VandenBroeck and Keller, Equation (23)

Percentage Accurate: 77.0% → 99.6%

Time: 17.5s

Alternatives: 24

Speedup: 1.4×

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: 77.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3.6 \cdot 10^{+36}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 2750:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\frac{\sin B}{F}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - x \cdot \frac{\cos B}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -3.6e+36)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 2750.0)
       (- (/ (pow (fma 2.0 x (fma F F 2.0)) -0.5) (/ (sin B) F)) t_0)
       (- (/ 1.0 (sin B)) (* x (/ (cos B) (sin B))))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -3.6e+36) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 2750.0) {
		tmp = (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) / (sin(B) / F)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - (x * (cos(B) / sin(B)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -3.6e+36)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 2750.0)
		tmp = Float64(Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) / Float64(sin(B) / F)) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x * Float64(cos(B) / 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, -3.6e+36], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 2750.0], N[(N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[(N[Sin[B], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x * N[(N[Cos[B], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.5999999999999997e36

   1. Initial program 54.6%

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

   3. Simplified 67.7%

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

   5. Taylor expanded in F around -inf 99.7%

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

   if -3.5999999999999997e36 < F < 2750

   1. Initial program 99.4%

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

   3. Simplified 99.5%

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

      1. associate-*r/ 99.6%

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

      2. associate-*l/ 99.5%

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

      3. *-commutative 99.5%

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

      4. clear-num 99.5%

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

      5. un-div-inv 99.5%

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

      6. fma-define 99.5%

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

      7. fma-undefine 99.5%

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

      8. *-commutative 99.5%

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

      9. fma-define 99.5%

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

      10. fma-define 99.5%

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

   6. Applied egg-rr 99.5%

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

   if 2750 < F

   1. Initial program 64.6%

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

   3. Simplified 80.7%

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

   5. Taylor expanded in F around inf 99.7%

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

      1. associate-/l* 99.8%

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

   7. Simplified 99.8%

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

Alternative 2: 99.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1 \cdot 10^{+37}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 2750:\\ \;\;\;\;F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - x \cdot \frac{\cos B}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1e+37)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 2750.0)
       (- (* F (/ (pow (fma x 2.0 (fma F F 2.0)) -0.5) (sin B))) t_0)
       (- (/ 1.0 (sin B)) (* x (/ (cos B) (sin B))))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1e+37) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 2750.0) {
		tmp = (F * (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) / sin(B))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - (x * (cos(B) / sin(B)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1e+37)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 2750.0)
		tmp = Float64(Float64(F * Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) / sin(B))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x * Float64(cos(B) / 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, -1e+37], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 2750.0], 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[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x * N[(N[Cos[B], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -9.99999999999999954e36

   1. Initial program 54.6%

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

   3. Simplified 67.7%

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

   5. Taylor expanded in F around -inf 99.7%

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

   if -9.99999999999999954e36 < F < 2750

   1. Initial program 99.4%

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

   3. Simplified 99.5%

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

   if 2750 < F

   1. Initial program 64.6%

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

   3. Simplified 80.7%

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

   5. Taylor expanded in F around inf 99.7%

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

      1. associate-/l* 99.8%

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

   7. Simplified 99.8%

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

Alternative 3: 99.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2 \cdot 10^{+38}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_1\\ \mathbf{elif}\;F \leq 2750:\\ \;\;\;\;F \cdot \left(t\_0 \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}\right) - t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 - x \cdot \frac{\cos B}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))) (t_1 (/ x (tan B))))
   (if (<= F -2e+38)
     (- (/ -1.0 (sin B)) t_1)
     (if (<= F 2750.0)
       (- (* F (* t_0 (sqrt (/ 1.0 (fma F F 2.0))))) t_1)
       (- t_0 (* x (/ (cos B) (sin B))))))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / sin(B);
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -2e+38) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= 2750.0) {
		tmp = (F * (t_0 * sqrt((1.0 / fma(F, F, 2.0))))) - t_1;
	} else {
		tmp = t_0 - (x * (cos(B) / sin(B)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / sin(B))
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -2e+38)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= 2750.0)
		tmp = Float64(Float64(F * Float64(t_0 * sqrt(Float64(1.0 / fma(F, F, 2.0))))) - t_1);
	else
		tmp = Float64(t_0 - Float64(x * Float64(cos(B) / sin(B))));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2e+38], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, 2750.0], N[(N[(F * N[(t$95$0 * N[Sqrt[N[(1.0 / N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(t$95$0 - N[(x * N[(N[Cos[B], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.99999999999999995e38

   1. Initial program 54.6%

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

   3. Simplified 67.7%

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

   5. Taylor expanded in F around -inf 99.7%

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

   if -1.99999999999999995e38 < F < 2750

   1. Initial program 99.4%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around 0 99.5%

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

      1. +-commutative 99.5%

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

      2. unpow2 99.5%

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

      3. fma-undefine 99.5%

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

   7. Simplified 99.5%

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

   if 2750 < F

   1. Initial program 64.6%

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

   3. Simplified 80.7%

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

   5. Taylor expanded in F around inf 99.7%

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

      1. associate-/l* 99.8%

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

   7. Simplified 99.8%

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

Alternative 4: 99.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1 \cdot 10^{+37}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 2750:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - x \cdot \frac{\cos B}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1e+37)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 2750.0)
       (- (* F (/ (sqrt (/ 1.0 (fma F F 2.0))) (sin B))) t_0)
       (- (/ 1.0 (sin B)) (* x (/ (cos B) (sin B))))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1e+37) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 2750.0) {
		tmp = (F * (sqrt((1.0 / fma(F, F, 2.0))) / sin(B))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - (x * (cos(B) / sin(B)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1e+37)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 2750.0)
		tmp = Float64(Float64(F * Float64(sqrt(Float64(1.0 / fma(F, F, 2.0))) / sin(B))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x * Float64(cos(B) / 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, -1e+37], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 2750.0], N[(N[(F * N[(N[Sqrt[N[(1.0 / N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x * N[(N[Cos[B], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -9.99999999999999954e36

   1. Initial program 54.6%

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

   3. Simplified 67.7%

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

   5. Taylor expanded in F around -inf 99.7%

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

   if -9.99999999999999954e36 < F < 2750

   1. Initial program 99.4%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around 0 99.5%

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

      1. associate-*l/ 99.5%

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

      2. *-lft-identity 99.5%

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

      3. +-commutative 99.5%

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

      4. unpow2 99.5%

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

      5. fma-undefine 99.5%

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

   7. Simplified 99.5%

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

   if 2750 < F

   1. Initial program 64.6%

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

   3. Simplified 80.7%

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

   5. Taylor expanded in F around inf 99.7%

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

      1. associate-/l* 99.8%

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

   7. Simplified 99.8%

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

Alternative 5: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -4.8 \cdot 10^{+36}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2750:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - x \cdot \frac{\cos B}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4.8e+36)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 2750.0)
     (+
      (* x (/ -1.0 (tan B)))
      (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)))
     (- (/ 1.0 (sin B)) (* x (/ (cos B) (sin B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.8e+36) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 2750.0) {
		tmp = (x * (-1.0 / tan(B))) + ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5));
	} else {
		tmp = (1.0 / sin(B)) - (x * (cos(B) / sin(B)));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-4.8d+36)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 2750.0d0) then
        tmp = (x * ((-1.0d0) / tan(b))) + ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)))
    else
        tmp = (1.0d0 / sin(b)) - (x * (cos(b) / sin(b)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.8e+36) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 2750.0) {
		tmp = (x * (-1.0 / Math.tan(B))) + ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5));
	} else {
		tmp = (1.0 / Math.sin(B)) - (x * (Math.cos(B) / Math.sin(B)));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -4.8e+36:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 2750.0:
		tmp = (x * (-1.0 / math.tan(B))) + ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5))
	else:
		tmp = (1.0 / math.sin(B)) - (x * (math.cos(B) / math.sin(B)))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -4.8e+36)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 2750.0)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x * Float64(cos(B) / sin(B))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -4.8e+36)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 2750.0)
		tmp = (x * (-1.0 / tan(B))) + ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5));
	else
		tmp = (1.0 / sin(B)) - (x * (cos(B) / sin(B)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -4.8e+36], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2750.0], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $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[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x * N[(N[Cos[B], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.79999999999999985e36

   1. Initial program 54.6%

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

   3. Simplified 67.7%

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

   5. Taylor expanded in F around -inf 99.7%

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

   if -4.79999999999999985e36 < F < 2750

   1. Initial program 99.4%

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

      1. metadata-eval 99.4%

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

      2. metadata-eval 99.4%

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

   4. Applied egg-rr 99.4%

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

   if 2750 < F

   1. Initial program 64.6%

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

   3. Simplified 80.7%

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

   5. Taylor expanded in F around inf 99.7%

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

      1. associate-/l* 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.8 \cdot 10^{+36}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2750:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - x \cdot \frac{\cos B}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.39:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 3.4 \cdot 10^{-156}:\\ \;\;\;\;F \cdot \left(\frac{1}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\right) - t\_0\\ \mathbf{elif}\;F \leq 2200:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - x \cdot \frac{\cos B}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -0.39)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 3.4e-156)
       (- (* F (* (/ 1.0 B) (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))))) t_0)
       (if (<= F 2200.0)
         (- (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)) (/ x B))
         (- (/ 1.0 (sin B)) (* x (/ (cos B) (sin B)))))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -0.39) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 3.4e-156) {
		tmp = (F * ((1.0 / B) * sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_0;
	} else if (F <= 2200.0) {
		tmp = ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B);
	} else {
		tmp = (1.0 / sin(B)) - (x * (cos(B) / 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 <= (-0.39d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 3.4d-156) then
        tmp = (f * ((1.0d0 / b) * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))))) - t_0
    else if (f <= 2200.0d0) then
        tmp = ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0))) - (x / b)
    else
        tmp = (1.0d0 / sin(b)) - (x * (cos(b) / 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 <= -0.39) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 3.4e-156) {
		tmp = (F * ((1.0 / B) * Math.sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_0;
	} else if (F <= 2200.0) {
		tmp = ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B);
	} else {
		tmp = (1.0 / Math.sin(B)) - (x * (Math.cos(B) / Math.sin(B)));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -0.39:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 3.4e-156:
		tmp = (F * ((1.0 / B) * math.sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_0
	elif F <= 2200.0:
		tmp = ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B)
	else:
		tmp = (1.0 / math.sin(B)) - (x * (math.cos(B) / math.sin(B)))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -0.39)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 3.4e-156)
		tmp = Float64(Float64(F * Float64(Float64(1.0 / B) * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0)))))) - t_0);
	elseif (F <= 2200.0)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x * Float64(cos(B) / sin(B))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -0.39)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 3.4e-156)
		tmp = (F * ((1.0 / B) * sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_0;
	elseif (F <= 2200.0)
		tmp = ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5)) - (x / B);
	else
		tmp = (1.0 / sin(B)) - (x * (cos(B) / 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, -0.39], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 3.4e-156], N[(N[(F * N[(N[(1.0 / B), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 2200.0], 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], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x * N[(N[Cos[B], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.39000000000000001

   1. Initial program 55.9%

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

   3. Simplified 68.6%

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

   5. Taylor expanded in F around -inf 99.5%

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

   if -0.39000000000000001 < F < 3.3999999999999999e-156

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in F around 0 98.7%

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

      1. *-commutative 98.7%

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

   7. Simplified 98.7%

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

   8. Taylor expanded in B around 0 88.9%

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

   if 3.3999999999999999e-156 < F < 2200

   1. Initial program 99.3%

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

   3. Taylor expanded in B around 0 83.9%

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

      1. associate-*r/ 83.9%

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

      2. neg-mul-1 83.9%

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

   5. Simplified 83.9%

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

      1. metadata-eval 99.3%

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

      2. metadata-eval 99.3%

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

   7. Applied egg-rr 83.9%

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

   if 2200 < F

   1. Initial program 64.6%

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

   3. Simplified 80.7%

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

   5. Taylor expanded in F around inf 99.7%

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

      1. associate-/l* 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.39:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 3.4 \cdot 10^{-156}:\\ \;\;\;\;F \cdot \left(\frac{1}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\right) - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2200:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - x \cdot \frac{\cos B}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 0.6:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - x \cdot \frac{\cos B}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.4)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 0.6)
       (- (/ (* F (sqrt 0.5)) (sin B)) t_0)
       (- (/ 1.0 (sin B)) (* x (/ (cos B) (sin B))))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.4) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 0.6) {
		tmp = ((F * sqrt(0.5)) / sin(B)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - (x * (cos(B) / sin(B)));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.4d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 0.6d0) then
        tmp = ((f * sqrt(0.5d0)) / sin(b)) - t_0
    else
        tmp = (1.0d0 / sin(b)) - (x * (cos(b) / sin(b)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.4) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 0.6) {
		tmp = ((F * Math.sqrt(0.5)) / Math.sin(B)) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - (x * (Math.cos(B) / Math.sin(B)));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.4:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 0.6:
		tmp = ((F * math.sqrt(0.5)) / math.sin(B)) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - (x * (math.cos(B) / math.sin(B)))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.4)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 0.6)
		tmp = Float64(Float64(Float64(F * sqrt(0.5)) / sin(B)) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x * Float64(cos(B) / sin(B))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.4)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 0.6)
		tmp = ((F * sqrt(0.5)) / sin(B)) - t_0;
	else
		tmp = (1.0 / sin(B)) - (x * (cos(B) / sin(B)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.4], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.6], N[(N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x * N[(N[Cos[B], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.3999999999999999

   1. Initial program 55.9%

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

   3. Simplified 68.6%

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

   5. Taylor expanded in F around -inf 99.5%

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

   if -1.3999999999999999 < F < 0.599999999999999978

   1. Initial program 99.4%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in F around 0 98.9%

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

      1. *-commutative 98.9%

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

   7. Simplified 98.9%

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

   8. Taylor expanded in x around 0 98.9%

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

   if 0.599999999999999978 < F

   1. Initial program 65.7%

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

   3. Simplified 81.3%

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

   5. Taylor expanded in F around inf 98.5%

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

      1. associate-/l* 98.7%

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

   7. Simplified 98.7%

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

Alternative 8: 87.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1650000:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq -2.45 \cdot 10^{-79}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + x \cdot 2\right)}} - x}{B}\\ \mathbf{elif}\;F \leq 6.5 \cdot 10^{-157}:\\ \;\;\;\;\frac{x \cdot \left(-\cos B\right)}{\sin B}\\ \mathbf{elif}\;F \leq 3.4 \cdot 10^{-6}:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{2 + x \cdot 2}}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1650000.0)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F -2.45e-79)
       (/ (- (* F (sqrt (/ 1.0 (+ 2.0 (+ (* F F) (* x 2.0)))))) x) B)
       (if (<= F 6.5e-157)
         (/ (* x (- (cos B))) (sin B))
         (if (<= F 3.4e-6)
           (- (* F (/ (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))) (sin B))) (/ x B))
           (- (/ 1.0 (sin B)) t_0)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1650000.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= -2.45e-79) {
		tmp = ((F * sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0)))))) - x) / B;
	} else if (F <= 6.5e-157) {
		tmp = (x * -cos(B)) / sin(B);
	} else if (F <= 3.4e-6) {
		tmp = (F * (sqrt((1.0 / (2.0 + (x * 2.0)))) / sin(B))) - (x / B);
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1650000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= (-2.45d-79)) then
        tmp = ((f * sqrt((1.0d0 / (2.0d0 + ((f * f) + (x * 2.0d0)))))) - x) / b
    else if (f <= 6.5d-157) then
        tmp = (x * -cos(b)) / sin(b)
    else if (f <= 3.4d-6) then
        tmp = (f * (sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))) / sin(b))) - (x / b)
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1650000.0) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= -2.45e-79) {
		tmp = ((F * Math.sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0)))))) - x) / B;
	} else if (F <= 6.5e-157) {
		tmp = (x * -Math.cos(B)) / Math.sin(B);
	} else if (F <= 3.4e-6) {
		tmp = (F * (Math.sqrt((1.0 / (2.0 + (x * 2.0)))) / Math.sin(B))) - (x / B);
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1650000.0:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= -2.45e-79:
		tmp = ((F * math.sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0)))))) - x) / B
	elif F <= 6.5e-157:
		tmp = (x * -math.cos(B)) / math.sin(B)
	elif F <= 3.4e-6:
		tmp = (F * (math.sqrt((1.0 / (2.0 + (x * 2.0)))) / math.sin(B))) - (x / B)
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1650000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= -2.45e-79)
		tmp = Float64(Float64(Float64(F * sqrt(Float64(1.0 / Float64(2.0 + Float64(Float64(F * F) + Float64(x * 2.0)))))) - x) / B);
	elseif (F <= 6.5e-157)
		tmp = Float64(Float64(x * Float64(-cos(B))) / sin(B));
	elseif (F <= 3.4e-6)
		tmp = Float64(Float64(F * Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0)))) / sin(B))) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1650000.0)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= -2.45e-79)
		tmp = ((F * sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0)))))) - x) / B;
	elseif (F <= 6.5e-157)
		tmp = (x * -cos(B)) / sin(B);
	elseif (F <= 3.4e-6)
		tmp = (F * (sqrt((1.0 / (2.0 + (x * 2.0)))) / sin(B))) - (x / B);
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1650000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -2.45e-79], N[(N[(N[(F * N[Sqrt[N[(1.0 / N[(2.0 + N[(N[(F * F), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 6.5e-157], N[(N[(x * (-N[Cos[B], $MachinePrecision])), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.4e-6], N[(N[(F * N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -1.65e6

   1. Initial program 55.2%

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

   3. Simplified 68.1%

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

   5. Taylor expanded in F around -inf 99.7%

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

   if -1.65e6 < F < -2.45e-79

   1. Initial program 99.3%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in B around 0 99.2%

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

      1. unpow2 99.2%

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

   7. Applied egg-rr 99.2%

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

   if -2.45e-79 < F < 6.5000000000000002e-157

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in F around 0 99.6%

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

      1. *-commutative 99.6%

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

   7. Simplified 99.6%

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

      1. sqrt-div 99.5%

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

      2. metadata-eval 99.5%

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

      3. +-commutative 99.5%

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

      4. *-commutative 99.5%

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

      5. fma-define 99.5%

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

   9. Applied egg-rr 99.5%

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

   10. Taylor expanded in F around 0 84.5%

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

       1. mul-1-neg 84.5%

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

   12. Simplified 84.5%

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

   if 6.5000000000000002e-157 < F < 3.40000000000000006e-6

   1. Initial program 99.3%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in F around 0 99.4%

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

      1. *-commutative 99.4%

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

   7. Simplified 99.4%

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

   8. Taylor expanded in B around 0 83.2%

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

   if 3.40000000000000006e-6 < F

   1. Initial program 65.7%

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

   3. Simplified 81.3%

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

   5. Taylor expanded in F around inf 98.7%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1650000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -2.45 \cdot 10^{-79}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + x \cdot 2\right)}} - x}{B}\\ \mathbf{elif}\;F \leq 6.5 \cdot 10^{-157}:\\ \;\;\;\;\frac{x \cdot \left(-\cos B\right)}{\sin B}\\ \mathbf{elif}\;F \leq 3.4 \cdot 10^{-6}:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{2 + x \cdot 2}}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.47:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 2.2 \cdot 10^{-156}:\\ \;\;\;\;F \cdot \left(\frac{1}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\right) - t\_0\\ \mathbf{elif}\;F \leq 600:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -0.47)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 2.2e-156)
       (- (* F (* (/ 1.0 B) (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))))) t_0)
       (if (<= F 600.0)
         (- (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)) (/ x B))
         (- (/ 1.0 (sin B)) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -0.47) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 2.2e-156) {
		tmp = (F * ((1.0 / B) * sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_0;
	} else if (F <= 600.0) {
		tmp = ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B);
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-0.47d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 2.2d-156) then
        tmp = (f * ((1.0d0 / b) * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))))) - t_0
    else if (f <= 600.0d0) then
        tmp = ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0))) - (x / b)
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -0.47) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 2.2e-156) {
		tmp = (F * ((1.0 / B) * Math.sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_0;
	} else if (F <= 600.0) {
		tmp = ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B);
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -0.47:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 2.2e-156:
		tmp = (F * ((1.0 / B) * math.sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_0
	elif F <= 600.0:
		tmp = ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B)
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -0.47)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 2.2e-156)
		tmp = Float64(Float64(F * Float64(Float64(1.0 / B) * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0)))))) - t_0);
	elseif (F <= 600.0)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -0.47)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 2.2e-156)
		tmp = (F * ((1.0 / B) * sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_0;
	elseif (F <= 600.0)
		tmp = ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5)) - (x / B);
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.47], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 2.2e-156], N[(N[(F * N[(N[(1.0 / B), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 600.0], 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], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.46999999999999997

   1. Initial program 55.9%

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

   3. Simplified 68.6%

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

   5. Taylor expanded in F around -inf 99.5%

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

   if -0.46999999999999997 < F < 2.1999999999999999e-156

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in F around 0 98.7%

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

      1. *-commutative 98.7%

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

   7. Simplified 98.7%

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

   8. Taylor expanded in B around 0 88.9%

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

   if 2.1999999999999999e-156 < F < 600

   1. Initial program 99.3%

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

   3. Taylor expanded in B around 0 83.9%

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

      1. associate-*r/ 83.9%

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

      2. neg-mul-1 83.9%

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

   5. Simplified 83.9%

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

      1. metadata-eval 99.3%

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

      2. metadata-eval 99.3%

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

   7. Applied egg-rr 83.9%

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

   if 600 < F

   1. Initial program 64.6%

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

   3. Simplified 80.7%

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

   5. Taylor expanded in F around inf 99.8%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.47:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.2 \cdot 10^{-156}:\\ \;\;\;\;F \cdot \left(\frac{1}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\right) - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 600:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 84.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1650000:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq -5.1 \cdot 10^{-79}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + x \cdot 2\right)}} - x}{B}\\ \mathbf{elif}\;F \leq 2.2 \cdot 10^{-156}:\\ \;\;\;\;\frac{x \cdot \left(-\cos B\right)}{\sin B}\\ \mathbf{elif}\;F \leq 1.35 \cdot 10^{-5}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1650000.0)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F -5.1e-79)
       (/ (- (* F (sqrt (/ 1.0 (+ 2.0 (+ (* F F) (* x 2.0)))))) x) B)
       (if (<= F 2.2e-156)
         (/ (* x (- (cos B))) (sin B))
         (if (<= F 1.35e-5)
           (* (/ F (sin B)) (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))))
           (- (/ 1.0 (sin B)) t_0)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1650000.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= -5.1e-79) {
		tmp = ((F * sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0)))))) - x) / B;
	} else if (F <= 2.2e-156) {
		tmp = (x * -cos(B)) / sin(B);
	} else if (F <= 1.35e-5) {
		tmp = (F / sin(B)) * sqrt((1.0 / (2.0 + (x * 2.0))));
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1650000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= (-5.1d-79)) then
        tmp = ((f * sqrt((1.0d0 / (2.0d0 + ((f * f) + (x * 2.0d0)))))) - x) / b
    else if (f <= 2.2d-156) then
        tmp = (x * -cos(b)) / sin(b)
    else if (f <= 1.35d-5) then
        tmp = (f / sin(b)) * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1650000.0) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= -5.1e-79) {
		tmp = ((F * Math.sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0)))))) - x) / B;
	} else if (F <= 2.2e-156) {
		tmp = (x * -Math.cos(B)) / Math.sin(B);
	} else if (F <= 1.35e-5) {
		tmp = (F / Math.sin(B)) * Math.sqrt((1.0 / (2.0 + (x * 2.0))));
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1650000.0:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= -5.1e-79:
		tmp = ((F * math.sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0)))))) - x) / B
	elif F <= 2.2e-156:
		tmp = (x * -math.cos(B)) / math.sin(B)
	elif F <= 1.35e-5:
		tmp = (F / math.sin(B)) * math.sqrt((1.0 / (2.0 + (x * 2.0))))
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1650000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= -5.1e-79)
		tmp = Float64(Float64(Float64(F * sqrt(Float64(1.0 / Float64(2.0 + Float64(Float64(F * F) + Float64(x * 2.0)))))) - x) / B);
	elseif (F <= 2.2e-156)
		tmp = Float64(Float64(x * Float64(-cos(B))) / sin(B));
	elseif (F <= 1.35e-5)
		tmp = Float64(Float64(F / sin(B)) * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0)))));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1650000.0)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= -5.1e-79)
		tmp = ((F * sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0)))))) - x) / B;
	elseif (F <= 2.2e-156)
		tmp = (x * -cos(B)) / sin(B);
	elseif (F <= 1.35e-5)
		tmp = (F / sin(B)) * sqrt((1.0 / (2.0 + (x * 2.0))));
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1650000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -5.1e-79], N[(N[(N[(F * N[Sqrt[N[(1.0 / N[(2.0 + N[(N[(F * F), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 2.2e-156], N[(N[(x * (-N[Cos[B], $MachinePrecision])), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.35e-5], N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -1.65e6

   1. Initial program 55.2%

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

   3. Simplified 68.1%

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

   5. Taylor expanded in F around -inf 99.7%

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

   if -1.65e6 < F < -5.0999999999999999e-79

   1. Initial program 99.3%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in B around 0 99.2%

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

      1. unpow2 99.2%

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

   7. Applied egg-rr 99.2%

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

   if -5.0999999999999999e-79 < F < 2.1999999999999999e-156

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in F around 0 99.6%

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

      1. *-commutative 99.6%

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

   7. Simplified 99.6%

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

      1. sqrt-div 99.5%

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

      2. metadata-eval 99.5%

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

      3. +-commutative 99.5%

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

      4. *-commutative 99.5%

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

      5. fma-define 99.5%

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

   9. Applied egg-rr 99.5%

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

   10. Taylor expanded in F around 0 84.5%

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

       1. mul-1-neg 84.5%

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

   12. Simplified 84.5%

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

   if 2.1999999999999999e-156 < F < 1.3499999999999999e-5

   1. Initial program 99.3%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in F around 0 99.4%

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

      1. *-commutative 99.4%

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

   7. Simplified 99.4%

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

      1. sqrt-div 99.1%

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

      2. metadata-eval 99.1%

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

      3. +-commutative 99.1%

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

      4. *-commutative 99.1%

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

      5. fma-define 99.1%

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

   9. Applied egg-rr 99.1%

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

   10. Taylor expanded in F around inf 62.4%

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

   if 1.3499999999999999e-5 < F

   1. Initial program 65.7%

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

   3. Simplified 81.3%

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

   5. Taylor expanded in F around inf 98.7%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1650000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -5.1 \cdot 10^{-79}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + x \cdot 2\right)}} - x}{B}\\ \mathbf{elif}\;F \leq 2.2 \cdot 10^{-156}:\\ \;\;\;\;\frac{x \cdot \left(-\cos B\right)}{\sin B}\\ \mathbf{elif}\;F \leq 1.35 \cdot 10^{-5}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 91.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{2 + x \cdot 2}}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.43:\\ \;\;\;\;\frac{-1}{\sin B} - t\_1\\ \mathbf{elif}\;F \leq 4.8 \cdot 10^{-156}:\\ \;\;\;\;F \cdot \left(\frac{1}{B} \cdot t\_0\right) - t\_1\\ \mathbf{elif}\;F \leq 0.105:\\ \;\;\;\;F \cdot \frac{t\_0}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))) (t_1 (/ x (tan B))))
   (if (<= F -0.43)
     (- (/ -1.0 (sin B)) t_1)
     (if (<= F 4.8e-156)
       (- (* F (* (/ 1.0 B) t_0)) t_1)
       (if (<= F 0.105)
         (- (* F (/ t_0 (sin B))) (/ x B))
         (- (/ 1.0 (sin B)) t_1))))))

C

double code(double F, double B, double x) {
	double t_0 = sqrt((1.0 / (2.0 + (x * 2.0))));
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -0.43) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= 4.8e-156) {
		tmp = (F * ((1.0 / B) * t_0)) - t_1;
	} else if (F <= 0.105) {
		tmp = (F * (t_0 / sin(B))) - (x / B);
	} else {
		tmp = (1.0 / sin(B)) - t_1;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))
    t_1 = x / tan(b)
    if (f <= (-0.43d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_1
    else if (f <= 4.8d-156) then
        tmp = (f * ((1.0d0 / b) * t_0)) - t_1
    else if (f <= 0.105d0) then
        tmp = (f * (t_0 / sin(b))) - (x / b)
    else
        tmp = (1.0d0 / sin(b)) - t_1
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = Math.sqrt((1.0 / (2.0 + (x * 2.0))));
	double t_1 = x / Math.tan(B);
	double tmp;
	if (F <= -0.43) {
		tmp = (-1.0 / Math.sin(B)) - t_1;
	} else if (F <= 4.8e-156) {
		tmp = (F * ((1.0 / B) * t_0)) - t_1;
	} else if (F <= 0.105) {
		tmp = (F * (t_0 / Math.sin(B))) - (x / B);
	} else {
		tmp = (1.0 / Math.sin(B)) - t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = math.sqrt((1.0 / (2.0 + (x * 2.0))))
	t_1 = x / math.tan(B)
	tmp = 0
	if F <= -0.43:
		tmp = (-1.0 / math.sin(B)) - t_1
	elif F <= 4.8e-156:
		tmp = (F * ((1.0 / B) * t_0)) - t_1
	elif F <= 0.105:
		tmp = (F * (t_0 / math.sin(B))) - (x / B)
	else:
		tmp = (1.0 / math.sin(B)) - t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -0.43)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= 4.8e-156)
		tmp = Float64(Float64(F * Float64(Float64(1.0 / B) * t_0)) - t_1);
	elseif (F <= 0.105)
		tmp = Float64(Float64(F * Float64(t_0 / sin(B))) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = sqrt((1.0 / (2.0 + (x * 2.0))));
	t_1 = x / tan(B);
	tmp = 0.0;
	if (F <= -0.43)
		tmp = (-1.0 / sin(B)) - t_1;
	elseif (F <= 4.8e-156)
		tmp = (F * ((1.0 / B) * t_0)) - t_1;
	elseif (F <= 0.105)
		tmp = (F * (t_0 / sin(B))) - (x / B);
	else
		tmp = (1.0 / sin(B)) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.43], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, 4.8e-156], N[(N[(F * N[(N[(1.0 / B), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, 0.105], N[(N[(F * N[(t$95$0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.429999999999999993

   1. Initial program 55.9%

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

   3. Simplified 68.6%

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

   5. Taylor expanded in F around -inf 99.5%

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

   if -0.429999999999999993 < F < 4.8e-156

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in F around 0 98.7%

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

      1. *-commutative 98.7%

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

   7. Simplified 98.7%

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

   8. Taylor expanded in B around 0 88.9%

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

   if 4.8e-156 < F < 0.104999999999999996

   1. Initial program 99.3%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in F around 0 99.4%

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

      1. *-commutative 99.4%

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

   7. Simplified 99.4%

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

   8. Taylor expanded in B around 0 83.2%

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

   if 0.104999999999999996 < F

   1. Initial program 65.7%

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

   3. Simplified 81.3%

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

   5. Taylor expanded in F around inf 98.7%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.43:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 4.8 \cdot 10^{-156}:\\ \;\;\;\;F \cdot \left(\frac{1}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\right) - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.105:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{2 + x \cdot 2}}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 84.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1650000:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq -1.12 \cdot 10^{-78}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + x \cdot 2\right)}} - x}{B}\\ \mathbf{elif}\;F \leq 4.6 \cdot 10^{-156}:\\ \;\;\;\;\frac{x \cdot \left(-\cos B\right)}{\sin B}\\ \mathbf{elif}\;F \leq 0.0015:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1650000.0)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F -1.12e-78)
       (/ (- (* F (sqrt (/ 1.0 (+ 2.0 (+ (* F F) (* x 2.0)))))) x) B)
       (if (<= F 4.6e-156)
         (/ (* x (- (cos B))) (sin B))
         (if (<= F 0.0015)
           (/ (* F (sqrt 0.5)) (sin B))
           (- (/ 1.0 (sin B)) t_0)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1650000.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= -1.12e-78) {
		tmp = ((F * sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0)))))) - x) / B;
	} else if (F <= 4.6e-156) {
		tmp = (x * -cos(B)) / sin(B);
	} else if (F <= 0.0015) {
		tmp = (F * sqrt(0.5)) / sin(B);
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1650000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= (-1.12d-78)) then
        tmp = ((f * sqrt((1.0d0 / (2.0d0 + ((f * f) + (x * 2.0d0)))))) - x) / b
    else if (f <= 4.6d-156) then
        tmp = (x * -cos(b)) / sin(b)
    else if (f <= 0.0015d0) then
        tmp = (f * sqrt(0.5d0)) / sin(b)
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1650000.0) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= -1.12e-78) {
		tmp = ((F * Math.sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0)))))) - x) / B;
	} else if (F <= 4.6e-156) {
		tmp = (x * -Math.cos(B)) / Math.sin(B);
	} else if (F <= 0.0015) {
		tmp = (F * Math.sqrt(0.5)) / Math.sin(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1650000.0:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= -1.12e-78:
		tmp = ((F * math.sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0)))))) - x) / B
	elif F <= 4.6e-156:
		tmp = (x * -math.cos(B)) / math.sin(B)
	elif F <= 0.0015:
		tmp = (F * math.sqrt(0.5)) / math.sin(B)
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1650000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= -1.12e-78)
		tmp = Float64(Float64(Float64(F * sqrt(Float64(1.0 / Float64(2.0 + Float64(Float64(F * F) + Float64(x * 2.0)))))) - x) / B);
	elseif (F <= 4.6e-156)
		tmp = Float64(Float64(x * Float64(-cos(B))) / sin(B));
	elseif (F <= 0.0015)
		tmp = Float64(Float64(F * sqrt(0.5)) / sin(B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1650000.0)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= -1.12e-78)
		tmp = ((F * sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0)))))) - x) / B;
	elseif (F <= 4.6e-156)
		tmp = (x * -cos(B)) / sin(B);
	elseif (F <= 0.0015)
		tmp = (F * sqrt(0.5)) / sin(B);
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1650000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -1.12e-78], N[(N[(N[(F * N[Sqrt[N[(1.0 / N[(2.0 + N[(N[(F * F), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 4.6e-156], N[(N[(x * (-N[Cos[B], $MachinePrecision])), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.0015], N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -1.65e6

   1. Initial program 55.2%

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

   3. Simplified 68.1%

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

   5. Taylor expanded in F around -inf 99.7%

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

   if -1.65e6 < F < -1.12000000000000005e-78

   1. Initial program 99.3%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in B around 0 99.2%

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

      1. unpow2 99.2%

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

   7. Applied egg-rr 99.2%

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

   if -1.12000000000000005e-78 < F < 4.5999999999999999e-156

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in F around 0 99.6%

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

      1. *-commutative 99.6%

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

   7. Simplified 99.6%

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

      1. sqrt-div 99.5%

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

      2. metadata-eval 99.5%

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

      3. +-commutative 99.5%

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

      4. *-commutative 99.5%

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

      5. fma-define 99.5%

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

   9. Applied egg-rr 99.5%

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

   10. Taylor expanded in F around 0 84.5%

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

       1. mul-1-neg 84.5%

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

   12. Simplified 84.5%

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

   if 4.5999999999999999e-156 < F < 0.0015

   1. Initial program 99.3%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in F around 0 99.4%

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

      1. *-commutative 99.4%

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

   7. Simplified 99.4%

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

      1. sqrt-div 99.1%

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

      2. metadata-eval 99.1%

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

      3. +-commutative 99.1%

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

      4. *-commutative 99.1%

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

      5. fma-define 99.1%

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

   9. Applied egg-rr 99.1%

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

   10. Taylor expanded in x around 0 62.3%

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

   if 0.0015 < F

   1. Initial program 65.7%

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

   3. Simplified 81.3%

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

   5. Taylor expanded in F around inf 98.7%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1650000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.12 \cdot 10^{-78}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + x \cdot 2\right)}} - x}{B}\\ \mathbf{elif}\;F \leq 4.6 \cdot 10^{-156}:\\ \;\;\;\;\frac{x \cdot \left(-\cos B\right)}{\sin B}\\ \mathbf{elif}\;F \leq 0.0015:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 78.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1650000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -9.8 \cdot 10^{-79}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + x \cdot 2\right)}} - x}{B}\\ \mathbf{elif}\;F \leq 4.8 \cdot 10^{-156}:\\ \;\;\;\;\frac{x \cdot \left(-\cos B\right)}{\sin B}\\ \mathbf{elif}\;F \leq 0.007:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1650000.0)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F -9.8e-79)
     (/ (- (* F (sqrt (/ 1.0 (+ 2.0 (+ (* F F) (* x 2.0)))))) x) B)
     (if (<= F 4.8e-156)
       (/ (* x (- (cos B))) (sin B))
       (if (<= F 0.007)
         (/ (* F (sqrt 0.5)) (sin B))
         (- (/ 1.0 (sin B)) (/ x B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1650000.0) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= -9.8e-79) {
		tmp = ((F * sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0)))))) - x) / B;
	} else if (F <= 4.8e-156) {
		tmp = (x * -cos(B)) / sin(B);
	} else if (F <= 0.007) {
		tmp = (F * sqrt(0.5)) / sin(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 <= (-1650000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= (-9.8d-79)) then
        tmp = ((f * sqrt((1.0d0 / (2.0d0 + ((f * f) + (x * 2.0d0)))))) - x) / b
    else if (f <= 4.8d-156) then
        tmp = (x * -cos(b)) / sin(b)
    else if (f <= 0.007d0) then
        tmp = (f * sqrt(0.5d0)) / sin(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 <= -1650000.0) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= -9.8e-79) {
		tmp = ((F * Math.sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0)))))) - x) / B;
	} else if (F <= 4.8e-156) {
		tmp = (x * -Math.cos(B)) / Math.sin(B);
	} else if (F <= 0.007) {
		tmp = (F * Math.sqrt(0.5)) / Math.sin(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1650000.0:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= -9.8e-79:
		tmp = ((F * math.sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0)))))) - x) / B
	elif F <= 4.8e-156:
		tmp = (x * -math.cos(B)) / math.sin(B)
	elif F <= 0.007:
		tmp = (F * math.sqrt(0.5)) / math.sin(B)
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1650000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= -9.8e-79)
		tmp = Float64(Float64(Float64(F * sqrt(Float64(1.0 / Float64(2.0 + Float64(Float64(F * F) + Float64(x * 2.0)))))) - x) / B);
	elseif (F <= 4.8e-156)
		tmp = Float64(Float64(x * Float64(-cos(B))) / sin(B));
	elseif (F <= 0.007)
		tmp = Float64(Float64(F * sqrt(0.5)) / sin(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 <= -1650000.0)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= -9.8e-79)
		tmp = ((F * sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0)))))) - x) / B;
	elseif (F <= 4.8e-156)
		tmp = (x * -cos(B)) / sin(B);
	elseif (F <= 0.007)
		tmp = (F * sqrt(0.5)) / sin(B);
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1650000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -9.8e-79], N[(N[(N[(F * N[Sqrt[N[(1.0 / N[(2.0 + N[(N[(F * F), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 4.8e-156], N[(N[(x * (-N[Cos[B], $MachinePrecision])), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.007], N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[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 < -1.65e6

   1. Initial program 55.2%

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

   3. Simplified 68.1%

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

   5. Taylor expanded in F around -inf 99.7%

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

   if -1.65e6 < F < -9.8000000000000001e-79

   1. Initial program 99.3%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in B around 0 99.2%

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

      1. unpow2 99.2%

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

   7. Applied egg-rr 99.2%

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

   if -9.8000000000000001e-79 < F < 4.8e-156

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in F around 0 99.6%

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

      1. *-commutative 99.6%

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

   7. Simplified 99.6%

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

      1. sqrt-div 99.5%

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

      2. metadata-eval 99.5%

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

      3. +-commutative 99.5%

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

      4. *-commutative 99.5%

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

      5. fma-define 99.5%

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

   9. Applied egg-rr 99.5%

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

   10. Taylor expanded in F around 0 84.5%

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

       1. mul-1-neg 84.5%

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

   12. Simplified 84.5%

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

   if 4.8e-156 < F < 0.00700000000000000015

   1. Initial program 99.3%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in F around 0 99.4%

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

      1. *-commutative 99.4%

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

   7. Simplified 99.4%

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

      1. sqrt-div 99.1%

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

      2. metadata-eval 99.1%

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

      3. +-commutative 99.1%

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

      4. *-commutative 99.1%

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

      5. fma-define 99.1%

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

   9. Applied egg-rr 99.1%

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

   10. Taylor expanded in x around 0 62.3%

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

   if 0.00700000000000000015 < F

   1. Initial program 65.7%

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

   3. Taylor expanded in B around 0 35.6%

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

      1. associate-*r/ 35.6%

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

      2. neg-mul-1 35.6%

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

   5. Simplified 35.6%

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

   6. Taylor expanded in F around inf 68.7%

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

      1. mul-1-neg 68.7%

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

      2. distribute-frac-neg 68.7%

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

      3. +-commutative 68.7%

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

      4. distribute-frac-neg 68.7%

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

      5. unsub-neg 68.7%

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

   8. Simplified 68.7%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1650000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -9.8 \cdot 10^{-79}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + x \cdot 2\right)}} - x}{B}\\ \mathbf{elif}\;F \leq 4.8 \cdot 10^{-156}:\\ \;\;\;\;\frac{x \cdot \left(-\cos B\right)}{\sin B}\\ \mathbf{elif}\;F \leq 0.007:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 71.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.05 \cdot 10^{+43}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -8.5 \cdot 10^{-79}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + x \cdot 2\right)}} - x}{B}\\ \mathbf{elif}\;F \leq 4.8 \cdot 10^{-156}:\\ \;\;\;\;\frac{x \cdot \left(-\cos B\right)}{\sin B}\\ \mathbf{elif}\;F \leq 0.000195:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.05e+43)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F -8.5e-79)
     (/ (- (* F (sqrt (/ 1.0 (+ 2.0 (+ (* F F) (* x 2.0)))))) x) B)
     (if (<= F 4.8e-156)
       (/ (* x (- (cos B))) (sin B))
       (if (<= F 0.000195)
         (/ (* F (sqrt 0.5)) (sin B))
         (- (/ 1.0 (sin B)) (/ x B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.05e+43) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -8.5e-79) {
		tmp = ((F * sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0)))))) - x) / B;
	} else if (F <= 4.8e-156) {
		tmp = (x * -cos(B)) / sin(B);
	} else if (F <= 0.000195) {
		tmp = (F * sqrt(0.5)) / sin(B);
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.05d+43)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-8.5d-79)) then
        tmp = ((f * sqrt((1.0d0 / (2.0d0 + ((f * f) + (x * 2.0d0)))))) - x) / b
    else if (f <= 4.8d-156) then
        tmp = (x * -cos(b)) / sin(b)
    else if (f <= 0.000195d0) then
        tmp = (f * sqrt(0.5d0)) / sin(b)
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.05e+43) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -8.5e-79) {
		tmp = ((F * Math.sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0)))))) - x) / B;
	} else if (F <= 4.8e-156) {
		tmp = (x * -Math.cos(B)) / Math.sin(B);
	} else if (F <= 0.000195) {
		tmp = (F * Math.sqrt(0.5)) / Math.sin(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.05e+43:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -8.5e-79:
		tmp = ((F * math.sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0)))))) - x) / B
	elif F <= 4.8e-156:
		tmp = (x * -math.cos(B)) / math.sin(B)
	elif F <= 0.000195:
		tmp = (F * math.sqrt(0.5)) / math.sin(B)
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.05e+43)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -8.5e-79)
		tmp = Float64(Float64(Float64(F * sqrt(Float64(1.0 / Float64(2.0 + Float64(Float64(F * F) + Float64(x * 2.0)))))) - x) / B);
	elseif (F <= 4.8e-156)
		tmp = Float64(Float64(x * Float64(-cos(B))) / sin(B));
	elseif (F <= 0.000195)
		tmp = Float64(Float64(F * sqrt(0.5)) / sin(B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.05e+43)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -8.5e-79)
		tmp = ((F * sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0)))))) - x) / B;
	elseif (F <= 4.8e-156)
		tmp = (x * -cos(B)) / sin(B);
	elseif (F <= 0.000195)
		tmp = (F * sqrt(0.5)) / sin(B);
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.05e+43], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -8.5e-79], N[(N[(N[(F * N[Sqrt[N[(1.0 / N[(2.0 + N[(N[(F * F), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 4.8e-156], N[(N[(x * (-N[Cos[B], $MachinePrecision])), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.000195], N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[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 < -1.05000000000000001e43

   1. Initial program 54.5%

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

   3. Simplified 66.7%

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

   5. Taylor expanded in F around -inf 99.7%

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

   6. Taylor expanded in B around 0 75.8%

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

   if -1.05000000000000001e43 < F < -8.50000000000000029e-79

   1. Initial program 93.0%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in B around 0 92.3%

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

      1. unpow2 92.3%

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

   7. Applied egg-rr 92.3%

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

   if -8.50000000000000029e-79 < F < 4.8e-156

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in F around 0 99.6%

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

      1. *-commutative 99.6%

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

   7. Simplified 99.6%

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

      1. sqrt-div 99.5%

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

      2. metadata-eval 99.5%

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

      3. +-commutative 99.5%

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

      4. *-commutative 99.5%

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

      5. fma-define 99.5%

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

   9. Applied egg-rr 99.5%

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

   10. Taylor expanded in F around 0 84.5%

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

       1. mul-1-neg 84.5%

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

   12. Simplified 84.5%

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

   if 4.8e-156 < F < 1.94999999999999996e-4

   1. Initial program 99.3%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in F around 0 99.4%

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

      1. *-commutative 99.4%

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

   7. Simplified 99.4%

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

      1. sqrt-div 99.1%

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

      2. metadata-eval 99.1%

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

      3. +-commutative 99.1%

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

      4. *-commutative 99.1%

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

      5. fma-define 99.1%

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

   9. Applied egg-rr 99.1%

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

   10. Taylor expanded in x around 0 62.3%

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

   if 1.94999999999999996e-4 < F

   1. Initial program 65.7%

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

   3. Taylor expanded in B around 0 35.6%

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

      1. associate-*r/ 35.6%

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

      2. neg-mul-1 35.6%

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

   5. Simplified 35.6%

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

   6. Taylor expanded in F around inf 68.7%

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

      1. mul-1-neg 68.7%

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

      2. distribute-frac-neg 68.7%

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

      3. +-commutative 68.7%

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

      4. distribute-frac-neg 68.7%

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

      5. unsub-neg 68.7%

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

   8. Simplified 68.7%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.05 \cdot 10^{+43}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -8.5 \cdot 10^{-79}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + x \cdot 2\right)}} - x}{B}\\ \mathbf{elif}\;F \leq 4.8 \cdot 10^{-156}:\\ \;\;\;\;\frac{x \cdot \left(-\cos B\right)}{\sin B}\\ \mathbf{elif}\;F \leq 0.000195:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 64.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{2 + x \cdot 2}}\\ \mathbf{if}\;F \leq -0.39:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -2.8 \cdot 10^{-79}:\\ \;\;\;\;t\_0 \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 5.7 \cdot 10^{-140}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 4.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{F \cdot t\_0 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))))
   (if (<= F -0.39)
     (- (/ -1.0 (sin B)) (/ x B))
     (if (<= F -2.8e-79)
       (- (* t_0 (/ F B)) (/ x B))
       (if (<= F 5.7e-140)
         (- (/ -1.0 B) (/ x (tan B)))
         (if (<= F 4.2e-9)
           (/ (- (* F t_0) x) B)
           (- (/ 1.0 (sin B)) (/ x B))))))))

C

double code(double F, double B, double x) {
	double t_0 = sqrt((1.0 / (2.0 + (x * 2.0))));
	double tmp;
	if (F <= -0.39) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -2.8e-79) {
		tmp = (t_0 * (F / B)) - (x / B);
	} else if (F <= 5.7e-140) {
		tmp = (-1.0 / B) - (x / tan(B));
	} else if (F <= 4.2e-9) {
		tmp = ((F * t_0) - 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) :: t_0
    real(8) :: tmp
    t_0 = sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))
    if (f <= (-0.39d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-2.8d-79)) then
        tmp = (t_0 * (f / b)) - (x / b)
    else if (f <= 5.7d-140) then
        tmp = ((-1.0d0) / b) - (x / tan(b))
    else if (f <= 4.2d-9) then
        tmp = ((f * t_0) - 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 t_0 = Math.sqrt((1.0 / (2.0 + (x * 2.0))));
	double tmp;
	if (F <= -0.39) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -2.8e-79) {
		tmp = (t_0 * (F / B)) - (x / B);
	} else if (F <= 5.7e-140) {
		tmp = (-1.0 / B) - (x / Math.tan(B));
	} else if (F <= 4.2e-9) {
		tmp = ((F * t_0) - x) / B;
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = math.sqrt((1.0 / (2.0 + (x * 2.0))))
	tmp = 0
	if F <= -0.39:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -2.8e-79:
		tmp = (t_0 * (F / B)) - (x / B)
	elif F <= 5.7e-140:
		tmp = (-1.0 / B) - (x / math.tan(B))
	elif F <= 4.2e-9:
		tmp = ((F * t_0) - x) / B
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))
	tmp = 0.0
	if (F <= -0.39)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -2.8e-79)
		tmp = Float64(Float64(t_0 * Float64(F / B)) - Float64(x / B));
	elseif (F <= 5.7e-140)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / tan(B)));
	elseif (F <= 4.2e-9)
		tmp = Float64(Float64(Float64(F * t_0) - 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)
	t_0 = sqrt((1.0 / (2.0 + (x * 2.0))));
	tmp = 0.0;
	if (F <= -0.39)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -2.8e-79)
		tmp = (t_0 * (F / B)) - (x / B);
	elseif (F <= 5.7e-140)
		tmp = (-1.0 / B) - (x / tan(B));
	elseif (F <= 4.2e-9)
		tmp = ((F * t_0) - x) / B;
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[F, -0.39], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -2.8e-79], N[(N[(t$95$0 * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.7e-140], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.2e-9], N[(N[(N[(F * t$95$0), $MachinePrecision] - x), $MachinePrecision] / B), $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 < -0.39000000000000001

   1. Initial program 55.9%

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

   3. Simplified 68.6%

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

   5. Taylor expanded in F around -inf 99.5%

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

   6. Taylor expanded in B around 0 75.5%

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

   if -0.39000000000000001 < F < -2.80000000000000012e-79

   1. Initial program 99.4%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in B around 0 99.3%

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

   6. Taylor expanded in F around 0 91.9%

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

   if -2.80000000000000012e-79 < F < 5.7e-140

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in F around -inf 45.5%

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

   6. Taylor expanded in B around 0 60.3%

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

   if 5.7e-140 < F < 4.20000000000000039e-9

   1. Initial program 99.4%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in B around 0 57.5%

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

   6. Taylor expanded in F around 0 57.5%

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

   if 4.20000000000000039e-9 < F

   1. Initial program 66.7%

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

   3. Taylor expanded in B around 0 37.6%

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

      1. associate-*r/ 37.6%

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

      2. neg-mul-1 37.6%

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

   5. Simplified 37.6%

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

   6. Taylor expanded in F around inf 66.9%

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

      1. mul-1-neg 66.9%

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

      2. distribute-frac-neg 66.9%

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

      3. +-commutative 66.9%

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

      4. distribute-frac-neg 66.9%

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

      5. unsub-neg 66.9%

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

   8. Simplified 66.9%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.39:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -2.8 \cdot 10^{-79}:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 5.7 \cdot 10^{-140}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 4.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 64.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{if}\;F \leq -0.39:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.7 \cdot 10^{-79}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 6 \cdot 10^{-141}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 4.2 \cdot 10^{-9}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- (* F (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))) x) B)))
   (if (<= F -0.39)
     (- (/ -1.0 (sin B)) (/ x B))
     (if (<= F -1.7e-79)
       t_0
       (if (<= F 6e-141)
         (- (/ -1.0 B) (/ x (tan B)))
         (if (<= F 4.2e-9) t_0 (- (/ 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 tmp;
	if (F <= -0.39) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -1.7e-79) {
		tmp = t_0;
	} else if (F <= 6e-141) {
		tmp = (-1.0 / B) - (x / tan(B));
	} else if (F <= 4.2e-9) {
		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 = ((f * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))) - x) / b
    if (f <= (-0.39d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-1.7d-79)) then
        tmp = t_0
    else if (f <= 6d-141) then
        tmp = ((-1.0d0) / b) - (x / tan(b))
    else if (f <= 4.2d-9) 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 = ((F * Math.sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B;
	double tmp;
	if (F <= -0.39) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -1.7e-79) {
		tmp = t_0;
	} else if (F <= 6e-141) {
		tmp = (-1.0 / B) - (x / Math.tan(B));
	} else if (F <= 4.2e-9) {
		tmp = t_0;
	} 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
	tmp = 0
	if F <= -0.39:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -1.7e-79:
		tmp = t_0
	elif F <= 6e-141:
		tmp = (-1.0 / B) - (x / math.tan(B))
	elif F <= 4.2e-9:
		tmp = t_0
	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)
	tmp = 0.0
	if (F <= -0.39)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -1.7e-79)
		tmp = t_0;
	elseif (F <= 6e-141)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / tan(B)));
	elseif (F <= 4.2e-9)
		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 = ((F * sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B;
	tmp = 0.0;
	if (F <= -0.39)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -1.7e-79)
		tmp = t_0;
	elseif (F <= 6e-141)
		tmp = (-1.0 / B) - (x / tan(B));
	elseif (F <= 4.2e-9)
		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[(N[(F * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]}, If[LessEqual[F, -0.39], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.7e-79], t$95$0, If[LessEqual[F, 6e-141], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.2e-9], t$95$0, N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.39000000000000001

   1. Initial program 55.9%

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

   3. Simplified 68.6%

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

   5. Taylor expanded in F around -inf 99.5%

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

   6. Taylor expanded in B around 0 75.5%

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

   if -0.39000000000000001 < F < -1.69999999999999988e-79 or 5.99999999999999967e-141 < F < 4.20000000000000039e-9

   1. Initial program 99.4%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in B around 0 67.0%

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

   6. Taylor expanded in F around 0 65.3%

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

   if -1.69999999999999988e-79 < F < 5.99999999999999967e-141

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in F around -inf 45.5%

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

   6. Taylor expanded in B around 0 60.3%

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

   if 4.20000000000000039e-9 < F

   1. Initial program 66.7%

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

   3. Taylor expanded in B around 0 37.6%

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

      1. associate-*r/ 37.6%

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

      2. neg-mul-1 37.6%

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

   5. Simplified 37.6%

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

   6. Taylor expanded in F around inf 66.9%

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

      1. mul-1-neg 66.9%

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

      2. distribute-frac-neg 66.9%

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

      3. +-commutative 66.9%

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

      4. distribute-frac-neg 66.9%

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

      5. unsub-neg 66.9%

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

   8. Simplified 66.9%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.39:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.7 \cdot 10^{-79}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{elif}\;F \leq 6 \cdot 10^{-141}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 4.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 56.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;x \leq -6.6 \cdot 10^{-161}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-266}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-110}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ -1.0 B) (/ x (tan B)))))
   (if (<= x -6.6e-161)
     t_0
     (if (<= x 1.6e-266)
       (/ -1.0 (sin B))
       (if (<= x 1.8e-110) (- (/ 1.0 (sin B)) (/ x B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 / B) - (x / tan(B));
	double tmp;
	if (x <= -6.6e-161) {
		tmp = t_0;
	} else if (x <= 1.6e-266) {
		tmp = -1.0 / sin(B);
	} else if (x <= 1.8e-110) {
		tmp = (1.0 / sin(B)) - (x / B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-1.0d0) / b) - (x / tan(b))
    if (x <= (-6.6d-161)) then
        tmp = t_0
    else if (x <= 1.6d-266) then
        tmp = (-1.0d0) / sin(b)
    else if (x <= 1.8d-110) then
        tmp = (1.0d0 / sin(b)) - (x / b)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (-1.0 / B) - (x / Math.tan(B));
	double tmp;
	if (x <= -6.6e-161) {
		tmp = t_0;
	} else if (x <= 1.6e-266) {
		tmp = -1.0 / Math.sin(B);
	} else if (x <= 1.8e-110) {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (-1.0 / B) - (x / math.tan(B))
	tmp = 0
	if x <= -6.6e-161:
		tmp = t_0
	elif x <= 1.6e-266:
		tmp = -1.0 / math.sin(B)
	elif x <= 1.8e-110:
		tmp = (1.0 / math.sin(B)) - (x / B)
	else:
		tmp = t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 / B) - Float64(x / tan(B)))
	tmp = 0.0
	if (x <= -6.6e-161)
		tmp = t_0;
	elseif (x <= 1.6e-266)
		tmp = Float64(-1.0 / sin(B));
	elseif (x <= 1.8e-110)
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (-1.0 / B) - (x / tan(B));
	tmp = 0.0;
	if (x <= -6.6e-161)
		tmp = t_0;
	elseif (x <= 1.6e-266)
		tmp = -1.0 / sin(B);
	elseif (x <= 1.8e-110)
		tmp = (1.0 / sin(B)) - (x / B);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.6e-161], t$95$0, If[LessEqual[x, 1.6e-266], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.8e-110], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.5999999999999997e-161 or 1.79999999999999997e-110 < x

   1. Initial program 85.1%

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

   3. Simplified 93.7%

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

   5. Taylor expanded in F around -inf 76.2%

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

   6. Taylor expanded in B around 0 78.1%

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

   if -6.5999999999999997e-161 < x < 1.6e-266

   1. Initial program 74.0%

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

   3. Simplified 77.3%

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

   5. Taylor expanded in F around -inf 32.6%

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

      1. *-commutative 32.6%

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

   7. Simplified 32.6%

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

   8. Taylor expanded in x around 0 32.8%

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

   if 1.6e-266 < x < 1.79999999999999997e-110

   1. Initial program 62.2%

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

   3. Taylor expanded in B around 0 56.9%

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

      1. associate-*r/ 56.9%

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

      2. neg-mul-1 56.9%

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

   5. Simplified 56.9%

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

   6. Taylor expanded in F around inf 41.7%

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

      1. mul-1-neg 41.7%

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

      2. distribute-frac-neg 41.7%

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

      3. +-commutative 41.7%

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

      4. distribute-frac-neg 41.7%

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

      5. unsub-neg 41.7%

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

   8. Simplified 41.7%

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

Alternative 18: 56.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 1.5 \cdot 10^{-22}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + x \cdot 2\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 1.5e-22)
   (/ (- (* F (sqrt (/ 1.0 (+ 2.0 (+ (* F F) (* x 2.0)))))) x) B)
   (- (/ -1.0 B) (/ x (tan B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 1.5e-22) {
		tmp = ((F * sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0)))))) - x) / B;
	} else {
		tmp = (-1.0 / B) - (x / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (b <= 1.5d-22) then
        tmp = ((f * sqrt((1.0d0 / (2.0d0 + ((f * f) + (x * 2.0d0)))))) - x) / b
    else
        tmp = ((-1.0d0) / b) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (B <= 1.5e-22) {
		tmp = ((F * Math.sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0)))))) - x) / B;
	} else {
		tmp = (-1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if B <= 1.5e-22:
		tmp = ((F * math.sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0)))))) - x) / B
	else:
		tmp = (-1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 1.5e-22)
		tmp = Float64(Float64(Float64(F * sqrt(Float64(1.0 / Float64(2.0 + Float64(Float64(F * F) + Float64(x * 2.0)))))) - x) / B);
	else
		tmp = Float64(Float64(-1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (B <= 1.5e-22)
		tmp = ((F * sqrt((1.0 / (2.0 + ((F * F) + (x * 2.0)))))) - x) / B;
	else
		tmp = (-1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 1.5e-22], N[(N[(N[(F * N[Sqrt[N[(1.0 / N[(2.0 + N[(N[(F * F), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.5e-22

   1. Initial program 76.6%

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

   3. Simplified 86.9%

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

   5. Taylor expanded in B around 0 53.5%

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

      1. unpow2 53.5%

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

   7. Applied egg-rr 53.5%

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

   if 1.5e-22 < B

   1. Initial program 86.3%

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

   3. Simplified 86.2%

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

   5. Taylor expanded in F around -inf 56.0%

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

   6. Taylor expanded in B around 0 53.9%

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

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.5 \cdot 10^{-22}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + \left(F \cdot F + x \cdot 2\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 56.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{-159} \lor \neg \left(x \leq 1.05 \cdot 10^{-150}\right):\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (or (<= x -2.05e-159) (not (<= x 1.05e-150)))
   (- (/ -1.0 B) (/ x (tan B)))
   (/ -1.0 (sin B))))

C

double code(double F, double B, double x) {
	double tmp;
	if ((x <= -2.05e-159) || !(x <= 1.05e-150)) {
		tmp = (-1.0 / B) - (x / tan(B));
	} else {
		tmp = -1.0 / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-2.05d-159)) .or. (.not. (x <= 1.05d-150))) then
        tmp = ((-1.0d0) / b) - (x / tan(b))
    else
        tmp = (-1.0d0) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if ((x <= -2.05e-159) || !(x <= 1.05e-150)) {
		tmp = (-1.0 / B) - (x / Math.tan(B));
	} else {
		tmp = -1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if (x <= -2.05e-159) or not (x <= 1.05e-150):
		tmp = (-1.0 / B) - (x / math.tan(B))
	else:
		tmp = -1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if ((x <= -2.05e-159) || !(x <= 1.05e-150))
		tmp = Float64(Float64(-1.0 / B) - Float64(x / tan(B)));
	else
		tmp = Float64(-1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if ((x <= -2.05e-159) || ~((x <= 1.05e-150)))
		tmp = (-1.0 / B) - (x / tan(B));
	else
		tmp = -1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[Or[LessEqual[x, -2.05e-159], N[Not[LessEqual[x, 1.05e-150]], $MachinePrecision]], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.05000000000000007e-159 or 1.0500000000000001e-150 < x

   1. Initial program 83.6%

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

   3. Simplified 91.8%

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

   5. Taylor expanded in F around -inf 73.1%

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

   6. Taylor expanded in B around 0 75.4%

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

   if -2.05000000000000007e-159 < x < 1.0500000000000001e-150

   1. Initial program 70.5%

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

   3. Simplified 76.1%

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

   5. Taylor expanded in F around -inf 27.6%

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

      1. *-commutative 27.6%

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

   7. Simplified 27.6%

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

   8. Taylor expanded in x around 0 27.8%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{-159} \lor \neg \left(x \leq 1.05 \cdot 10^{-150}\right):\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 44.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.0102:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.0102)
   (/ -1.0 (sin B))
   (if (<= F 6.2e-25) (/ x (- B)) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.0102) {
		tmp = -1.0 / sin(B);
	} else if (F <= 6.2e-25) {
		tmp = x / -B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-0.0102d0)) then
        tmp = (-1.0d0) / sin(b)
    else if (f <= 6.2d-25) then
        tmp = x / -b
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.0102) {
		tmp = -1.0 / Math.sin(B);
	} else if (F <= 6.2e-25) {
		tmp = x / -B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -0.0102:
		tmp = -1.0 / math.sin(B)
	elif F <= 6.2e-25:
		tmp = x / -B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.0102)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 6.2e-25)
		tmp = Float64(x / Float64(-B));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -0.0102)
		tmp = -1.0 / sin(B);
	elseif (F <= 6.2e-25)
		tmp = x / -B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.0102], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 6.2e-25], N[(x / (-B)), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.010200000000000001

   1. Initial program 56.5%

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

   3. Simplified 69.1%

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

   5. Taylor expanded in F around -inf 98.0%

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

      1. *-commutative 98.0%

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

   7. Simplified 98.0%

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

   8. Taylor expanded in x around 0 55.9%

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

   if -0.010200000000000001 < F < 6.19999999999999989e-25

   1. Initial program 99.4%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in B around 0 48.3%

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

   6. Taylor expanded in F around 0 32.0%

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

      1. associate-*r/ 32.0%

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

      2. neg-mul-1 32.0%

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

   8. Simplified 32.0%

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

   if 6.19999999999999989e-25 < F

   1. Initial program 68.6%

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

   3. Simplified 82.9%

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

   5. Taylor expanded in B around 0 35.1%

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

   6. Taylor expanded in F around inf 36.8%

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

4. Final simplification 39.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.0102:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 44.1% accurate, 21.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -6.2 \cdot 10^{-57}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 7.8 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -6.2e-57)
   (/ (- -1.0 x) B)
   (if (<= F 7.8e-25) (/ x (- B)) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -6.2e-57) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 7.8e-25) {
		tmp = x / -B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-6.2d-57)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 7.8d-25) then
        tmp = x / -b
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -6.2e-57) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 7.8e-25) {
		tmp = x / -B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -6.2e-57:
		tmp = (-1.0 - x) / B
	elif F <= 7.8e-25:
		tmp = x / -B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -6.2e-57)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 7.8e-25)
		tmp = Float64(x / Float64(-B));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -6.2e-57)
		tmp = (-1.0 - x) / B;
	elseif (F <= 7.8e-25)
		tmp = x / -B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -6.2e-57], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 7.8e-25], N[(x / (-B)), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -6.19999999999999952e-57

   1. Initial program 59.3%

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

   3. Simplified 71.1%

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

   5. Taylor expanded in B around 0 38.0%

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

   6. Taylor expanded in F around -inf 44.7%

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

   if -6.19999999999999952e-57 < F < 7.8e-25

   1. Initial program 99.4%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in B around 0 46.0%

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

   6. Taylor expanded in F around 0 32.5%

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

      1. associate-*r/ 32.5%

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

      2. neg-mul-1 32.5%

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

   8. Simplified 32.5%

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

   if 7.8e-25 < F

   1. Initial program 68.6%

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

   3. Simplified 82.9%

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

   5. Taylor expanded in B around 0 35.1%

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

   6. Taylor expanded in F around inf 36.8%

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

4. Final simplification 37.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6.2 \cdot 10^{-57}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 7.8 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 37.0% accurate, 32.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.55 \cdot 10^{-60}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.55e-60) (/ (- -1.0 x) B) (/ x (- B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.55e-60) {
		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 <= (-1.55d-60)) 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 <= -1.55e-60) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = x / -B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.55e-60:
		tmp = (-1.0 - x) / B
	else:
		tmp = x / -B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.55e-60)
		tmp = Float64(Float64(-1.0 - x) / B);
	else
		tmp = Float64(x / Float64(-B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.55e-60)
		tmp = (-1.0 - x) / B;
	else
		tmp = x / -B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.55e-60], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[(x / (-B)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -1.54999999999999994e-60

   1. Initial program 59.3%

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

   3. Simplified 71.1%

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

   5. Taylor expanded in B around 0 38.0%

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

   6. Taylor expanded in F around -inf 44.7%

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

   if -1.54999999999999994e-60 < F

   1. Initial program 87.7%

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

   3. Simplified 93.2%

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

   5. Taylor expanded in B around 0 41.8%

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

   6. Taylor expanded in F around 0 28.6%

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

      1. associate-*r/ 28.6%

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

      2. neg-mul-1 28.6%

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

   8. Simplified 28.6%

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

4. Final simplification 33.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.55 \cdot 10^{-60}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 29.9% accurate, 35.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.9 \cdot 10^{+174}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.9e+174) (/ -1.0 B) (/ x (- B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.9e+174) {
		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 <= (-1.9d+174)) 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 <= -1.9e+174) {
		tmp = -1.0 / B;
	} else {
		tmp = x / -B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.9e+174:
		tmp = -1.0 / B
	else:
		tmp = x / -B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.9e+174)
		tmp = Float64(-1.0 / B);
	else
		tmp = Float64(x / Float64(-B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.9e+174)
		tmp = -1.0 / B;
	else
		tmp = x / -B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.9e+174], N[(-1.0 / B), $MachinePrecision], N[(x / (-B)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -1.9000000000000001e174

   1. Initial program 35.4%

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

   3. Simplified 45.2%

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

   5. Taylor expanded in B around 0 12.4%

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

   6. Taylor expanded in F around -inf 39.5%

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

   7. Taylor expanded in x around 0 28.3%

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

   if -1.9000000000000001e174 < F

   1. Initial program 85.7%

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

   3. Simplified 92.6%

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

   5. Taylor expanded in B around 0 44.7%

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

   6. Taylor expanded in F around 0 28.3%

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

      1. associate-*r/ 28.3%

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

      2. neg-mul-1 28.3%

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

   8. Simplified 28.3%

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

4. Final simplification 28.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.9 \cdot 10^{+174}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 10.6% 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 79.4%

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

3. Simplified 86.7%

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

5. Taylor expanded in B around 0 40.7%

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

6. Taylor expanded in F around -inf 26.8%

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

7. Taylor expanded in x around 0 9.9%

   \[\leadsto \color{blue}{\frac{-1}{B}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024177 
(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))))))