F <= -1.65e-307:
		tmp = t_0 + ((F / B) * (-1.0 / F))
	elif F <= 2.35e-138:
		tmp = ((F * t_1) - x) / B
	elif F <= 0.0235:
		tmp = t_0 + (-1.0 / B)
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	t_1 = sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))
	tmp = 0.0
	if (F <= -0.235)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -4.5e-145)
		tmp = Float64(Float64(Float64(0.3333333333333333 * Float64(B * x)) + Float64(t_1 * Float64(F / B))) - Float64(x / B));
	elseif (F <= -1.65e-307)
		tmp = Float64(t_0 + Float64(Float64(F / B) * Float64(-1.0 / F)));
	elseif (F <= 2.35e-138)
		tmp = Float64(Float64(Float64(F * t_1) - x) / B);
	elseif (F <= 0.0235)
		tmp = Float64(t_0 + Float64(-1.0 / B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x * (-1.0 / tan(B));
	t_1 = sqrt((1.0 / (2.0 + (x * 2.0))));
	tmp = 0.0;
	if (F <= -0.235)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -4.5e-145)
		tmp = ((0.3333333333333333 * (B * x)) + (t_1 * (F / B))) - (x / B);
	elseif (F <= -1.65e-307)
		tmp = t_0 + ((F / B) * (-1.0 / F));
	elseif (F <= 2.35e-138)
		tmp = ((F * t_1) - x) / B;
	elseif (F <= 0.0235)
		tmp = t_0 + (-1.0 / B);
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[F, -0.235], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -4.5e-145], N[(N[(N[(0.3333333333333333 * N[(B * x), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(F / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.65e-307], N[(t$95$0 + N[(N[(F / B), $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.35e-138], N[(N[(N[(F * t$95$1), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 0.0235], N[(t$95$0 + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if F < -0.23499999999999999

   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. Taylor expanded in F around -inf 98.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 79.9%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{-1}{\sin B} \]

   if -0.23499999999999999 < F < -4.5000000000000001e-145

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative 99.5%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      2. unsub-neg 99.5%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      3. associate-*l/ 99.4%

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      4. associate-*r/ 99.6%

         \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      5. *-commutative 99.6%

         \[\leadsto \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \cdot F} - x \cdot \frac{1}{\tan B} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]

   4. Taylor expanded in F around 0 99.1%

      \[\leadsto F \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 \cdot x + 2}}\right)} - \frac{x}{\tan B} \]

   5. Taylor expanded in B around 0 65.9%

      \[\leadsto \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 \cdot x + 2}}} - \frac{x}{\tan B} \]

   6. Taylor expanded in B around 0 49.9%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \left(B \cdot x\right) + \sqrt{\frac{1}{2 + 2 \cdot x}} \cdot \frac{F}{B}\right) - \frac{x}{B}} \]

   if -4.5000000000000001e-145 < F < -1.65e-307

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 31.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\frac{-1}{F}} \]

   3. Taylor expanded in B around 0 60.5%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot \frac{-1}{F} \]

   if -1.65e-307 < F < 2.3500000000000001e-138

   1. Initial program 99.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative 99.7%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      2. unsub-neg 99.7%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      3. associate-*l/ 99.7%

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      4. associate-*r/ 99.7%

         \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      5. *-commutative 99.7%

         \[\leadsto \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \cdot F} - x \cdot \frac{1}{\tan B} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]

   4. Taylor expanded in F around 0 99.9%

      \[\leadsto F \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 \cdot x + 2}}\right)} - \frac{x}{\tan B} \]

   5. Taylor expanded in B around 0 74.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{2 + 2 \cdot x}} \cdot F - x}{B}} \]

   if 2.3500000000000001e-138 < F < 0.0235

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 40.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 54.3%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{B}} \]

   if 0.0235 < F

   1. Initial program 47.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 32.0%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in F around inf 83.4%

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.235:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -4.5 \cdot 10^{-145}:\\ \;\;\;\;\left(0.3333333333333333 \cdot \left(B \cdot x\right) + \sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B}\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.65 \cdot 10^{-307}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B} \cdot \frac{-1}{F}\\ \mathbf{elif}\;F \leq 2.35 \cdot 10^{-138}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{elif}\;F \leq 0.0235:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 19: 71.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + x \cdot 2\\ \mathbf{if}\;F \leq -0.235:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -3.4 \cdot 10^{-109}:\\ \;\;\;\;\left(0.3333333333333333 \cdot \left(B \cdot x\right) + \sqrt{\frac{1}{t_0}} \cdot \frac{F}{B}\right) - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B \cdot \left(F + 0.5 \cdot \frac{t_0}{F}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (+ 2.0 (* x 2.0))))
   (if (<= F -0.235)
     (- (/ -1.0 (sin B)) (/ x B))
     (if (<= F -3.4e-109)
       (-
        (+ (* 0.3333333333333333 (* B x)) (* (sqrt (/ 1.0 t_0)) (/ F B)))
        (/ x B))
       (+ (* x (/ -1.0 (tan B))) (/ F (* B (+ F (* 0.5 (/ t_0 F))))))))))

C

double code(double F, double B, double x) {
	double t_0 = 2.0 + (x * 2.0);
	double tmp;
	if (F <= -0.235) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -3.4e-109) {
		tmp = ((0.3333333333333333 * (B * x)) + (sqrt((1.0 / t_0)) * (F / B))) - (x / B);
	} else {
		tmp = (x * (-1.0 / tan(B))) + (F / (B * (F + (0.5 * (t_0 / F)))));
	}
	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 = 2.0d0 + (x * 2.0d0)
    if (f <= (-0.235d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-3.4d-109)) then
        tmp = ((0.3333333333333333d0 * (b * x)) + (sqrt((1.0d0 / t_0)) * (f / b))) - (x / b)
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (f / (b * (f + (0.5d0 * (t_0 / f)))))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = 2.0 + (x * 2.0);
	double tmp;
	if (F <= -0.235) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -3.4e-109) {
		tmp = ((0.3333333333333333 * (B * x)) + (Math.sqrt((1.0 / t_0)) * (F / B))) - (x / B);
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (F / (B * (F + (0.5 * (t_0 / F)))));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = 2.0 + (x * 2.0)
	tmp = 0
	if F <= -0.235:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -3.4e-109:
		tmp = ((0.3333333333333333 * (B * x)) + (math.sqrt((1.0 / t_0)) * (F / B))) - (x / B)
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (F / (B * (F + (0.5 * (t_0 / F)))))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(2.0 + Float64(x * 2.0))
	tmp = 0.0
	if (F <= -0.235)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -3.4e-109)
		tmp = Float64(Float64(Float64(0.3333333333333333 * Float64(B * x)) + Float64(sqrt(Float64(1.0 / t_0)) * Float64(F / B))) - Float64(x / B));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(F / Float64(B * Float64(F + Float64(0.5 * Float64(t_0 / F))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = 2.0 + (x * 2.0);
	tmp = 0.0;
	if (F <= -0.235)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -3.4e-109)
		tmp = ((0.3333333333333333 * (B * x)) + (sqrt((1.0 / t_0)) * (F / B))) - (x / B);
	else
		tmp = (x * (-1.0 / tan(B))) + (F / (B * (F + (0.5 * (t_0 / F)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.235], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -3.4e-109], N[(N[(N[(0.3333333333333333 * N[(B * x), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 / t$95$0), $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(F / N[(B * N[(F + N[(0.5 * N[(t$95$0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.23499999999999999

   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. Taylor expanded in F around -inf 98.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 79.9%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{-1}{\sin B} \]

   if -0.23499999999999999 < F < -3.40000000000000012e-109

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative 99.5%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      2. unsub-neg 99.5%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      3. associate-*l/ 99.4%

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      4. associate-*r/ 99.7%

         \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      5. *-commutative 99.7%

         \[\leadsto \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \cdot F} - x \cdot \frac{1}{\tan B} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]

   4. Taylor expanded in F around 0 99.1%

      \[\leadsto F \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 \cdot x + 2}}\right)} - \frac{x}{\tan B} \]

   5. Taylor expanded in B around 0 68.0%

      \[\leadsto \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 \cdot x + 2}}} - \frac{x}{\tan B} \]

   6. Taylor expanded in B around 0 53.2%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \left(B \cdot x\right) + \sqrt{\frac{1}{2 + 2 \cdot x}} \cdot \frac{F}{B}\right) - \frac{x}{B}} \]

   if -3.40000000000000012e-109 < F

   1. Initial program 75.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. associate-*l/ 85.2%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} \]

      2. +-commutative 85.2%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot {\color{blue}{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)}}{\sin B} \]

      3. *-commutative 85.2%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot {\left(\color{blue}{x \cdot 2} + \left(F \cdot F + 2\right)\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \]

      4. fma-udef 85.2%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot {\color{blue}{\left(\mathsf{fma}\left(x, 2, F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)}}{\sin B} \]

      5. fma-def 85.2%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \]

      6. metadata-eval 85.2%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(-\color{blue}{0.5}\right)}}{\sin B} \]

      7. metadata-eval 85.2%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{-0.5}}}{\sin B} \]

      8. associate-/l* 85.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}} \]

      9. fma-def 85.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F + 2}\right)\right)}^{-0.5}}} \]

      10. fma-udef 85.1%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\frac{\sin B}{{\color{blue}{\left(x \cdot 2 + \left(F \cdot F + 2\right)\right)}}^{-0.5}}} \]

      11. *-commutative 85.1%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\frac{\sin B}{{\left(\color{blue}{2 \cdot x} + \left(F \cdot F + 2\right)\right)}^{-0.5}}} \]

      12. fma-def 85.1%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\frac{\sin B}{{\color{blue}{\left(\mathsf{fma}\left(2, x, F \cdot F + 2\right)\right)}}^{-0.5}}} \]

      13. fma-def 85.1%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)\right)}^{-0.5}}} \]

   3. Applied egg-rr 85.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}} \]
   4. Step-by-step derivation

      1. clear-num 85.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\frac{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}{F}}} \]

      2. inv-pow 85.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{{\left(\frac{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}{F}\right)}^{-1}} \]

      3. div-inv 85.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + {\left(\frac{\color{blue}{\sin B \cdot \frac{1}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}}{F}\right)}^{-1} \]

      4. pow-flip 85.2%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + {\left(\frac{\sin B \cdot \color{blue}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(--0.5\right)}}}{F}\right)}^{-1} \]

      5. metadata-eval 85.2%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + {\left(\frac{\sin B \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{0.5}}}{F}\right)}^{-1} \]

      6. pow1/2 85.2%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + {\left(\frac{\sin B \cdot \color{blue}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}{F}\right)}^{-1} \]

   5. Applied egg-rr 85.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{{\left(\frac{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}{F}\right)}^{-1}} \]
   6. Step-by-step derivation

      1. unpow-1 85.2%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\frac{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}{F}}} \]

      2. associate-/l* 85.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\color{blue}{\frac{\sin B}{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}}} \]

   7. Simplified 85.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\frac{\sin B}{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}}} \]

   8. Taylor expanded in F around inf 86.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\frac{\sin B}{\frac{F}{\color{blue}{F + 0.5 \cdot \frac{2 \cdot x + 2}{F}}}}} \]

   9. Taylor expanded in B around 0 78.8%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{\left(F + 0.5 \cdot \frac{2 \cdot x + 2}{F}\right) \cdot B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.235:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -3.4 \cdot 10^{-109}:\\ \;\;\;\;\left(0.3333333333333333 \cdot \left(B \cdot x\right) + \sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B}\right) - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B \cdot \left(F + 0.5 \cdot \frac{2 + x \cdot 2}{F}\right)}\\ \end{array} \]

Alternative 20: 71.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -430:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -2.7 \cdot 10^{-110}:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B \cdot \left(F + 0.5 \cdot \frac{2 + x \cdot 2}{F}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -430.0)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F -2.7e-110)
     (- (* (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5) (/ F B)) (/ x B))
     (+
      (* x (/ -1.0 (tan B)))
      (/ F (* B (+ F (* 0.5 (/ (+ 2.0 (* x 2.0)) F)))))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -430.0) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -2.7e-110) {
		tmp = (pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else {
		tmp = (x * (-1.0 / tan(B))) + (F / (B * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))));
	}
	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 <= (-430.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-2.7d-110)) then
        tmp = ((((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)) * (f / b)) - (x / b)
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (f / (b * (f + (0.5d0 * ((2.0d0 + (x * 2.0d0)) / f)))))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -430.0) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -2.7e-110) {
		tmp = (Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (F / (B * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -430.0:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -2.7e-110:
		tmp = (math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B)
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (F / (B * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -430.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -2.7e-110)
		tmp = Float64(Float64((Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5) * Float64(F / B)) - Float64(x / B));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(F / Float64(B * Float64(F + Float64(0.5 * Float64(Float64(2.0 + Float64(x * 2.0)) / F))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -430.0)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -2.7e-110)
		tmp = ((((2.0 + (F * F)) + (x * 2.0)) ^ -0.5) * (F / B)) - (x / B);
	else
		tmp = (x * (-1.0 / tan(B))) + (F / (B * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -430.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -2.7e-110], N[(N[(N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(F / N[(B * N[(F + N[(0.5 * N[(N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -430

   1. Initial program 56.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 98.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 80.6%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{-1}{\sin B} \]

   if -430 < F < -2.6999999999999998e-110

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 82.6%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in B around 0 55.3%

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   if -2.6999999999999998e-110 < F

   1. Initial program 75.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. associate-*l/ 85.2%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} \]

      2. +-commutative 85.2%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot {\color{blue}{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)}}{\sin B} \]

      3. *-commutative 85.2%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot {\left(\color{blue}{x \cdot 2} + \left(F \cdot F + 2\right)\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \]

      4. fma-udef 85.2%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot {\color{blue}{\left(\mathsf{fma}\left(x, 2, F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)}}{\sin B} \]

      5. fma-def 85.2%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \]

      6. metadata-eval 85.2%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(-\color{blue}{0.5}\right)}}{\sin B} \]

      7. metadata-eval 85.2%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{-0.5}}}{\sin B} \]

      8. associate-/l* 85.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}} \]

      9. fma-def 85.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F + 2}\right)\right)}^{-0.5}}} \]

      10. fma-udef 85.1%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\frac{\sin B}{{\color{blue}{\left(x \cdot 2 + \left(F \cdot F + 2\right)\right)}}^{-0.5}}} \]

      11. *-commutative 85.1%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\frac{\sin B}{{\left(\color{blue}{2 \cdot x} + \left(F \cdot F + 2\right)\right)}^{-0.5}}} \]

      12. fma-def 85.1%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\frac{\sin B}{{\color{blue}{\left(\mathsf{fma}\left(2, x, F \cdot F + 2\right)\right)}}^{-0.5}}} \]

      13. fma-def 85.1%

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)\right)}^{-0.5}}} \]

   3. Applied egg-rr 85.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}} \]
   4. Step-by-step derivation

      1. clear-num 85.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\frac{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}{F}}} \]

      2. inv-pow 85.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{{\left(\frac{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}{F}\right)}^{-1}} \]

      3. div-inv 85.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + {\left(\frac{\color{blue}{\sin B \cdot \frac{1}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}}{F}\right)}^{-1} \]

      4. pow-flip 85.2%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + {\left(\frac{\sin B \cdot \color{blue}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(--0.5\right)}}}{F}\right)}^{-1} \]

      5. metadata-eval 85.2%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + {\left(\frac{\sin B \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{0.5}}}{F}\right)}^{-1} \]

      6. pow1/2 85.2%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + {\left(\frac{\sin B \cdot \color{blue}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}{F}\right)}^{-1} \]

   5. Applied egg-rr 85.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{{\left(\frac{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}{F}\right)}^{-1}} \]
   6. Step-by-step derivation

      1. unpow-1 85.2%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\frac{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}{F}}} \]

      2. associate-/l* 85.1%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\color{blue}{\frac{\sin B}{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}}} \]

   7. Simplified 85.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\frac{\sin B}{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}}} \]

   8. Taylor expanded in F around inf 86.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\frac{\sin B}{\frac{F}{\color{blue}{F + 0.5 \cdot \frac{2 \cdot x + 2}{F}}}}} \]

   9. Taylor expanded in B around 0 78.8%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{\left(F + 0.5 \cdot \frac{2 \cdot x + 2}{F}\right) \cdot B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -430:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -2.7 \cdot 10^{-110}:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B \cdot \left(F + 0.5 \cdot \frac{2 + x \cdot 2}{F}\right)}\\ \end{array} \]

Alternative 21: 63.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ t_1 := x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{if}\;F \leq -0.06:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -9.5 \cdot 10^{-147}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -3.8 \cdot 10^{-308}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 1.1 \cdot 10^{-138}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 0.49:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- (* F (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))) x) B))
        (t_1 (+ (* x (/ -1.0 (tan B))) (/ -1.0 B))))
   (if (<= F -0.06)
     (- (/ -1.0 (sin B)) (/ x B))
     (if (<= F -9.5e-147)
       t_0
       (if (<= F -3.8e-308)
         t_1
         (if (<= F 1.1e-138)
           t_0
           (if (<= F 0.49) t_1 (- (/ 1.0 (sin B)) (/ x B)))))))))

C

double code(double F, double B, double x) {
	double t_0 = ((F * sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B;
	double t_1 = (x * (-1.0 / tan(B))) + (-1.0 / B);
	double tmp;
	if (F <= -0.06) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -9.5e-147) {
		tmp = t_0;
	} else if (F <= -3.8e-308) {
		tmp = t_1;
	} else if (F <= 1.1e-138) {
		tmp = t_0;
	} else if (F <= 0.49) {
		tmp = t_1;
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((f * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))) - x) / b
    t_1 = (x * ((-1.0d0) / tan(b))) + ((-1.0d0) / b)
    if (f <= (-0.06d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-9.5d-147)) then
        tmp = t_0
    else if (f <= (-3.8d-308)) then
        tmp = t_1
    else if (f <= 1.1d-138) then
        tmp = t_0
    else if (f <= 0.49d0) then
        tmp = t_1
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = ((F * Math.sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B;
	double t_1 = (x * (-1.0 / Math.tan(B))) + (-1.0 / B);
	double tmp;
	if (F <= -0.06) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -9.5e-147) {
		tmp = t_0;
	} else if (F <= -3.8e-308) {
		tmp = t_1;
	} else if (F <= 1.1e-138) {
		tmp = t_0;
	} else if (F <= 0.49) {
		tmp = t_1;
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = ((F * math.sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B
	t_1 = (x * (-1.0 / math.tan(B))) + (-1.0 / B)
	tmp = 0
	if F <= -0.06:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -9.5e-147:
		tmp = t_0
	elif F <= -3.8e-308:
		tmp = t_1
	elif F <= 1.1e-138:
		tmp = t_0
	elif F <= 0.49:
		tmp = t_1
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(Float64(F * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))) - x) / B)
	t_1 = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(-1.0 / B))
	tmp = 0.0
	if (F <= -0.06)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -9.5e-147)
		tmp = t_0;
	elseif (F <= -3.8e-308)
		tmp = t_1;
	elseif (F <= 1.1e-138)
		tmp = t_0;
	elseif (F <= 0.49)
		tmp = t_1;
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = ((F * sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B;
	t_1 = (x * (-1.0 / tan(B))) + (-1.0 / B);
	tmp = 0.0;
	if (F <= -0.06)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -9.5e-147)
		tmp = t_0;
	elseif (F <= -3.8e-308)
		tmp = t_1;
	elseif (F <= 1.1e-138)
		tmp = t_0;
	elseif (F <= 0.49)
		tmp = t_1;
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(N[(F * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.06], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -9.5e-147], t$95$0, If[LessEqual[F, -3.8e-308], t$95$1, If[LessEqual[F, 1.1e-138], t$95$0, If[LessEqual[F, 0.49], t$95$1, N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.059999999999999998

   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. Taylor expanded in F around -inf 98.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 79.9%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{-1}{\sin B} \]

   if -0.059999999999999998 < F < -9.49999999999999986e-147 or -3.79999999999999975e-308 < F < 1.0999999999999999e-138

   1. Initial program 99.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative 99.6%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      2. unsub-neg 99.6%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      3. associate-*l/ 99.5%

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      4. associate-*r/ 99.7%

         \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      5. *-commutative 99.7%

         \[\leadsto \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \cdot F} - x \cdot \frac{1}{\tan B} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]

   4. Taylor expanded in F around 0 99.5%

      \[\leadsto F \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 \cdot x + 2}}\right)} - \frac{x}{\tan B} \]

   5. Taylor expanded in B around 0 60.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{2 + 2 \cdot x}} \cdot F - x}{B}} \]

   if -9.49999999999999986e-147 < F < -3.79999999999999975e-308 or 1.0999999999999999e-138 < F < 0.48999999999999999

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 35.3%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 56.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{B}} \]

   if 0.48999999999999999 < F

   1. Initial program 47.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 32.0%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in F around inf 83.4%

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.06:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -9.5 \cdot 10^{-147}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{elif}\;F \leq -3.8 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq 1.1 \cdot 10^{-138}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{elif}\;F \leq 0.49:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 22: 63.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ t_1 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -0.22:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.15 \cdot 10^{-145}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -2.2 \cdot 10^{-308}:\\ \;\;\;\;t_1 + \frac{F}{B} \cdot \frac{-1}{F}\\ \mathbf{elif}\;F \leq 1.3 \cdot 10^{-138}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 0.0275:\\ \;\;\;\;t_1 + \frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- (* F (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))) x) B))
        (t_1 (* x (/ -1.0 (tan B)))))
   (if (<= F -0.22)
     (- (/ -1.0 (sin B)) (/ x B))
     (if (<= F -1.15e-145)
       t_0
       (if (<= F -2.2e-308)
         (+ t_1 (* (/ F B) (/ -1.0 F)))
         (if (<= F 1.3e-138)
           t_0
           (if (<= F 0.0275)
             (+ t_1 (/ -1.0 B))
             (- (/ 1.0 (sin B)) (/ x B)))))))))

C

double code(double F, double B, double x) {
	double t_0 = ((F * sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B;
	double t_1 = x * (-1.0 / tan(B));
	double tmp;
	if (F <= -0.22) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -1.15e-145) {
		tmp = t_0;
	} else if (F <= -2.2e-308) {
		tmp = t_1 + ((F / B) * (-1.0 / F));
	} else if (F <= 1.3e-138) {
		tmp = t_0;
	} else if (F <= 0.0275) {
		tmp = t_1 + (-1.0 / B);
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((f * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))) - x) / b
    t_1 = x * ((-1.0d0) / tan(b))
    if (f <= (-0.22d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-1.15d-145)) then
        tmp = t_0
    else if (f <= (-2.2d-308)) then
        tmp = t_1 + ((f / b) * ((-1.0d0) / f))
    else if (f <= 1.3d-138) then
        tmp = t_0
    else if (f <= 0.0275d0) then
        tmp = t_1 + ((-1.0d0) / b)
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = ((F * Math.sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B;
	double t_1 = x * (-1.0 / Math.tan(B));
	double tmp;
	if (F <= -0.22) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -1.15e-145) {
		tmp = t_0;
	} else if (F <= -2.2e-308) {
		tmp = t_1 + ((F / B) * (-1.0 / F));
	} else if (F <= 1.3e-138) {
		tmp = t_0;
	} else if (F <= 0.0275) {
		tmp = t_1 + (-1.0 / B);
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = ((F * math.sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B
	t_1 = x * (-1.0 / math.tan(B))
	tmp = 0
	if F <= -0.22:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -1.15e-145:
		tmp = t_0
	elif F <= -2.2e-308:
		tmp = t_1 + ((F / B) * (-1.0 / F))
	elif F <= 1.3e-138:
		tmp = t_0
	elif F <= 0.0275:
		tmp = t_1 + (-1.0 / B)
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(Float64(F * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))) - x) / B)
	t_1 = Float64(x * Float64(-1.0 / tan(B)))
	tmp = 0.0
	if (F <= -0.22)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -1.15e-145)
		tmp = t_0;
	elseif (F <= -2.2e-308)
		tmp = Float64(t_1 + Float64(Float64(F / B) * Float64(-1.0 / F)));
	elseif (F <= 1.3e-138)
		tmp = t_0;
	elseif (F <= 0.0275)
		tmp = Float64(t_1 + Float64(-1.0 / B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = ((F * sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B;
	t_1 = x * (-1.0 / tan(B));
	tmp = 0.0;
	if (F <= -0.22)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -1.15e-145)
		tmp = t_0;
	elseif (F <= -2.2e-308)
		tmp = t_1 + ((F / B) * (-1.0 / F));
	elseif (F <= 1.3e-138)
		tmp = t_0;
	elseif (F <= 0.0275)
		tmp = t_1 + (-1.0 / B);
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(N[(F * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.22], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.15e-145], t$95$0, If[LessEqual[F, -2.2e-308], N[(t$95$1 + N[(N[(F / B), $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.3e-138], t$95$0, If[LessEqual[F, 0.0275], N[(t$95$1 + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -0.220000000000000001

   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. Taylor expanded in F around -inf 98.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 79.9%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{-1}{\sin B} \]

   if -0.220000000000000001 < F < -1.15000000000000004e-145 or -2.2000000000000002e-308 < F < 1.3e-138

   1. Initial program 99.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative 99.6%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      2. unsub-neg 99.6%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      3. associate-*l/ 99.5%

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      4. associate-*r/ 99.7%

         \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      5. *-commutative 99.7%

         \[\leadsto \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \cdot F} - x \cdot \frac{1}{\tan B} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]

   4. Taylor expanded in F around 0 99.5%

      \[\leadsto F \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 \cdot x + 2}}\right)} - \frac{x}{\tan B} \]

   5. Taylor expanded in B around 0 60.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{2 + 2 \cdot x}} \cdot F - x}{B}} \]

   if -1.15000000000000004e-145 < F < -2.2000000000000002e-308

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 31.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\frac{-1}{F}} \]

   3. Taylor expanded in B around 0 60.5%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot \frac{-1}{F} \]

   if 1.3e-138 < F < 0.0275000000000000001

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 40.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 54.3%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{B}} \]

   if 0.0275000000000000001 < F

   1. Initial program 47.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 32.0%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in F around inf 83.4%

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.22:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.15 \cdot 10^{-145}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{elif}\;F \leq -2.2 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B} \cdot \frac{-1}{F}\\ \mathbf{elif}\;F \leq 1.3 \cdot 10^{-138}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{elif}\;F \leq 0.0275:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 23: 60.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{if}\;F \leq -1.3 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.08 \cdot 10^{-29}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{B}{F}}\\ \mathbf{elif}\;F \leq -5.2 \cdot 10^{-305}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 1.36 \cdot 10^{-139}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 1.28 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (+ (* x (/ -1.0 (tan B))) (/ -1.0 B))))
   (if (<= F -1.3e-5)
     (- (/ -1.0 (sin B)) (/ x B))
     (if (<= F -1.08e-29)
       (/ (sqrt 0.5) (/ B F))
       (if (<= F -5.2e-305)
         t_0
         (if (<= F 1.36e-139)
           (/ (- x) B)
           (if (<= F 1.28e-5) t_0 (- (/ 1.0 (sin B)) (/ x B)))))))))

C

double code(double F, double B, double x) {
	double t_0 = (x * (-1.0 / tan(B))) + (-1.0 / B);
	double tmp;
	if (F <= -1.3e-5) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -1.08e-29) {
		tmp = sqrt(0.5) / (B / F);
	} else if (F <= -5.2e-305) {
		tmp = t_0;
	} else if (F <= 1.36e-139) {
		tmp = -x / B;
	} else if (F <= 1.28e-5) {
		tmp = t_0;
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * ((-1.0d0) / tan(b))) + ((-1.0d0) / b)
    if (f <= (-1.3d-5)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-1.08d-29)) then
        tmp = sqrt(0.5d0) / (b / f)
    else if (f <= (-5.2d-305)) then
        tmp = t_0
    else if (f <= 1.36d-139) then
        tmp = -x / b
    else if (f <= 1.28d-5) then
        tmp = t_0
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (x * (-1.0 / Math.tan(B))) + (-1.0 / B);
	double tmp;
	if (F <= -1.3e-5) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -1.08e-29) {
		tmp = Math.sqrt(0.5) / (B / F);
	} else if (F <= -5.2e-305) {
		tmp = t_0;
	} else if (F <= 1.36e-139) {
		tmp = -x / B;
	} else if (F <= 1.28e-5) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (x * (-1.0 / math.tan(B))) + (-1.0 / B)
	tmp = 0
	if F <= -1.3e-5:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -1.08e-29:
		tmp = math.sqrt(0.5) / (B / F)
	elif F <= -5.2e-305:
		tmp = t_0
	elif F <= 1.36e-139:
		tmp = -x / B
	elif F <= 1.28e-5:
		tmp = t_0
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(-1.0 / B))
	tmp = 0.0
	if (F <= -1.3e-5)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -1.08e-29)
		tmp = Float64(sqrt(0.5) / Float64(B / F));
	elseif (F <= -5.2e-305)
		tmp = t_0;
	elseif (F <= 1.36e-139)
		tmp = Float64(Float64(-x) / B);
	elseif (F <= 1.28e-5)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (x * (-1.0 / tan(B))) + (-1.0 / B);
	tmp = 0.0;
	if (F <= -1.3e-5)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -1.08e-29)
		tmp = sqrt(0.5) / (B / F);
	elseif (F <= -5.2e-305)
		tmp = t_0;
	elseif (F <= 1.36e-139)
		tmp = -x / B;
	elseif (F <= 1.28e-5)
		tmp = t_0;
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.3e-5], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.08e-29], N[(N[Sqrt[0.5], $MachinePrecision] / N[(B / F), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -5.2e-305], t$95$0, If[LessEqual[F, 1.36e-139], N[((-x) / B), $MachinePrecision], If[LessEqual[F, 1.28e-5], t$95$0, N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -1.29999999999999992e-5

   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. Taylor expanded in F around -inf 98.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 79.9%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{-1}{\sin B} \]

   if -1.29999999999999992e-5 < F < -1.07999999999999995e-29

   1. Initial program 99.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative 99.4%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      2. unsub-neg 99.4%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      3. associate-*l/ 99.2%

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      4. associate-*r/ 99.7%

         \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      5. *-commutative 99.7%

         \[\leadsto \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \cdot F} - x \cdot \frac{1}{\tan B} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]

   4. Taylor expanded in F around 0 98.1%

      \[\leadsto F \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 \cdot x + 2}}\right)} - \frac{x}{\tan B} \]

   5. Taylor expanded in x around 0 92.4%

      \[\leadsto \color{blue}{\frac{\sqrt{0.5} \cdot F}{\sin B}} \]

   6. Taylor expanded in B around 0 50.1%

      \[\leadsto \color{blue}{\frac{\sqrt{0.5} \cdot F}{B}} \]
   7. Step-by-step derivation

      1. associate-/l* 50.1%

         \[\leadsto \color{blue}{\frac{\sqrt{0.5}}{\frac{B}{F}}} \]

   8. Simplified 50.1%

      \[\leadsto \color{blue}{\frac{\sqrt{0.5}}{\frac{B}{F}}} \]

   if -1.07999999999999995e-29 < F < -5.2000000000000003e-305 or 1.36000000000000003e-139 < F < 1.2799999999999999e-5

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 35.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 51.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{B}} \]

   if -5.2000000000000003e-305 < F < 1.36000000000000003e-139

   1. Initial program 99.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 30.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 27.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   4. Step-by-step derivation

      1. associate-*r/ 27.9%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

      2. distribute-lft-in 27.9%

         \[\leadsto \frac{\color{blue}{-1 \cdot 1 + -1 \cdot x}}{B} \]

      3. metadata-eval 27.9%

         \[\leadsto \frac{\color{blue}{-1} + -1 \cdot x}{B} \]

      4. neg-mul-1 27.9%

         \[\leadsto \frac{-1 + \color{blue}{\left(-x\right)}}{B} \]

   5. Simplified 27.9%

      \[\leadsto \color{blue}{\frac{-1 + \left(-x\right)}{B}} \]

   6. Taylor expanded in x around inf 63.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   7. Step-by-step derivation

      1. associate-*r/ 63.6%

         \[\leadsto \color{blue}{\frac{-1 \cdot x}{B}} \]

      2. mul-1-neg 63.6%

         \[\leadsto \frac{\color{blue}{-x}}{B} \]

   8. Simplified 63.6%

      \[\leadsto \color{blue}{\frac{-x}{B}} \]

   if 1.2799999999999999e-5 < F

   1. Initial program 47.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 32.0%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in F around inf 83.4%

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.3 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.08 \cdot 10^{-29}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{B}{F}}\\ \mathbf{elif}\;F \leq -5.2 \cdot 10^{-305}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq 1.36 \cdot 10^{-139}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 1.28 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 24: 57.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.0019:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1 \cdot 10^{-110}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{B}\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{-30}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.0019)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F -1e-110)
     (/ (* F (sqrt 0.5)) B)
     (if (<= F 1.4e-30) (/ (- x) B) (- (/ 1.0 (sin B)) (/ x B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.0019) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -1e-110) {
		tmp = (F * sqrt(0.5)) / B;
	} else if (F <= 1.4e-30) {
		tmp = -x / B;
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-0.0019d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-1d-110)) then
        tmp = (f * sqrt(0.5d0)) / b
    else if (f <= 1.4d-30) then
        tmp = -x / b
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.0019) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -1e-110) {
		tmp = (F * Math.sqrt(0.5)) / B;
	} else if (F <= 1.4e-30) {
		tmp = -x / B;
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -0.0019:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -1e-110:
		tmp = (F * math.sqrt(0.5)) / B
	elif F <= 1.4e-30:
		tmp = -x / B
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.0019)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -1e-110)
		tmp = Float64(Float64(F * sqrt(0.5)) / B);
	elseif (F <= 1.4e-30)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -0.0019)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -1e-110)
		tmp = (F * sqrt(0.5)) / B;
	elseif (F <= 1.4e-30)
		tmp = -x / B;
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.0019], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1e-110], N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.4e-30], N[((-x) / B), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.0019

   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. Taylor expanded in F around -inf 98.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 79.9%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{-1}{\sin B} \]

   if -0.0019 < F < -1.0000000000000001e-110

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative 99.5%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      2. unsub-neg 99.5%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      3. associate-*l/ 99.4%

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      4. associate-*r/ 99.7%

         \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      5. *-commutative 99.7%

         \[\leadsto \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \cdot F} - x \cdot \frac{1}{\tan B} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]

   4. Taylor expanded in F around 0 99.2%

      \[\leadsto F \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 \cdot x + 2}}\right)} - \frac{x}{\tan B} \]

   5. Taylor expanded in x around 0 67.5%

      \[\leadsto \color{blue}{\frac{\sqrt{0.5} \cdot F}{\sin B}} \]

   6. Taylor expanded in B around 0 40.3%

      \[\leadsto \color{blue}{\frac{\sqrt{0.5} \cdot F}{B}} \]

   if -1.0000000000000001e-110 < F < 1.39999999999999994e-30

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 30.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 16.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   4. Step-by-step derivation

      1. associate-*r/ 16.8%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

      2. distribute-lft-in 16.8%

         \[\leadsto \frac{\color{blue}{-1 \cdot 1 + -1 \cdot x}}{B} \]

      3. metadata-eval 16.8%

         \[\leadsto \frac{\color{blue}{-1} + -1 \cdot x}{B} \]

      4. neg-mul-1 16.8%

         \[\leadsto \frac{-1 + \color{blue}{\left(-x\right)}}{B} \]

   5. Simplified 16.8%

      \[\leadsto \color{blue}{\frac{-1 + \left(-x\right)}{B}} \]

   6. Taylor expanded in x around inf 38.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   7. Step-by-step derivation

      1. associate-*r/ 38.9%

         \[\leadsto \color{blue}{\frac{-1 \cdot x}{B}} \]

      2. mul-1-neg 38.9%

         \[\leadsto \frac{\color{blue}{-x}}{B} \]

   8. Simplified 38.9%

      \[\leadsto \color{blue}{\frac{-x}{B}} \]

   if 1.39999999999999994e-30 < F

   1. Initial program 51.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 35.4%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in F around inf 80.2%

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.0019:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1 \cdot 10^{-110}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{B}\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{-30}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 25: 37.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.034:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -6 \cdot 10^{-114}:\\ \;\;\;\;\frac{F}{\frac{B}{\sqrt{0.5}}}\\ \mathbf{elif}\;F \leq 52000000000:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-1 + x}{B}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.034)
   (/ -1.0 (sin B))
   (if (<= F -6e-114)
     (/ F (/ B (sqrt 0.5)))
     (if (<= F 52000000000.0) (/ (- x) B) (fabs (/ (+ -1.0 x) B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.034) {
		tmp = -1.0 / sin(B);
	} else if (F <= -6e-114) {
		tmp = F / (B / sqrt(0.5));
	} else if (F <= 52000000000.0) {
		tmp = -x / B;
	} else {
		tmp = fabs(((-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 <= (-0.034d0)) then
        tmp = (-1.0d0) / sin(b)
    else if (f <= (-6d-114)) then
        tmp = f / (b / sqrt(0.5d0))
    else if (f <= 52000000000.0d0) then
        tmp = -x / b
    else
        tmp = abs((((-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 <= -0.034) {
		tmp = -1.0 / Math.sin(B);
	} else if (F <= -6e-114) {
		tmp = F / (B / Math.sqrt(0.5));
	} else if (F <= 52000000000.0) {
		tmp = -x / B;
	} else {
		tmp = Math.abs(((-1.0 + x) / B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -0.034:
		tmp = -1.0 / math.sin(B)
	elif F <= -6e-114:
		tmp = F / (B / math.sqrt(0.5))
	elif F <= 52000000000.0:
		tmp = -x / B
	else:
		tmp = math.fabs(((-1.0 + x) / B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.034)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= -6e-114)
		tmp = Float64(F / Float64(B / sqrt(0.5)));
	elseif (F <= 52000000000.0)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = abs(Float64(Float64(-1.0 + x) / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -0.034)
		tmp = -1.0 / sin(B);
	elseif (F <= -6e-114)
		tmp = F / (B / sqrt(0.5));
	elseif (F <= 52000000000.0)
		tmp = -x / B;
	else
		tmp = abs(((-1.0 + x) / B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.034], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -6e-114], N[(F / N[(B / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 52000000000.0], N[((-x) / B), $MachinePrecision], N[Abs[N[(N[(-1.0 + x), $MachinePrecision] / B), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.034000000000000002

   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. Taylor expanded in F around -inf 98.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 65.5%

      \[\leadsto \left(-\color{blue}{\left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)}\right) + \frac{-1}{\sin B} \]
   4. Step-by-step derivation

      1. *-commutative 65.5%

         \[\leadsto \left(-\left(\frac{x}{B} + -0.3333333333333333 \cdot \color{blue}{\left(x \cdot B\right)}\right)\right) + \frac{-1}{\sin B} \]

   5. Simplified 65.5%

      \[\leadsto \left(-\color{blue}{\left(\frac{x}{B} + -0.3333333333333333 \cdot \left(x \cdot B\right)\right)}\right) + \frac{-1}{\sin B} \]

   6. Taylor expanded in x around 0 60.9%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   if -0.034000000000000002 < F < -6.0000000000000003e-114

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative 99.5%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      2. unsub-neg 99.5%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      3. associate-*l/ 99.4%

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      4. associate-*r/ 99.7%

         \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      5. *-commutative 99.7%

         \[\leadsto \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \cdot F} - x \cdot \frac{1}{\tan B} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]

   4. Taylor expanded in F around 0 99.2%

      \[\leadsto F \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 \cdot x + 2}}\right)} - \frac{x}{\tan B} \]

   5. Taylor expanded in x around 0 67.5%

      \[\leadsto \color{blue}{\frac{\sqrt{0.5} \cdot F}{\sin B}} \]

   6. Taylor expanded in B around 0 40.3%

      \[\leadsto \color{blue}{\frac{\sqrt{0.5} \cdot F}{B}} \]
   7. Step-by-step derivation

      1. *-commutative 40.3%

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{0.5}}}{B} \]

      2. associate-/l* 40.2%

         \[\leadsto \color{blue}{\frac{F}{\frac{B}{\sqrt{0.5}}}} \]

   8. Simplified 40.2%

      \[\leadsto \color{blue}{\frac{F}{\frac{B}{\sqrt{0.5}}}} \]

   if -6.0000000000000003e-114 < F < 5.2e10

   1. Initial program 98.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 33.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 20.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   4. Step-by-step derivation

      1. associate-*r/ 20.4%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

      2. distribute-lft-in 20.4%

         \[\leadsto \frac{\color{blue}{-1 \cdot 1 + -1 \cdot x}}{B} \]

      3. metadata-eval 20.4%

         \[\leadsto \frac{\color{blue}{-1} + -1 \cdot x}{B} \]

      4. neg-mul-1 20.4%

         \[\leadsto \frac{-1 + \color{blue}{\left(-x\right)}}{B} \]

   5. Simplified 20.4%

      \[\leadsto \color{blue}{\frac{-1 + \left(-x\right)}{B}} \]

   6. Taylor expanded in x around inf 39.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   7. Step-by-step derivation

      1. associate-*r/ 39.9%

         \[\leadsto \color{blue}{\frac{-1 \cdot x}{B}} \]

      2. mul-1-neg 39.9%

         \[\leadsto \frac{\color{blue}{-x}}{B} \]

   8. Simplified 39.9%

      \[\leadsto \color{blue}{\frac{-x}{B}} \]

   if 5.2e10 < F

   1. Initial program 45.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 40.7%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 25.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   4. Step-by-step derivation

      1. associate-*r/ 25.6%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

      2. distribute-lft-in 25.6%

         \[\leadsto \frac{\color{blue}{-1 \cdot 1 + -1 \cdot x}}{B} \]

      3. metadata-eval 25.6%

         \[\leadsto \frac{\color{blue}{-1} + -1 \cdot x}{B} \]

      4. neg-mul-1 25.6%

         \[\leadsto \frac{-1 + \color{blue}{\left(-x\right)}}{B} \]

   5. Simplified 25.6%

      \[\leadsto \color{blue}{\frac{-1 + \left(-x\right)}{B}} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 11.7%

         \[\leadsto \color{blue}{\sqrt{\frac{-1 + \left(-x\right)}{B}} \cdot \sqrt{\frac{-1 + \left(-x\right)}{B}}} \]

      2. sqrt-unprod 18.3%

         \[\leadsto \color{blue}{\sqrt{\frac{-1 + \left(-x\right)}{B} \cdot \frac{-1 + \left(-x\right)}{B}}} \]

      3. pow2 18.3%

         \[\leadsto \sqrt{\color{blue}{{\left(\frac{-1 + \left(-x\right)}{B}\right)}^{2}}} \]

      4. +-commutative 18.3%

         \[\leadsto \sqrt{{\left(\frac{\color{blue}{\left(-x\right) + -1}}{B}\right)}^{2}} \]

      5. add-sqr-sqrt 8.4%

         \[\leadsto \sqrt{{\left(\frac{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}} + -1}{B}\right)}^{2}} \]

      6. sqrt-unprod 18.4%

         \[\leadsto \sqrt{{\left(\frac{\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}} + -1}{B}\right)}^{2}} \]

      7. sqr-neg 18.4%

         \[\leadsto \sqrt{{\left(\frac{\sqrt{\color{blue}{x \cdot x}} + -1}{B}\right)}^{2}} \]

      8. sqrt-unprod 9.9%

         \[\leadsto \sqrt{{\left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}} + -1}{B}\right)}^{2}} \]

      9. add-sqr-sqrt 18.3%

         \[\leadsto \sqrt{{\left(\frac{\color{blue}{x} + -1}{B}\right)}^{2}} \]

   7. Applied egg-rr 18.3%

      \[\leadsto \color{blue}{\sqrt{{\left(\frac{x + -1}{B}\right)}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 18.3%

         \[\leadsto \sqrt{\color{blue}{\frac{x + -1}{B} \cdot \frac{x + -1}{B}}} \]

      2. rem-sqrt-square 39.7%

         \[\leadsto \color{blue}{\left|\frac{x + -1}{B}\right|} \]

   9. Simplified 39.7%

      \[\leadsto \color{blue}{\left|\frac{x + -1}{B}\right|} \]
3. Recombined 4 regimes into one program.

4. Final simplification 45.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.034:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -6 \cdot 10^{-114}:\\ \;\;\;\;\frac{F}{\frac{B}{\sqrt{0.5}}}\\ \mathbf{elif}\;F \leq 52000000000:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-1 + x}{B}\right|\\ \end{array} \]

Alternative 26: 37.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.112:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -1.35 \cdot 10^{-110}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{B}\\ \mathbf{elif}\;F \leq 39000000000:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-1 + x}{B}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.112)
   (/ -1.0 (sin B))
   (if (<= F -1.35e-110)
     (/ (* F (sqrt 0.5)) B)
     (if (<= F 39000000000.0) (/ (- x) B) (fabs (/ (+ -1.0 x) B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.112) {
		tmp = -1.0 / sin(B);
	} else if (F <= -1.35e-110) {
		tmp = (F * sqrt(0.5)) / B;
	} else if (F <= 39000000000.0) {
		tmp = -x / B;
	} else {
		tmp = fabs(((-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 <= (-0.112d0)) then
        tmp = (-1.0d0) / sin(b)
    else if (f <= (-1.35d-110)) then
        tmp = (f * sqrt(0.5d0)) / b
    else if (f <= 39000000000.0d0) then
        tmp = -x / b
    else
        tmp = abs((((-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 <= -0.112) {
		tmp = -1.0 / Math.sin(B);
	} else if (F <= -1.35e-110) {
		tmp = (F * Math.sqrt(0.5)) / B;
	} else if (F <= 39000000000.0) {
		tmp = -x / B;
	} else {
		tmp = Math.abs(((-1.0 + x) / B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -0.112:
		tmp = -1.0 / math.sin(B)
	elif F <= -1.35e-110:
		tmp = (F * math.sqrt(0.5)) / B
	elif F <= 39000000000.0:
		tmp = -x / B
	else:
		tmp = math.fabs(((-1.0 + x) / B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.112)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= -1.35e-110)
		tmp = Float64(Float64(F * sqrt(0.5)) / B);
	elseif (F <= 39000000000.0)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = abs(Float64(Float64(-1.0 + x) / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -0.112)
		tmp = -1.0 / sin(B);
	elseif (F <= -1.35e-110)
		tmp = (F * sqrt(0.5)) / B;
	elseif (F <= 39000000000.0)
		tmp = -x / B;
	else
		tmp = abs(((-1.0 + x) / B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.112], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.35e-110], N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 39000000000.0], N[((-x) / B), $MachinePrecision], N[Abs[N[(N[(-1.0 + x), $MachinePrecision] / B), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.112000000000000002

   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. Taylor expanded in F around -inf 98.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 65.5%

      \[\leadsto \left(-\color{blue}{\left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)}\right) + \frac{-1}{\sin B} \]
   4. Step-by-step derivation

      1. *-commutative 65.5%

         \[\leadsto \left(-\left(\frac{x}{B} + -0.3333333333333333 \cdot \color{blue}{\left(x \cdot B\right)}\right)\right) + \frac{-1}{\sin B} \]

   5. Simplified 65.5%

      \[\leadsto \left(-\color{blue}{\left(\frac{x}{B} + -0.3333333333333333 \cdot \left(x \cdot B\right)\right)}\right) + \frac{-1}{\sin B} \]

   6. Taylor expanded in x around 0 60.9%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   if -0.112000000000000002 < F < -1.3499999999999999e-110

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative 99.5%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      2. unsub-neg 99.5%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      3. associate-*l/ 99.4%

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      4. associate-*r/ 99.7%

         \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      5. *-commutative 99.7%

         \[\leadsto \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \cdot F} - x \cdot \frac{1}{\tan B} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]

   4. Taylor expanded in F around 0 99.2%

      \[\leadsto F \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 \cdot x + 2}}\right)} - \frac{x}{\tan B} \]

   5. Taylor expanded in x around 0 67.5%

      \[\leadsto \color{blue}{\frac{\sqrt{0.5} \cdot F}{\sin B}} \]

   6. Taylor expanded in B around 0 40.3%

      \[\leadsto \color{blue}{\frac{\sqrt{0.5} \cdot F}{B}} \]

   if -1.3499999999999999e-110 < F < 3.9e10

   1. Initial program 98.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 33.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 20.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   4. Step-by-step derivation

      1. associate-*r/ 20.4%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

      2. distribute-lft-in 20.4%

         \[\leadsto \frac{\color{blue}{-1 \cdot 1 + -1 \cdot x}}{B} \]

      3. metadata-eval 20.4%

         \[\leadsto \frac{\color{blue}{-1} + -1 \cdot x}{B} \]

      4. neg-mul-1 20.4%

         \[\leadsto \frac{-1 + \color{blue}{\left(-x\right)}}{B} \]

   5. Simplified 20.4%

      \[\leadsto \color{blue}{\frac{-1 + \left(-x\right)}{B}} \]

   6. Taylor expanded in x around inf 39.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   7. Step-by-step derivation

      1. associate-*r/ 39.9%

         \[\leadsto \color{blue}{\frac{-1 \cdot x}{B}} \]

      2. mul-1-neg 39.9%

         \[\leadsto \frac{\color{blue}{-x}}{B} \]

   8. Simplified 39.9%

      \[\leadsto \color{blue}{\frac{-x}{B}} \]

   if 3.9e10 < F

   1. Initial program 45.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 40.7%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 25.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   4. Step-by-step derivation

      1. associate-*r/ 25.6%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

      2. distribute-lft-in 25.6%

         \[\leadsto \frac{\color{blue}{-1 \cdot 1 + -1 \cdot x}}{B} \]

      3. metadata-eval 25.6%

         \[\leadsto \frac{\color{blue}{-1} + -1 \cdot x}{B} \]

      4. neg-mul-1 25.6%

         \[\leadsto \frac{-1 + \color{blue}{\left(-x\right)}}{B} \]

   5. Simplified 25.6%

      \[\leadsto \color{blue}{\frac{-1 + \left(-x\right)}{B}} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 11.7%

         \[\leadsto \color{blue}{\sqrt{\frac{-1 + \left(-x\right)}{B}} \cdot \sqrt{\frac{-1 + \left(-x\right)}{B}}} \]

      2. sqrt-unprod 18.3%

         \[\leadsto \color{blue}{\sqrt{\frac{-1 + \left(-x\right)}{B} \cdot \frac{-1 + \left(-x\right)}{B}}} \]

      3. pow2 18.3%

         \[\leadsto \sqrt{\color{blue}{{\left(\frac{-1 + \left(-x\right)}{B}\right)}^{2}}} \]

      4. +-commutative 18.3%

         \[\leadsto \sqrt{{\left(\frac{\color{blue}{\left(-x\right) + -1}}{B}\right)}^{2}} \]

      5. add-sqr-sqrt 8.4%

         \[\leadsto \sqrt{{\left(\frac{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}} + -1}{B}\right)}^{2}} \]

      6. sqrt-unprod 18.4%

         \[\leadsto \sqrt{{\left(\frac{\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}} + -1}{B}\right)}^{2}} \]

      7. sqr-neg 18.4%

         \[\leadsto \sqrt{{\left(\frac{\sqrt{\color{blue}{x \cdot x}} + -1}{B}\right)}^{2}} \]

      8. sqrt-unprod 9.9%

         \[\leadsto \sqrt{{\left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}} + -1}{B}\right)}^{2}} \]

      9. add-sqr-sqrt 18.3%

         \[\leadsto \sqrt{{\left(\frac{\color{blue}{x} + -1}{B}\right)}^{2}} \]

   7. Applied egg-rr 18.3%

      \[\leadsto \color{blue}{\sqrt{{\left(\frac{x + -1}{B}\right)}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 18.3%

         \[\leadsto \sqrt{\color{blue}{\frac{x + -1}{B} \cdot \frac{x + -1}{B}}} \]

      2. rem-sqrt-square 39.7%

         \[\leadsto \color{blue}{\left|\frac{x + -1}{B}\right|} \]

   9. Simplified 39.7%

      \[\leadsto \color{blue}{\left|\frac{x + -1}{B}\right|} \]
3. Recombined 4 regimes into one program.

4. Final simplification 45.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.112:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -1.35 \cdot 10^{-110}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{B}\\ \mathbf{elif}\;F \leq 39000000000:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-1 + x}{B}\right|\\ \end{array} \]

Alternative 27: 43.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.0034:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1 \cdot 10^{-110}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{B}\\ \mathbf{elif}\;F \leq 16500000000:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-1 + x}{B}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.0034)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F -1e-110)
     (/ (* F (sqrt 0.5)) B)
     (if (<= F 16500000000.0) (/ (- x) B) (fabs (/ (+ -1.0 x) B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.0034) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -1e-110) {
		tmp = (F * sqrt(0.5)) / B;
	} else if (F <= 16500000000.0) {
		tmp = -x / B;
	} else {
		tmp = fabs(((-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 <= (-0.0034d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-1d-110)) then
        tmp = (f * sqrt(0.5d0)) / b
    else if (f <= 16500000000.0d0) then
        tmp = -x / b
    else
        tmp = abs((((-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 <= -0.0034) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -1e-110) {
		tmp = (F * Math.sqrt(0.5)) / B;
	} else if (F <= 16500000000.0) {
		tmp = -x / B;
	} else {
		tmp = Math.abs(((-1.0 + x) / B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -0.0034:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -1e-110:
		tmp = (F * math.sqrt(0.5)) / B
	elif F <= 16500000000.0:
		tmp = -x / B
	else:
		tmp = math.fabs(((-1.0 + x) / B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.0034)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -1e-110)
		tmp = Float64(Float64(F * sqrt(0.5)) / B);
	elseif (F <= 16500000000.0)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = abs(Float64(Float64(-1.0 + x) / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -0.0034)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -1e-110)
		tmp = (F * sqrt(0.5)) / B;
	elseif (F <= 16500000000.0)
		tmp = -x / B;
	else
		tmp = abs(((-1.0 + x) / B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.0034], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1e-110], N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 16500000000.0], N[((-x) / B), $MachinePrecision], N[Abs[N[(N[(-1.0 + x), $MachinePrecision] / B), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.00339999999999999981

   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. Taylor expanded in F around -inf 98.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 79.9%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{-1}{\sin B} \]

   if -0.00339999999999999981 < F < -1.0000000000000001e-110

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. +-commutative 99.5%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      2. unsub-neg 99.5%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      3. associate-*l/ 99.4%

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      4. associate-*r/ 99.7%

         \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      5. *-commutative 99.7%

         \[\leadsto \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \cdot F} - x \cdot \frac{1}{\tan B} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]

   4. Taylor expanded in F around 0 99.2%

      \[\leadsto F \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 \cdot x + 2}}\right)} - \frac{x}{\tan B} \]

   5. Taylor expanded in x around 0 67.5%

      \[\leadsto \color{blue}{\frac{\sqrt{0.5} \cdot F}{\sin B}} \]

   6. Taylor expanded in B around 0 40.3%

      \[\leadsto \color{blue}{\frac{\sqrt{0.5} \cdot F}{B}} \]

   if -1.0000000000000001e-110 < F < 1.65e10

   1. Initial program 98.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 33.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 20.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   4. Step-by-step derivation

      1. associate-*r/ 20.4%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

      2. distribute-lft-in 20.4%

         \[\leadsto \frac{\color{blue}{-1 \cdot 1 + -1 \cdot x}}{B} \]

      3. metadata-eval 20.4%

         \[\leadsto \frac{\color{blue}{-1} + -1 \cdot x}{B} \]

      4. neg-mul-1 20.4%

         \[\leadsto \frac{-1 + \color{blue}{\left(-x\right)}}{B} \]

   5. Simplified 20.4%

      \[\leadsto \color{blue}{\frac{-1 + \left(-x\right)}{B}} \]

   6. Taylor expanded in x around inf 39.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   7. Step-by-step derivation

      1. associate-*r/ 39.9%

         \[\leadsto \color{blue}{\frac{-1 \cdot x}{B}} \]

      2. mul-1-neg 39.9%

         \[\leadsto \frac{\color{blue}{-x}}{B} \]

   8. Simplified 39.9%

      \[\leadsto \color{blue}{\frac{-x}{B}} \]

   if 1.65e10 < F

   1. Initial program 45.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 40.7%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 25.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   4. Step-by-step derivation

      1. associate-*r/ 25.6%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

      2. distribute-lft-in 25.6%

         \[\leadsto \frac{\color{blue}{-1 \cdot 1 + -1 \cdot x}}{B} \]

      3. metadata-eval 25.6%

         \[\leadsto \frac{\color{blue}{-1} + -1 \cdot x}{B} \]

      4. neg-mul-1 25.6%

         \[\leadsto \frac{-1 + \color{blue}{\left(-x\right)}}{B} \]

   5. Simplified 25.6%

      \[\leadsto \color{blue}{\frac{-1 + \left(-x\right)}{B}} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 11.7%

         \[\leadsto \color{blue}{\sqrt{\frac{-1 + \left(-x\right)}{B}} \cdot \sqrt{\frac{-1 + \left(-x\right)}{B}}} \]

      2. sqrt-unprod 18.3%

         \[\leadsto \color{blue}{\sqrt{\frac{-1 + \left(-x\right)}{B} \cdot \frac{-1 + \left(-x\right)}{B}}} \]

      3. pow2 18.3%

         \[\leadsto \sqrt{\color{blue}{{\left(\frac{-1 + \left(-x\right)}{B}\right)}^{2}}} \]

      4. +-commutative 18.3%

         \[\leadsto \sqrt{{\left(\frac{\color{blue}{\left(-x\right) + -1}}{B}\right)}^{2}} \]

      5. add-sqr-sqrt 8.4%

         \[\leadsto \sqrt{{\left(\frac{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}} + -1}{B}\right)}^{2}} \]

      6. sqrt-unprod 18.4%

         \[\leadsto \sqrt{{\left(\frac{\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}} + -1}{B}\right)}^{2}} \]

      7. sqr-neg 18.4%

         \[\leadsto \sqrt{{\left(\frac{\sqrt{\color{blue}{x \cdot x}} + -1}{B}\right)}^{2}} \]

      8. sqrt-unprod 9.9%

         \[\leadsto \sqrt{{\left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}} + -1}{B}\right)}^{2}} \]

      9. add-sqr-sqrt 18.3%

         \[\leadsto \sqrt{{\left(\frac{\color{blue}{x} + -1}{B}\right)}^{2}} \]

   7. Applied egg-rr 18.3%

      \[\leadsto \color{blue}{\sqrt{{\left(\frac{x + -1}{B}\right)}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 18.3%

         \[\leadsto \sqrt{\color{blue}{\frac{x + -1}{B} \cdot \frac{x + -1}{B}}} \]

      2. rem-sqrt-square 39.7%

         \[\leadsto \color{blue}{\left|\frac{x + -1}{B}\right|} \]

   9. Simplified 39.7%

      \[\leadsto \color{blue}{\left|\frac{x + -1}{B}\right|} \]
3. Recombined 4 regimes into one program.

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.0034:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1 \cdot 10^{-110}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{B}\\ \mathbf{elif}\;F \leq 16500000000:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-1 + x}{B}\right|\\ \end{array} \]

Alternative 28: 36.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -5.2 \cdot 10^{-47}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 7400000000:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-1 + x}{B}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -5.2e-47)
   (/ -1.0 (sin B))
   (if (<= F 7400000000.0) (/ (- x) B) (fabs (/ (+ -1.0 x) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -5.2e-47) {
		tmp = -1.0 / sin(B);
	} else if (F <= 7400000000.0) {
		tmp = -x / B;
	} else {
		tmp = fabs(((-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 <= (-5.2d-47)) then
        tmp = (-1.0d0) / sin(b)
    else if (f <= 7400000000.0d0) then
        tmp = -x / b
    else
        tmp = abs((((-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 <= -5.2e-47) {
		tmp = -1.0 / Math.sin(B);
	} else if (F <= 7400000000.0) {
		tmp = -x / B;
	} else {
		tmp = Math.abs(((-1.0 + x) / B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -5.2e-47:
		tmp = -1.0 / math.sin(B)
	elif F <= 7400000000.0:
		tmp = -x / B
	else:
		tmp = math.fabs(((-1.0 + x) / B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -5.2e-47)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 7400000000.0)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = abs(Float64(Float64(-1.0 + x) / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -5.2e-47)
		tmp = -1.0 / sin(B);
	elseif (F <= 7400000000.0)
		tmp = -x / B;
	else
		tmp = abs(((-1.0 + x) / B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -5.2e-47], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7400000000.0], N[((-x) / B), $MachinePrecision], N[Abs[N[(N[(-1.0 + x), $MachinePrecision] / B), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.2e-47

   1. Initial program 63.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 85.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 56.7%

      \[\leadsto \left(-\color{blue}{\left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)}\right) + \frac{-1}{\sin B} \]
   4. Step-by-step derivation

      1. *-commutative 56.7%

         \[\leadsto \left(-\left(\frac{x}{B} + -0.3333333333333333 \cdot \color{blue}{\left(x \cdot B\right)}\right)\right) + \frac{-1}{\sin B} \]

   5. Simplified 56.7%

      \[\leadsto \left(-\color{blue}{\left(\frac{x}{B} + -0.3333333333333333 \cdot \left(x \cdot B\right)\right)}\right) + \frac{-1}{\sin B} \]

   6. Taylor expanded in x around 0 53.1%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   if -5.2e-47 < F < 7.4e9

   1. Initial program 98.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 34.5%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 19.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   4. Step-by-step derivation

      1. associate-*r/ 19.3%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

      2. distribute-lft-in 19.3%

         \[\leadsto \frac{\color{blue}{-1 \cdot 1 + -1 \cdot x}}{B} \]

      3. metadata-eval 19.3%

         \[\leadsto \frac{\color{blue}{-1} + -1 \cdot x}{B} \]

      4. neg-mul-1 19.3%

         \[\leadsto \frac{-1 + \color{blue}{\left(-x\right)}}{B} \]

   5. Simplified 19.3%

      \[\leadsto \color{blue}{\frac{-1 + \left(-x\right)}{B}} \]

   6. Taylor expanded in x around inf 37.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   7. Step-by-step derivation

      1. associate-*r/ 37.0%

         \[\leadsto \color{blue}{\frac{-1 \cdot x}{B}} \]

      2. mul-1-neg 37.0%

         \[\leadsto \frac{\color{blue}{-x}}{B} \]

   8. Simplified 37.0%

      \[\leadsto \color{blue}{\frac{-x}{B}} \]

   if 7.4e9 < F

   1. Initial program 45.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 40.7%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 25.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   4. Step-by-step derivation

      1. associate-*r/ 25.6%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

      2. distribute-lft-in 25.6%

         \[\leadsto \frac{\color{blue}{-1 \cdot 1 + -1 \cdot x}}{B} \]

      3. metadata-eval 25.6%

         \[\leadsto \frac{\color{blue}{-1} + -1 \cdot x}{B} \]

      4. neg-mul-1 25.6%

         \[\leadsto \frac{-1 + \color{blue}{\left(-x\right)}}{B} \]

   5. Simplified 25.6%

      \[\leadsto \color{blue}{\frac{-1 + \left(-x\right)}{B}} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 11.7%

         \[\leadsto \color{blue}{\sqrt{\frac{-1 + \left(-x\right)}{B}} \cdot \sqrt{\frac{-1 + \left(-x\right)}{B}}} \]

      2. sqrt-unprod 18.3%

         \[\leadsto \color{blue}{\sqrt{\frac{-1 + \left(-x\right)}{B} \cdot \frac{-1 + \left(-x\right)}{B}}} \]

      3. pow2 18.3%

         \[\leadsto \sqrt{\color{blue}{{\left(\frac{-1 + \left(-x\right)}{B}\right)}^{2}}} \]

      4. +-commutative 18.3%

         \[\leadsto \sqrt{{\left(\frac{\color{blue}{\left(-x\right) + -1}}{B}\right)}^{2}} \]

      5. add-sqr-sqrt 8.4%

         \[\leadsto \sqrt{{\left(\frac{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}} + -1}{B}\right)}^{2}} \]

      6. sqrt-unprod 18.4%

         \[\leadsto \sqrt{{\left(\frac{\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}} + -1}{B}\right)}^{2}} \]

      7. sqr-neg 18.4%

         \[\leadsto \sqrt{{\left(\frac{\sqrt{\color{blue}{x \cdot x}} + -1}{B}\right)}^{2}} \]

      8. sqrt-unprod 9.9%

         \[\leadsto \sqrt{{\left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}} + -1}{B}\right)}^{2}} \]

      9. add-sqr-sqrt 18.3%

         \[\leadsto \sqrt{{\left(\frac{\color{blue}{x} + -1}{B}\right)}^{2}} \]

   7. Applied egg-rr 18.3%

      \[\leadsto \color{blue}{\sqrt{{\left(\frac{x + -1}{B}\right)}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 18.3%

         \[\leadsto \sqrt{\color{blue}{\frac{x + -1}{B} \cdot \frac{x + -1}{B}}} \]

      2. rem-sqrt-square 39.7%

         \[\leadsto \color{blue}{\left|\frac{x + -1}{B}\right|} \]

   9. Simplified 39.7%

      \[\leadsto \color{blue}{\left|\frac{x + -1}{B}\right|} \]
3. Recombined 3 regimes into one program.

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5.2 \cdot 10^{-47}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 7400000000:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-1 + x}{B}\right|\\ \end{array} \]

Alternative 29: 36.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -5.2 \cdot 10^{-47}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -5.2e-47) (/ -1.0 (sin B)) (/ (- x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -5.2e-47) {
		tmp = -1.0 / sin(B);
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-5.2d-47)) then
        tmp = (-1.0d0) / sin(b)
    else
        tmp = -x / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -5.2e-47) {
		tmp = -1.0 / Math.sin(B);
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -5.2e-47:
		tmp = -1.0 / math.sin(B)
	else:
		tmp = -x / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -5.2e-47)
		tmp = Float64(-1.0 / sin(B));
	else
		tmp = Float64(Float64(-x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -5.2e-47)
		tmp = -1.0 / sin(B);
	else
		tmp = -x / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -5.2e-47], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[((-x) / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -5.2e-47

   1. Initial program 63.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 85.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 56.7%

      \[\leadsto \left(-\color{blue}{\left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)}\right) + \frac{-1}{\sin B} \]
   4. Step-by-step derivation

      1. *-commutative 56.7%

         \[\leadsto \left(-\left(\frac{x}{B} + -0.3333333333333333 \cdot \color{blue}{\left(x \cdot B\right)}\right)\right) + \frac{-1}{\sin B} \]

   5. Simplified 56.7%

      \[\leadsto \left(-\color{blue}{\left(\frac{x}{B} + -0.3333333333333333 \cdot \left(x \cdot B\right)\right)}\right) + \frac{-1}{\sin B} \]

   6. Taylor expanded in x around 0 53.1%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   if -5.2e-47 < F

   1. Initial program 77.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 37.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 21.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   4. Step-by-step derivation

      1. associate-*r/ 21.8%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

      2. distribute-lft-in 21.8%

         \[\leadsto \frac{\color{blue}{-1 \cdot 1 + -1 \cdot x}}{B} \]

      3. metadata-eval 21.8%

         \[\leadsto \frac{\color{blue}{-1} + -1 \cdot x}{B} \]

      4. neg-mul-1 21.8%

         \[\leadsto \frac{-1 + \color{blue}{\left(-x\right)}}{B} \]

   5. Simplified 21.8%

      \[\leadsto \color{blue}{\frac{-1 + \left(-x\right)}{B}} \]

   6. Taylor expanded in x around inf 32.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   7. Step-by-step derivation

      1. associate-*r/ 32.9%

         \[\leadsto \color{blue}{\frac{-1 \cdot x}{B}} \]

      2. mul-1-neg 32.9%

         \[\leadsto \frac{\color{blue}{-x}}{B} \]

   8. Simplified 32.9%

      \[\leadsto \color{blue}{\frac{-x}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5.2 \cdot 10^{-47}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \]

Alternative 30: 36.9% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{-47}:\\ \;\;\;\;B \cdot -0.16666666666666666 + \frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -5e-47)
   (+ (* B -0.16666666666666666) (/ (- -1.0 x) B))
   (/ (- x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -5e-47) {
		tmp = (B * -0.16666666666666666) + ((-1.0 - x) / B);
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-5d-47)) then
        tmp = (b * (-0.16666666666666666d0)) + (((-1.0d0) - x) / b)
    else
        tmp = -x / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -5e-47) {
		tmp = (B * -0.16666666666666666) + ((-1.0 - x) / B);
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -5e-47:
		tmp = (B * -0.16666666666666666) + ((-1.0 - x) / B)
	else:
		tmp = -x / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -5e-47)
		tmp = Float64(Float64(B * -0.16666666666666666) + Float64(Float64(-1.0 - x) / B));
	else
		tmp = Float64(Float64(-x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -5e-47)
		tmp = (B * -0.16666666666666666) + ((-1.0 - x) / B);
	else
		tmp = -x / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -5e-47], N[(N[(B * -0.16666666666666666), $MachinePrecision] + N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], N[((-x) / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -5.00000000000000011e-47

   1. Initial program 63.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 85.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 69.3%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{-1}{\sin B} \]

   4. Taylor expanded in B around 0 43.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B} + -0.16666666666666666 \cdot B} \]
   5. Step-by-step derivation

      1. +-commutative 43.7%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot B + -1 \cdot \frac{1 + x}{B}} \]

      2. mul-1-neg 43.7%

         \[\leadsto -0.16666666666666666 \cdot B + \color{blue}{\left(-\frac{1 + x}{B}\right)} \]

      3. unsub-neg 43.7%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot B - \frac{1 + x}{B}} \]

      4. *-commutative 43.7%

         \[\leadsto \color{blue}{B \cdot -0.16666666666666666} - \frac{1 + x}{B} \]

      5. +-commutative 43.7%

         \[\leadsto B \cdot -0.16666666666666666 - \frac{\color{blue}{x + 1}}{B} \]

   6. Simplified 43.7%

      \[\leadsto \color{blue}{B \cdot -0.16666666666666666 - \frac{x + 1}{B}} \]

   if -5.00000000000000011e-47 < F

   1. Initial program 77.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 37.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 21.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   4. Step-by-step derivation

      1. associate-*r/ 21.8%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

      2. distribute-lft-in 21.8%

         \[\leadsto \frac{\color{blue}{-1 \cdot 1 + -1 \cdot x}}{B} \]

      3. metadata-eval 21.8%

         \[\leadsto \frac{\color{blue}{-1} + -1 \cdot x}{B} \]

      4. neg-mul-1 21.8%

         \[\leadsto \frac{-1 + \color{blue}{\left(-x\right)}}{B} \]

   5. Simplified 21.8%

      \[\leadsto \color{blue}{\frac{-1 + \left(-x\right)}{B}} \]

   6. Taylor expanded in x around inf 32.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   7. Step-by-step derivation

      1. associate-*r/ 32.9%

         \[\leadsto \color{blue}{\frac{-1 \cdot x}{B}} \]

      2. mul-1-neg 32.9%

         \[\leadsto \frac{\color{blue}{-x}}{B} \]

   8. Simplified 32.9%

      \[\leadsto \color{blue}{\frac{-x}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{-47}:\\ \;\;\;\;B \cdot -0.16666666666666666 + \frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \]

Alternative 31: 36.8% accurate, 45.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -4.1 \cdot 10^{-47}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4.1e-47) (/ (- -1.0 x) B) (/ (- x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.1e-47) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-4.1d-47)) then
        tmp = ((-1.0d0) - x) / b
    else
        tmp = -x / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.1e-47) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -4.1e-47:
		tmp = (-1.0 - x) / B
	else:
		tmp = -x / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -4.1e-47)
		tmp = Float64(Float64(-1.0 - x) / B);
	else
		tmp = Float64(Float64(-x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -4.1e-47)
		tmp = (-1.0 - x) / B;
	else
		tmp = -x / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -4.1e-47], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[((-x) / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -4.10000000000000002e-47

   1. Initial program 63.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 85.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 43.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   4. Step-by-step derivation

      1. associate-*r/ 43.7%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

      2. distribute-lft-in 43.7%

         \[\leadsto \frac{\color{blue}{-1 \cdot 1 + -1 \cdot x}}{B} \]

      3. metadata-eval 43.7%

         \[\leadsto \frac{\color{blue}{-1} + -1 \cdot x}{B} \]

      4. neg-mul-1 43.7%

         \[\leadsto \frac{-1 + \color{blue}{\left(-x\right)}}{B} \]

   5. Simplified 43.7%

      \[\leadsto \color{blue}{\frac{-1 + \left(-x\right)}{B}} \]

   if -4.10000000000000002e-47 < F

   1. Initial program 77.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 37.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 21.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   4. Step-by-step derivation

      1. associate-*r/ 21.8%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

      2. distribute-lft-in 21.8%

         \[\leadsto \frac{\color{blue}{-1 \cdot 1 + -1 \cdot x}}{B} \]

      3. metadata-eval 21.8%

         \[\leadsto \frac{\color{blue}{-1} + -1 \cdot x}{B} \]

      4. neg-mul-1 21.8%

         \[\leadsto \frac{-1 + \color{blue}{\left(-x\right)}}{B} \]

   5. Simplified 21.8%

      \[\leadsto \color{blue}{\frac{-1 + \left(-x\right)}{B}} \]

   6. Taylor expanded in x around inf 32.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   7. Step-by-step derivation

      1. associate-*r/ 32.9%

         \[\leadsto \color{blue}{\frac{-1 \cdot x}{B}} \]

      2. mul-1-neg 32.9%

         \[\leadsto \frac{\color{blue}{-x}}{B} \]

   8. Simplified 32.9%

      \[\leadsto \color{blue}{\frac{-x}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.1 \cdot 10^{-47}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \]

Alternative 32: 30.0% accurate, 53.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.85 \cdot 10^{+109}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.85e+109) (/ -1.0 B) (/ (- x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.85e+109) {
		tmp = -1.0 / B;
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.85d+109)) then
        tmp = (-1.0d0) / b
    else
        tmp = -x / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.85e+109) {
		tmp = -1.0 / B;
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.85e+109:
		tmp = -1.0 / B
	else:
		tmp = -x / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.85e+109)
		tmp = Float64(-1.0 / B);
	else
		tmp = Float64(Float64(-x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.85e+109)
		tmp = -1.0 / B;
	else
		tmp = -x / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.85e+109], N[(-1.0 / B), $MachinePrecision], N[((-x) / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -1.8500000000000001e109

   1. Initial program 38.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 99.7%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 54.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   4. Step-by-step derivation

      1. associate-*r/ 54.6%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

      2. distribute-lft-in 54.6%

         \[\leadsto \frac{\color{blue}{-1 \cdot 1 + -1 \cdot x}}{B} \]

      3. metadata-eval 54.6%

         \[\leadsto \frac{\color{blue}{-1} + -1 \cdot x}{B} \]

      4. neg-mul-1 54.6%

         \[\leadsto \frac{-1 + \color{blue}{\left(-x\right)}}{B} \]

   5. Simplified 54.6%

      \[\leadsto \color{blue}{\frac{-1 + \left(-x\right)}{B}} \]

   6. Taylor expanded in x around 0 37.0%

      \[\leadsto \color{blue}{\frac{-1}{B}} \]

   if -1.8500000000000001e109 < F

   1. Initial program 80.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf 41.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   3. Taylor expanded in B around 0 23.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   4. Step-by-step derivation

      1. associate-*r/ 23.0%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

      2. distribute-lft-in 23.0%

         \[\leadsto \frac{\color{blue}{-1 \cdot 1 + -1 \cdot x}}{B} \]

      3. metadata-eval 23.0%

         \[\leadsto \frac{\color{blue}{-1} + -1 \cdot x}{B} \]

      4. neg-mul-1 23.0%

         \[\leadsto \frac{-1 + \color{blue}{\left(-x\right)}}{B} \]

   5. Simplified 23.0%

      \[\leadsto \color{blue}{\frac{-1 + \left(-x\right)}{B}} \]

   6. Taylor expanded in x around inf 30.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   7. Step-by-step derivation

      1. associate-*r/ 30.4%

         \[\leadsto \color{blue}{\frac{-1 \cdot x}{B}} \]

      2. mul-1-neg 30.4%

         \[\leadsto \frac{\color{blue}{-x}}{B} \]

   8. Simplified 30.4%

      \[\leadsto \color{blue}{\frac{-x}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.85 \cdot 10^{+109}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \]

Alternative 33: 10.1% accurate, 108.0× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (/ -1.0 B))

C

double code(double F, double B, double x) {
	return -1.0 / B;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = (-1.0d0) / b
end function

Java

public static double code(double F, double B, double x) {
	return -1.0 / B;
}

Python

def code(F, B, x):
	return -1.0 / B

Julia

function code(F, B, x)
	return Float64(-1.0 / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = -1.0 / B;
end

Wolfram

code[F_, B_, x_] := N[(-1.0 / B), $MachinePrecision]

Derivation

1. Initial program 73.1%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

2. Taylor expanded in F around -inf 52.3%

   \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

3. Taylor expanded in B around 0 28.7%

   \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
4. Step-by-step derivation

   1. associate-*r/ 28.7%

      \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

   2. distribute-lft-in 28.7%

      \[\leadsto \frac{\color{blue}{-1 \cdot 1 + -1 \cdot x}}{B} \]

   3. metadata-eval 28.7%

      \[\leadsto \frac{\color{blue}{-1} + -1 \cdot x}{B} \]

   4. neg-mul-1 28.7%

      \[\leadsto \frac{-1 + \color{blue}{\left(-x\right)}}{B} \]

5. Simplified 28.7%

   \[\leadsto \color{blue}{\frac{-1 + \left(-x\right)}{B}} \]

6. Taylor expanded in x around 0 11.0%

   \[\leadsto \color{blue}{\frac{-1}{B}} \]

7. Final simplification 11.0%

   \[\leadsto \frac{-1}{B} \]

Reproduce

herbie shell --seed 2023250 
(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))))))