VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.3% → 99.6%

Time: 20.7s

Alternatives: 26

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1 \cdot 10^{+32}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 1050:\\ \;\;\;\;F \cdot {\left(\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}\right)}^{-1} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1e+32)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1050.0)
       (-
        (* F (pow (/ (sin B) (pow (fma 2.0 x (fma F F 2.0)) -0.5)) -1.0))
        t_0)
       (/ (- 1.0 (* 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+32) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1050.0) {
		tmp = (F * pow((sin(B) / pow(fma(2.0, x, fma(F, F, 2.0)), -0.5)), -1.0)) - t_0;
	} else {
		tmp = (1.0 - (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+32)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1050.0)
		tmp = Float64(Float64(F * (Float64(sin(B) / (fma(2.0, x, fma(F, F, 2.0)) ^ -0.5)) ^ -1.0)) - t_0);
	else
		tmp = Float64(Float64(1.0 - Float64(x * 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+32], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1050.0], N[(N[(F * N[Power[N[(N[Sin[B], $MachinePrecision] / N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 - N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.00000000000000005e32

   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 71.2%

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

      1. clear-num 71.3%

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

      2. inv-pow 71.3%

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

      3. fma-def 71.3%

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

      4. fma-udef 71.3%

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

      5. *-commutative 71.3%

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

      6. fma-def 71.3%

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

      7. fma-def 71.3%

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

   6. Applied egg-rr 71.3%

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

   7. Taylor expanded in F around -inf 99.8%

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

   if -1.00000000000000005e32 < F < 1050

   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. Step-by-step derivation

      1. clear-num 99.7%

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

      2. inv-pow 99.7%

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

      3. fma-def 99.7%

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

      4. fma-udef 99.7%

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

      5. *-commutative 99.7%

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

      6. fma-def 99.7%

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

      7. fma-def 99.7%

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

   6. Applied egg-rr 99.7%

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

   if 1050 < F

   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 75.4%

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

   5. Taylor expanded in F around inf 99.6%

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

      1. associate-/r* 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in F around 0 99.8%

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

      1. div-sub 99.8%

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

   10. Simplified 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{+32}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1050:\\ \;\;\;\;F \cdot {\left(\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}\right)}^{-1} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1 \cdot 10^{+32}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 5.3 \cdot 10^{+20}:\\ \;\;\;\;F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1e+32)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 5.3e+20)
       (- (* F (/ (pow (fma x 2.0 (fma F F 2.0)) -0.5) (sin B))) t_0)
       (/ (- 1.0 (* 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+32) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 5.3e+20) {
		tmp = (F * (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) / sin(B))) - t_0;
	} else {
		tmp = (1.0 - (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+32)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 5.3e+20)
		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 - Float64(x * 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+32], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 5.3e+20], 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[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.00000000000000005e32

   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 71.2%

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

      1. clear-num 71.3%

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

      2. inv-pow 71.3%

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

      3. fma-def 71.3%

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

      4. fma-udef 71.3%

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

      5. *-commutative 71.3%

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

      6. fma-def 71.3%

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

      7. fma-def 71.3%

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

   6. Applied egg-rr 71.3%

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

   7. Taylor expanded in F around -inf 99.8%

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

   if -1.00000000000000005e32 < F < 5.3e20

   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

   if 5.3e20 < F

   1. Initial program 52.5%

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

   3. Simplified 74.3%

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

   5. Taylor expanded in F around inf 99.6%

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

      1. associate-/r* 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in F around 0 99.8%

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

      1. div-sub 99.8%

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

   10. Simplified 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{+32}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 5.3 \cdot 10^{+20}:\\ \;\;\;\;F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos B\\ \mathbf{if}\;F \leq -50000000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1050:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{t_0}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t_0}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (cos B))))
   (if (<= F -50000000000.0)
     (- (/ -1.0 (sin B)) (/ x (tan B)))
     (if (<= F 1050.0)
       (-
        (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5))
        (/ t_0 (sin B)))
       (/ (- 1.0 t_0) (sin B))))))

C

double code(double F, double B, double x) {
	double t_0 = x * cos(B);
	double tmp;
	if (F <= -50000000000.0) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 1050.0) {
		tmp = ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (t_0 / sin(B));
	} else {
		tmp = (1.0 - t_0) / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * cos(b)
    if (f <= (-50000000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 1050.0d0) then
        tmp = ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0))) - (t_0 / sin(b))
    else
        tmp = (1.0d0 - t_0) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x * Math.cos(B);
	double tmp;
	if (F <= -50000000000.0) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 1050.0) {
		tmp = ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (t_0 / Math.sin(B));
	} else {
		tmp = (1.0 - t_0) / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * math.cos(B)
	tmp = 0
	if F <= -50000000000.0:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 1050.0:
		tmp = ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (t_0 / math.sin(B))
	else:
		tmp = (1.0 - t_0) / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * cos(B))
	tmp = 0.0
	if (F <= -50000000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 1050.0)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)) - Float64(t_0 / sin(B)));
	else
		tmp = Float64(Float64(1.0 - t_0) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x * cos(B);
	tmp = 0.0;
	if (F <= -50000000000.0)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 1050.0)
		tmp = ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5)) - (t_0 / sin(B));
	else
		tmp = (1.0 - t_0) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -50000000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1050.0], 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[(t$95$0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -5e10

   1. Initial program 60.6%

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

   3. Simplified 72.1%

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

      1. clear-num 72.2%

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

      2. inv-pow 72.2%

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

      3. fma-def 72.2%

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

      4. fma-udef 72.2%

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

      5. *-commutative 72.2%

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

      6. fma-def 72.2%

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

      7. fma-def 72.2%

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

   6. Applied egg-rr 72.2%

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

   7. Taylor expanded in F around -inf 99.8%

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

   if -5e10 < F < 1050

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 99.5%

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

   if 1050 < F

   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 75.4%

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

   5. Taylor expanded in F around inf 99.6%

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

      1. associate-/r* 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in F around 0 99.8%

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

      1. div-sub 99.8%

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

   10. Simplified 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -50000000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1050:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x \cdot \cos B}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.4 \cdot 10^{+27}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1050:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} + x \cdot \frac{-1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.4e+27)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 1050.0)
     (+
      (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5))
      (* x (/ -1.0 (tan B))))
     (/ (- 1.0 (* x (cos B))) (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.4e+27) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 1050.0) {
		tmp = ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) + (x * (-1.0 / tan(B)));
	} else {
		tmp = (1.0 - (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 <= (-3.4d+27)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 1050.0d0) then
        tmp = ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0))) + (x * ((-1.0d0) / tan(b)))
    else
        tmp = (1.0d0 - (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 <= -3.4e+27) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 1050.0) {
		tmp = ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) + (x * (-1.0 / Math.tan(B)));
	} else {
		tmp = (1.0 - (x * Math.cos(B))) / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.4e+27:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 1050.0:
		tmp = ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) + (x * (-1.0 / math.tan(B)))
	else:
		tmp = (1.0 - (x * math.cos(B))) / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.4e+27)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 1050.0)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)) + Float64(x * Float64(-1.0 / tan(B))));
	else
		tmp = Float64(Float64(1.0 - Float64(x * cos(B))) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -3.4e+27)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 1050.0)
		tmp = ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5)) + (x * (-1.0 / tan(B)));
	else
		tmp = (1.0 - (x * cos(B))) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.4e+27], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1050.0], 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 * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.4e27

   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 71.2%

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

      1. clear-num 71.3%

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

      2. inv-pow 71.3%

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

      3. fma-def 71.3%

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

      4. fma-udef 71.3%

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

      5. *-commutative 71.3%

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

      6. fma-def 71.3%

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

      7. fma-def 71.3%

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

   6. Applied egg-rr 71.3%

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

   7. Taylor expanded in F around -inf 99.8%

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

   if -3.4e27 < F < 1050

   1. Initial program 99.5%

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

   if 1050 < F

   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 75.4%

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

   5. Taylor expanded in F around inf 99.6%

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

      1. associate-/r* 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in F around 0 99.8%

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

      1. div-sub 99.8%

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

   10. Simplified 99.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.4 \cdot 10^{+27}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1050:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} + x \cdot \frac{-1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{+16}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1050:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} + \frac{-1}{\frac{\tan B}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1e+16)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 1050.0)
     (+
      (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5))
      (/ -1.0 (/ (tan B) x)))
     (/ (- 1.0 (* x (cos B))) (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1e+16) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 1050.0) {
		tmp = ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) + (-1.0 / (tan(B) / x));
	} else {
		tmp = (1.0 - (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 <= (-1d+16)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 1050.0d0) then
        tmp = ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0))) + ((-1.0d0) / (tan(b) / x))
    else
        tmp = (1.0d0 - (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 <= -1e+16) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 1050.0) {
		tmp = ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) + (-1.0 / (Math.tan(B) / x));
	} else {
		tmp = (1.0 - (x * Math.cos(B))) / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1e+16:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 1050.0:
		tmp = ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) + (-1.0 / (math.tan(B) / x))
	else:
		tmp = (1.0 - (x * math.cos(B))) / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1e+16)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 1050.0)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)) + Float64(-1.0 / Float64(tan(B) / x)));
	else
		tmp = Float64(Float64(1.0 - Float64(x * cos(B))) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1e+16)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 1050.0)
		tmp = ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5)) + (-1.0 / (tan(B) / x));
	else
		tmp = (1.0 - (x * cos(B))) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1e+16], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1050.0], 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[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1e16

   1. Initial program 60.6%

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

   3. Simplified 72.1%

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

      1. clear-num 72.2%

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

      2. inv-pow 72.2%

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

      3. fma-def 72.2%

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

      4. fma-udef 72.2%

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

      5. *-commutative 72.2%

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

      6. fma-def 72.2%

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

      7. fma-def 72.2%

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

   6. Applied egg-rr 72.2%

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

   7. Taylor expanded in F around -inf 99.8%

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

   if -1e16 < F < 1050

   1. Initial program 99.5%

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

      1. div-inv 99.6%

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

      2. clear-num 99.5%

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

   4. Applied egg-rr 99.5%

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

   if 1050 < F

   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 75.4%

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

   5. Taylor expanded in F around inf 99.6%

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

      1. associate-/r* 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in F around 0 99.8%

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

      1. div-sub 99.8%

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

   10. Simplified 99.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{+16}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1050:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} + \frac{-1}{\frac{\tan B}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 1.45:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \end{array} \]

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 1.45)
       (- (* F (/ (sqrt 0.5) (sin B))) t_0)
       (/ (- 1.0 (* 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 <= 1.45) {
		tmp = (F * (sqrt(0.5) / sin(B))) - t_0;
	} else {
		tmp = (1.0 - (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 <= 1.45d0) then
        tmp = (f * (sqrt(0.5d0) / sin(b))) - t_0
    else
        tmp = (1.0d0 - (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 <= 1.45) {
		tmp = (F * (Math.sqrt(0.5) / Math.sin(B))) - t_0;
	} else {
		tmp = (1.0 - (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 <= 1.45:
		tmp = (F * (math.sqrt(0.5) / math.sin(B))) - t_0
	else:
		tmp = (1.0 - (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 <= 1.45)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / sin(B))) - t_0);
	else
		tmp = Float64(Float64(1.0 - Float64(x * 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 <= 1.45)
		tmp = (F * (sqrt(0.5) / sin(B))) - t_0;
	else
		tmp = (1.0 - (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, 1.45], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 - N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.3999999999999999

   1. Initial program 62.4%

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

   3. Simplified 73.3%

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

      1. clear-num 73.4%

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

      2. inv-pow 73.4%

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

      3. fma-def 73.4%

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

      4. fma-udef 73.4%

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

      5. *-commutative 73.4%

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

      6. fma-def 73.4%

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

      7. fma-def 73.4%

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

   6. Applied egg-rr 73.4%

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

   7. Taylor expanded in F around -inf 98.1%

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

   if -1.3999999999999999 < F < 1.44999999999999996

   1. Initial program 99.5%

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

   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 99.2%

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

   6. Taylor expanded in x around 0 99.2%

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

   if 1.44999999999999996 < F

   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 75.4%

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

   5. Taylor expanded in F around inf 99.6%

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

      1. associate-/r* 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in F around 0 99.8%

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

      1. div-sub 99.8%

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

   10. Simplified 99.8%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.45:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -205000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 52:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -205000.0)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 52.0)
     (+
      (/ -1.0 (/ (tan B) x))
      (* (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5) (/ F B)))
     (/ (- 1.0 (* x (cos B))) (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -205000.0) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 52.0) {
		tmp = (-1.0 / (tan(B) / x)) + (pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B));
	} else {
		tmp = (1.0 - (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 <= (-205000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 52.0d0) then
        tmp = ((-1.0d0) / (tan(b) / x)) + ((((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)) * (f / b))
    else
        tmp = (1.0d0 - (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 <= -205000.0) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 52.0) {
		tmp = (-1.0 / (Math.tan(B) / x)) + (Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B));
	} else {
		tmp = (1.0 - (x * Math.cos(B))) / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -205000.0:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 52.0:
		tmp = (-1.0 / (math.tan(B) / x)) + (math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B))
	else:
		tmp = (1.0 - (x * math.cos(B))) / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -205000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 52.0)
		tmp = Float64(Float64(-1.0 / Float64(tan(B) / x)) + Float64((Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5) * Float64(F / B)));
	else
		tmp = Float64(Float64(1.0 - Float64(x * cos(B))) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -205000.0)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 52.0)
		tmp = (-1.0 / (tan(B) / x)) + ((((2.0 + (F * F)) + (x * 2.0)) ^ -0.5) * (F / B));
	else
		tmp = (1.0 - (x * cos(B))) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -205000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 52.0], N[(N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -205000

   1. Initial program 60.6%

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

   3. Simplified 72.1%

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

      1. clear-num 72.2%

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

      2. inv-pow 72.2%

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

      3. fma-def 72.2%

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

      4. fma-udef 72.2%

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

      5. *-commutative 72.2%

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

      6. fma-def 72.2%

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

      7. fma-def 72.2%

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

   6. Applied egg-rr 72.2%

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

   7. Taylor expanded in F around -inf 99.8%

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

   if -205000 < F < 52

   1. Initial program 99.5%

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

      1. div-inv 99.6%

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

      2. clear-num 99.5%

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in B around 0 85.3%

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

   if 52 < F

   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 75.4%

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

   5. Taylor expanded in F around inf 99.6%

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

      1. associate-/r* 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in F around 0 99.8%

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

      1. div-sub 99.8%

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

   10. Simplified 99.8%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -205000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 52:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2 \cdot 10^{-11}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 0.6:\\ \;\;\;\;F \cdot \left(\frac{1}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -2e-11)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 0.6)
       (- (* F (* (/ 1.0 B) (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))))) t_0)
       (/ (- 1.0 (* x (cos B))) (sin B))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -2e-11) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 0.6) {
		tmp = (F * ((1.0 / B) * sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_0;
	} else {
		tmp = (1.0 - (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 <= (-2d-11)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 0.6d0) then
        tmp = (f * ((1.0d0 / b) * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))))) - t_0
    else
        tmp = (1.0d0 - (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 <= -2e-11) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 0.6) {
		tmp = (F * ((1.0 / B) * Math.sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_0;
	} else {
		tmp = (1.0 - (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 <= -2e-11:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 0.6:
		tmp = (F * ((1.0 / B) * math.sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_0
	else:
		tmp = (1.0 - (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 <= -2e-11)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 0.6)
		tmp = Float64(Float64(F * Float64(Float64(1.0 / B) * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0)))))) - t_0);
	else
		tmp = Float64(Float64(1.0 - Float64(x * 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 <= -2e-11)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 0.6)
		tmp = (F * ((1.0 / B) * sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_0;
	else
		tmp = (1.0 - (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, -2e-11], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.6], 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], N[(N[(1.0 - N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.99999999999999988e-11

   1. Initial program 63.5%

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

   3. Simplified 74.1%

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

      1. clear-num 74.2%

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

      2. inv-pow 74.2%

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

      3. fma-def 74.2%

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

      4. fma-udef 74.2%

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

      5. *-commutative 74.2%

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

      6. fma-def 74.2%

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

      7. fma-def 74.2%

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

   6. Applied egg-rr 74.2%

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

   7. Taylor expanded in F around -inf 95.6%

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

   if -1.99999999999999988e-11 < F < 0.599999999999999978

   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 99.2%

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

   6. Taylor expanded in B around 0 86.0%

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

   if 0.599999999999999978 < F

   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 75.4%

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

   5. Taylor expanded in F around inf 99.6%

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

      1. associate-/r* 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in F around 0 99.8%

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

      1. div-sub 99.8%

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

   10. Simplified 99.8%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{-11}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.6:\\ \;\;\;\;F \cdot \left(\frac{1}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\right) - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2 \cdot 10^{-11}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 0.27:\\ \;\;\;\;\frac{F}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -2e-11)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 0.27)
       (- (* (/ F B) (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))) t_0)
       (/ (- 1.0 (* x (cos B))) (sin B))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -2e-11) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 0.27) {
		tmp = ((F / B) * sqrt((1.0 / (2.0 + (x * 2.0))))) - t_0;
	} else {
		tmp = (1.0 - (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 <= (-2d-11)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 0.27d0) then
        tmp = ((f / b) * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))) - t_0
    else
        tmp = (1.0d0 - (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 <= -2e-11) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 0.27) {
		tmp = ((F / B) * Math.sqrt((1.0 / (2.0 + (x * 2.0))))) - t_0;
	} else {
		tmp = (1.0 - (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 <= -2e-11:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 0.27:
		tmp = ((F / B) * math.sqrt((1.0 / (2.0 + (x * 2.0))))) - t_0
	else:
		tmp = (1.0 - (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 <= -2e-11)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 0.27)
		tmp = Float64(Float64(Float64(F / B) * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))) - t_0);
	else
		tmp = Float64(Float64(1.0 - Float64(x * 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 <= -2e-11)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 0.27)
		tmp = ((F / B) * sqrt((1.0 / (2.0 + (x * 2.0))))) - t_0;
	else
		tmp = (1.0 - (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, -2e-11], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.27], N[(N[(N[(F / B), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 - N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.99999999999999988e-11

   1. Initial program 63.5%

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

   3. Simplified 74.1%

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

      1. clear-num 74.2%

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

      2. inv-pow 74.2%

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

      3. fma-def 74.2%

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

      4. fma-udef 74.2%

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

      5. *-commutative 74.2%

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

      6. fma-def 74.2%

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

      7. fma-def 74.2%

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

   6. Applied egg-rr 74.2%

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

   7. Taylor expanded in F around -inf 95.6%

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

   if -1.99999999999999988e-11 < F < 0.27000000000000002

   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 99.2%

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

   6. Taylor expanded in B around 0 86.0%

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

   if 0.27000000000000002 < F

   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 75.4%

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

   5. Taylor expanded in F around inf 99.6%

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

      1. associate-/r* 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in F around 0 99.8%

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

      1. div-sub 99.8%

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

   10. Simplified 99.8%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{-11}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.27:\\ \;\;\;\;\frac{F}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 85.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.4 \cdot 10^{-58}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{-73}:\\ \;\;\;\;\frac{-x}{\frac{\sin B}{\cos B}}\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{-30}:\\ \;\;\;\;\frac{F}{\frac{\sin B}{\sqrt{0.5}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.4e-58)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 1.8e-73)
     (/ (- x) (/ (sin B) (cos B)))
     (if (<= F 5.2e-30)
       (/ F (/ (sin B) (sqrt 0.5)))
       (/ (- 1.0 (* x (cos B))) (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.4e-58) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 1.8e-73) {
		tmp = -x / (sin(B) / cos(B));
	} else if (F <= 5.2e-30) {
		tmp = F / (sin(B) / sqrt(0.5));
	} else {
		tmp = (1.0 - (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 <= (-3.4d-58)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 1.8d-73) then
        tmp = -x / (sin(b) / cos(b))
    else if (f <= 5.2d-30) then
        tmp = f / (sin(b) / sqrt(0.5d0))
    else
        tmp = (1.0d0 - (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 <= -3.4e-58) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 1.8e-73) {
		tmp = -x / (Math.sin(B) / Math.cos(B));
	} else if (F <= 5.2e-30) {
		tmp = F / (Math.sin(B) / Math.sqrt(0.5));
	} else {
		tmp = (1.0 - (x * Math.cos(B))) / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.4e-58:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 1.8e-73:
		tmp = -x / (math.sin(B) / math.cos(B))
	elif F <= 5.2e-30:
		tmp = F / (math.sin(B) / math.sqrt(0.5))
	else:
		tmp = (1.0 - (x * math.cos(B))) / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.4e-58)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 1.8e-73)
		tmp = Float64(Float64(-x) / Float64(sin(B) / cos(B)));
	elseif (F <= 5.2e-30)
		tmp = Float64(F / Float64(sin(B) / sqrt(0.5)));
	else
		tmp = Float64(Float64(1.0 - Float64(x * cos(B))) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -3.4e-58)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 1.8e-73)
		tmp = -x / (sin(B) / cos(B));
	elseif (F <= 5.2e-30)
		tmp = F / (sin(B) / sqrt(0.5));
	else
		tmp = (1.0 - (x * cos(B))) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.4e-58], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.8e-73], N[((-x) / N[(N[Sin[B], $MachinePrecision] / N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.2e-30], N[(F / N[(N[Sin[B], $MachinePrecision] / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.39999999999999973e-58

   1. Initial program 67.2%

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

   3. Simplified 76.8%

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

      1. clear-num 76.9%

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

      2. inv-pow 76.9%

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

      3. fma-def 76.9%

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

      4. fma-udef 76.9%

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

      5. *-commutative 76.9%

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

      6. fma-def 76.9%

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

      7. fma-def 76.9%

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

   6. Applied egg-rr 76.9%

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

   7. Taylor expanded in F around -inf 91.2%

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

   if -3.39999999999999973e-58 < F < 1.8e-73

   1. Initial program 99.5%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in F around 0 99.7%

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

   6. Taylor expanded in B around 0 89.6%

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

   7. Taylor expanded in F around 0 76.8%

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

      1. mul-1-neg 76.8%

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

      2. associate-/l* 76.8%

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

      3. distribute-neg-frac 76.8%

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

   9. Simplified 76.8%

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

   if 1.8e-73 < F < 5.19999999999999973e-30

   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 99.7%

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

   5. Taylor expanded in F around 0 99.4%

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

   6. Taylor expanded in x around 0 99.1%

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

      1. associate-/l* 99.7%

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

   8. Simplified 99.7%

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

   if 5.19999999999999973e-30 < F

   1. Initial program 59.1%

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

   3. Simplified 77.8%

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

   5. Taylor expanded in F around inf 96.2%

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

      1. associate-/r* 96.2%

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

   7. Simplified 96.2%

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

   8. Taylor expanded in F around 0 96.4%

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

      1. div-sub 96.4%

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

   10. Simplified 96.4%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.4 \cdot 10^{-58}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{-73}:\\ \;\;\;\;\frac{-x}{\frac{\sin B}{\cos B}}\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{-30}:\\ \;\;\;\;\frac{F}{\frac{\sin B}{\sqrt{0.5}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 72.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.3 \cdot 10^{-18}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 5.5 \cdot 10^{-72}:\\ \;\;\;\;x \cdot \frac{-\cos B}{\sin B}\\ \mathbf{elif}\;F \leq 5.6 \cdot 10^{-30}:\\ \;\;\;\;\frac{F}{\frac{\sin B}{\sqrt{0.5}}}\\ \mathbf{elif}\;F \leq 1.6 \cdot 10^{+96}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.3e-18)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 5.5e-72)
     (* x (/ (- (cos B)) (sin B)))
     (if (<= F 5.6e-30)
       (/ F (/ (sin B) (sqrt 0.5)))
       (if (<= F 1.6e+96)
         (- (/ 1.0 B) (/ x (tan B)))
         (/ (- 1.0 x) (sin B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.3e-18) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 5.5e-72) {
		tmp = x * (-cos(B) / sin(B));
	} else if (F <= 5.6e-30) {
		tmp = F / (sin(B) / sqrt(0.5));
	} else if (F <= 1.6e+96) {
		tmp = (1.0 / B) - (x / tan(B));
	} else {
		tmp = (1.0 - x) / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-2.3d-18)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 5.5d-72) then
        tmp = x * (-cos(b) / sin(b))
    else if (f <= 5.6d-30) then
        tmp = f / (sin(b) / sqrt(0.5d0))
    else if (f <= 1.6d+96) then
        tmp = (1.0d0 / b) - (x / tan(b))
    else
        tmp = (1.0d0 - x) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.3e-18) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 5.5e-72) {
		tmp = x * (-Math.cos(B) / Math.sin(B));
	} else if (F <= 5.6e-30) {
		tmp = F / (Math.sin(B) / Math.sqrt(0.5));
	} else if (F <= 1.6e+96) {
		tmp = (1.0 / B) - (x / Math.tan(B));
	} else {
		tmp = (1.0 - x) / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.3e-18:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 5.5e-72:
		tmp = x * (-math.cos(B) / math.sin(B))
	elif F <= 5.6e-30:
		tmp = F / (math.sin(B) / math.sqrt(0.5))
	elif F <= 1.6e+96:
		tmp = (1.0 / B) - (x / math.tan(B))
	else:
		tmp = (1.0 - x) / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.3e-18)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 5.5e-72)
		tmp = Float64(x * Float64(Float64(-cos(B)) / sin(B)));
	elseif (F <= 5.6e-30)
		tmp = Float64(F / Float64(sin(B) / sqrt(0.5)));
	elseif (F <= 1.6e+96)
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(1.0 - x) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.3e-18)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 5.5e-72)
		tmp = x * (-cos(B) / sin(B));
	elseif (F <= 5.6e-30)
		tmp = F / (sin(B) / sqrt(0.5));
	elseif (F <= 1.6e+96)
		tmp = (1.0 / B) - (x / tan(B));
	else
		tmp = (1.0 - x) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.3e-18], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.5e-72], N[(x * N[((-N[Cos[B], $MachinePrecision]) / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.6e-30], N[(F / N[(N[Sin[B], $MachinePrecision] / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.6e+96], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -2.3000000000000001e-18

   1. Initial program 64.0%

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

   3. Taylor expanded in F around -inf 94.2%

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

   4. Taylor expanded in B around 0 72.4%

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

   if -2.3000000000000001e-18 < F < 5.49999999999999994e-72

   1. Initial program 99.5%

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

   3. Taylor expanded in F around -inf 35.8%

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

   4. Taylor expanded in x around inf 75.5%

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

      1. mul-1-neg 75.5%

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

      2. associate-*r/ 75.5%

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

      3. distribute-rgt-neg-in 75.5%

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

   6. Simplified 75.5%

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

   if 5.49999999999999994e-72 < F < 5.59999999999999977e-30

   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 99.7%

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

   5. Taylor expanded in F around 0 99.4%

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

   6. Taylor expanded in x around 0 99.1%

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

      1. associate-/l* 99.7%

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

   8. Simplified 99.7%

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

   if 5.59999999999999977e-30 < F < 1.60000000000000003e96

   1. Initial program 92.5%

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

   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 90.0%

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

      1. associate-/r* 90.0%

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

   7. Simplified 90.0%

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

   8. Taylor expanded in B around 0 72.5%

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

   if 1.60000000000000003e96 < F

   1. Initial program 40.7%

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

   3. Simplified 65.9%

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

   5. Taylor expanded in F around inf 99.6%

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

      1. associate-/r* 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in F around 0 99.8%

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

      1. div-sub 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in B around 0 85.9%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.3 \cdot 10^{-18}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 5.5 \cdot 10^{-72}:\\ \;\;\;\;x \cdot \frac{-\cos B}{\sin B}\\ \mathbf{elif}\;F \leq 5.6 \cdot 10^{-30}:\\ \;\;\;\;\frac{F}{\frac{\sin B}{\sqrt{0.5}}}\\ \mathbf{elif}\;F \leq 1.6 \cdot 10^{+96}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 72.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.3 \cdot 10^{-18}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.08 \cdot 10^{-72}:\\ \;\;\;\;\frac{-x}{\frac{\sin B}{\cos B}}\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{-30}:\\ \;\;\;\;\frac{F}{\frac{\sin B}{\sqrt{0.5}}}\\ \mathbf{elif}\;F \leq 1.6 \cdot 10^{+96}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.3e-18)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 1.08e-72)
     (/ (- x) (/ (sin B) (cos B)))
     (if (<= F 5.2e-30)
       (/ F (/ (sin B) (sqrt 0.5)))
       (if (<= F 1.6e+96)
         (- (/ 1.0 B) (/ x (tan B)))
         (/ (- 1.0 x) (sin B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.3e-18) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 1.08e-72) {
		tmp = -x / (sin(B) / cos(B));
	} else if (F <= 5.2e-30) {
		tmp = F / (sin(B) / sqrt(0.5));
	} else if (F <= 1.6e+96) {
		tmp = (1.0 / B) - (x / tan(B));
	} else {
		tmp = (1.0 - x) / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-2.3d-18)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 1.08d-72) then
        tmp = -x / (sin(b) / cos(b))
    else if (f <= 5.2d-30) then
        tmp = f / (sin(b) / sqrt(0.5d0))
    else if (f <= 1.6d+96) then
        tmp = (1.0d0 / b) - (x / tan(b))
    else
        tmp = (1.0d0 - x) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.3e-18) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 1.08e-72) {
		tmp = -x / (Math.sin(B) / Math.cos(B));
	} else if (F <= 5.2e-30) {
		tmp = F / (Math.sin(B) / Math.sqrt(0.5));
	} else if (F <= 1.6e+96) {
		tmp = (1.0 / B) - (x / Math.tan(B));
	} else {
		tmp = (1.0 - x) / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.3e-18:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 1.08e-72:
		tmp = -x / (math.sin(B) / math.cos(B))
	elif F <= 5.2e-30:
		tmp = F / (math.sin(B) / math.sqrt(0.5))
	elif F <= 1.6e+96:
		tmp = (1.0 / B) - (x / math.tan(B))
	else:
		tmp = (1.0 - x) / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.3e-18)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 1.08e-72)
		tmp = Float64(Float64(-x) / Float64(sin(B) / cos(B)));
	elseif (F <= 5.2e-30)
		tmp = Float64(F / Float64(sin(B) / sqrt(0.5)));
	elseif (F <= 1.6e+96)
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(1.0 - x) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.3e-18)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 1.08e-72)
		tmp = -x / (sin(B) / cos(B));
	elseif (F <= 5.2e-30)
		tmp = F / (sin(B) / sqrt(0.5));
	elseif (F <= 1.6e+96)
		tmp = (1.0 / B) - (x / tan(B));
	else
		tmp = (1.0 - x) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.3e-18], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.08e-72], N[((-x) / N[(N[Sin[B], $MachinePrecision] / N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.2e-30], N[(F / N[(N[Sin[B], $MachinePrecision] / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.6e+96], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -2.3000000000000001e-18

   1. Initial program 64.0%

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

   3. Taylor expanded in F around -inf 94.2%

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

   4. Taylor expanded in B around 0 72.4%

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

   if -2.3000000000000001e-18 < F < 1.07999999999999998e-72

   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 99.7%

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

   6. Taylor expanded in B around 0 88.4%

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

   7. Taylor expanded in F around 0 75.5%

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

      1. mul-1-neg 75.5%

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

      2. associate-/l* 75.5%

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

      3. distribute-neg-frac 75.5%

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

   9. Simplified 75.5%

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

   if 1.07999999999999998e-72 < F < 5.19999999999999973e-30

   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 99.7%

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

   5. Taylor expanded in F around 0 99.4%

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

   6. Taylor expanded in x around 0 99.1%

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

      1. associate-/l* 99.7%

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

   8. Simplified 99.7%

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

   if 5.19999999999999973e-30 < F < 1.60000000000000003e96

   1. Initial program 92.5%

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

   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 90.0%

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

      1. associate-/r* 90.0%

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

   7. Simplified 90.0%

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

   8. Taylor expanded in B around 0 72.5%

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

   if 1.60000000000000003e96 < F

   1. Initial program 40.7%

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

   3. Simplified 65.9%

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

   5. Taylor expanded in F around inf 99.6%

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

      1. associate-/r* 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in F around 0 99.8%

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

      1. div-sub 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in B around 0 85.9%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.3 \cdot 10^{-18}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.08 \cdot 10^{-72}:\\ \;\;\;\;\frac{-x}{\frac{\sin B}{\cos B}}\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{-30}:\\ \;\;\;\;\frac{F}{\frac{\sin B}{\sqrt{0.5}}}\\ \mathbf{elif}\;F \leq 1.6 \cdot 10^{+96}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 78.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3.2 \cdot 10^{-58}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{-74}:\\ \;\;\;\;\frac{-x}{\frac{\sin B}{\cos B}}\\ \mathbf{elif}\;F \leq 4.9 \cdot 10^{-30}:\\ \;\;\;\;\frac{F}{\frac{\sin B}{\sqrt{0.5}}}\\ \mathbf{elif}\;F \leq 1.65 \cdot 10^{+96}:\\ \;\;\;\;\frac{1}{B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -3.2e-58)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.4e-74)
       (/ (- x) (/ (sin B) (cos B)))
       (if (<= F 4.9e-30)
         (/ F (/ (sin B) (sqrt 0.5)))
         (if (<= F 1.65e+96) (- (/ 1.0 B) t_0) (/ (- 1.0 x) (sin B))))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -3.2e-58) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.4e-74) {
		tmp = -x / (sin(B) / cos(B));
	} else if (F <= 4.9e-30) {
		tmp = F / (sin(B) / sqrt(0.5));
	} else if (F <= 1.65e+96) {
		tmp = (1.0 / B) - t_0;
	} else {
		tmp = (1.0 - x) / 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 <= (-3.2d-58)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.4d-74) then
        tmp = -x / (sin(b) / cos(b))
    else if (f <= 4.9d-30) then
        tmp = f / (sin(b) / sqrt(0.5d0))
    else if (f <= 1.65d+96) then
        tmp = (1.0d0 / b) - t_0
    else
        tmp = (1.0d0 - x) / 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 <= -3.2e-58) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.4e-74) {
		tmp = -x / (Math.sin(B) / Math.cos(B));
	} else if (F <= 4.9e-30) {
		tmp = F / (Math.sin(B) / Math.sqrt(0.5));
	} else if (F <= 1.65e+96) {
		tmp = (1.0 / B) - t_0;
	} else {
		tmp = (1.0 - x) / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -3.2e-58:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.4e-74:
		tmp = -x / (math.sin(B) / math.cos(B))
	elif F <= 4.9e-30:
		tmp = F / (math.sin(B) / math.sqrt(0.5))
	elif F <= 1.65e+96:
		tmp = (1.0 / B) - t_0
	else:
		tmp = (1.0 - x) / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -3.2e-58)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.4e-74)
		tmp = Float64(Float64(-x) / Float64(sin(B) / cos(B)));
	elseif (F <= 4.9e-30)
		tmp = Float64(F / Float64(sin(B) / sqrt(0.5)));
	elseif (F <= 1.65e+96)
		tmp = Float64(Float64(1.0 / B) - t_0);
	else
		tmp = Float64(Float64(1.0 - x) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -3.2e-58)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.4e-74)
		tmp = -x / (sin(B) / cos(B));
	elseif (F <= 4.9e-30)
		tmp = F / (sin(B) / sqrt(0.5));
	elseif (F <= 1.65e+96)
		tmp = (1.0 / B) - t_0;
	else
		tmp = (1.0 - x) / 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, -3.2e-58], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.4e-74], N[((-x) / N[(N[Sin[B], $MachinePrecision] / N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.9e-30], N[(F / N[(N[Sin[B], $MachinePrecision] / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.65e+96], N[(N[(1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -3.2000000000000001e-58

   1. Initial program 67.2%

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

   3. Simplified 76.8%

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

      1. clear-num 76.9%

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

      2. inv-pow 76.9%

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

      3. fma-def 76.9%

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

      4. fma-udef 76.9%

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

      5. *-commutative 76.9%

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

      6. fma-def 76.9%

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

      7. fma-def 76.9%

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

   6. Applied egg-rr 76.9%

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

   7. Taylor expanded in F around -inf 91.2%

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

   if -3.2000000000000001e-58 < F < 1.39999999999999994e-74

   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 99.7%

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

   5. Taylor expanded in F around 0 99.7%

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

   6. Taylor expanded in B around 0 89.6%

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

   7. Taylor expanded in F around 0 76.8%

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

      1. mul-1-neg 76.8%

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

      2. associate-/l* 76.8%

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

      3. distribute-neg-frac 76.8%

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

   9. Simplified 76.8%

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

   if 1.39999999999999994e-74 < F < 4.89999999999999971e-30

   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 99.7%

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

   5. Taylor expanded in F around 0 99.4%

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

   6. Taylor expanded in x around 0 99.1%

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

      1. associate-/l* 99.7%

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

   8. Simplified 99.7%

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

   if 4.89999999999999971e-30 < F < 1.64999999999999992e96

   1. Initial program 92.5%

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

   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 90.0%

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

      1. associate-/r* 90.0%

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

   7. Simplified 90.0%

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

   8. Taylor expanded in B around 0 72.5%

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

   if 1.64999999999999992e96 < F

   1. Initial program 40.7%

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

   3. Simplified 65.9%

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

   5. Taylor expanded in F around inf 99.6%

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

      1. associate-/r* 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in F around 0 99.8%

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

      1. div-sub 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in B around 0 85.9%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.2 \cdot 10^{-58}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{-74}:\\ \;\;\;\;\frac{-x}{\frac{\sin B}{\cos B}}\\ \mathbf{elif}\;F \leq 4.9 \cdot 10^{-30}:\\ \;\;\;\;\frac{F}{\frac{\sin B}{\sqrt{0.5}}}\\ \mathbf{elif}\;F \leq 1.65 \cdot 10^{+96}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 64.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{+88}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -8.6 \cdot 10^{-280}:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq 780:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2e+88)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F -8.6e-280)
     (- (/ -1.0 B) (* x (/ 1.0 (tan B))))
     (if (<= F 780.0)
       (-
        (* (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5) (/ F B))
        (+ (/ x B) (* -0.3333333333333333 (* B x))))
       (/ (- 1.0 x) (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2e+88) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -8.6e-280) {
		tmp = (-1.0 / B) - (x * (1.0 / tan(B)));
	} else if (F <= 780.0) {
		tmp = (pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - ((x / B) + (-0.3333333333333333 * (B * x)));
	} else {
		tmp = (1.0 - x) / 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 <= (-2d+88)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-8.6d-280)) then
        tmp = ((-1.0d0) / b) - (x * (1.0d0 / tan(b)))
    else if (f <= 780.0d0) then
        tmp = ((((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)) * (f / b)) - ((x / b) + ((-0.3333333333333333d0) * (b * x)))
    else
        tmp = (1.0d0 - x) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2e+88) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -8.6e-280) {
		tmp = (-1.0 / B) - (x * (1.0 / Math.tan(B)));
	} else if (F <= 780.0) {
		tmp = (Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - ((x / B) + (-0.3333333333333333 * (B * x)));
	} else {
		tmp = (1.0 - x) / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2e+88:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -8.6e-280:
		tmp = (-1.0 / B) - (x * (1.0 / math.tan(B)))
	elif F <= 780.0:
		tmp = (math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - ((x / B) + (-0.3333333333333333 * (B * x)))
	else:
		tmp = (1.0 - x) / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2e+88)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -8.6e-280)
		tmp = Float64(Float64(-1.0 / B) - Float64(x * Float64(1.0 / tan(B))));
	elseif (F <= 780.0)
		tmp = Float64(Float64((Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5) * Float64(F / B)) - Float64(Float64(x / B) + Float64(-0.3333333333333333 * Float64(B * x))));
	else
		tmp = Float64(Float64(1.0 - x) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2e+88)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -8.6e-280)
		tmp = (-1.0 / B) - (x * (1.0 / tan(B)));
	elseif (F <= 780.0)
		tmp = ((((2.0 + (F * F)) + (x * 2.0)) ^ -0.5) * (F / B)) - ((x / B) + (-0.3333333333333333 * (B * x)));
	else
		tmp = (1.0 - x) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2e+88], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -8.6e-280], N[(N[(-1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 780.0], N[(N[(N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(N[(x / B), $MachinePrecision] + N[(-0.3333333333333333 * N[(B * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.99999999999999992e88

   1. Initial program 52.4%

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

   3. Taylor expanded in F around -inf 99.7%

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

   4. Taylor expanded in B around 0 79.9%

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

   if -1.99999999999999992e88 < F < -8.5999999999999997e-280

   1. Initial program 99.5%

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

   3. Taylor expanded in F around -inf 52.9%

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

   4. Taylor expanded in B around 0 60.7%

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

   if -8.5999999999999997e-280 < F < 780

   1. Initial program 99.4%

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

   3. Taylor expanded in B around 0 63.0%

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

   4. Taylor expanded in B around 0 58.4%

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

   if 780 < F

   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 75.4%

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

   5. Taylor expanded in F around inf 99.6%

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

      1. associate-/r* 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in F around 0 99.8%

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

      1. div-sub 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in B around 0 81.9%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{+88}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -8.6 \cdot 10^{-280}:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq 780:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 64.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.8 \cdot 10^{+88}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -9 \cdot 10^{-280}:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq 0.125:\\ \;\;\;\;\left(\frac{F}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} + \left(B \cdot x\right) \cdot 0.3333333333333333\right) - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.8e+88)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F -9e-280)
     (- (/ -1.0 B) (* x (/ 1.0 (tan B))))
     (if (<= F 0.125)
       (-
        (+
         (* (/ F B) (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))))
         (* (* B x) 0.3333333333333333))
        (/ x B))
       (/ (- 1.0 x) (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.8e+88) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -9e-280) {
		tmp = (-1.0 / B) - (x * (1.0 / tan(B)));
	} else if (F <= 0.125) {
		tmp = (((F / B) * sqrt((1.0 / (2.0 + (x * 2.0))))) + ((B * x) * 0.3333333333333333)) - (x / B);
	} else {
		tmp = (1.0 - x) / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-3.8d+88)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-9d-280)) then
        tmp = ((-1.0d0) / b) - (x * (1.0d0 / tan(b)))
    else if (f <= 0.125d0) then
        tmp = (((f / b) * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))) + ((b * x) * 0.3333333333333333d0)) - (x / b)
    else
        tmp = (1.0d0 - x) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.8e+88) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -9e-280) {
		tmp = (-1.0 / B) - (x * (1.0 / Math.tan(B)));
	} else if (F <= 0.125) {
		tmp = (((F / B) * Math.sqrt((1.0 / (2.0 + (x * 2.0))))) + ((B * x) * 0.3333333333333333)) - (x / B);
	} else {
		tmp = (1.0 - x) / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.8e+88:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -9e-280:
		tmp = (-1.0 / B) - (x * (1.0 / math.tan(B)))
	elif F <= 0.125:
		tmp = (((F / B) * math.sqrt((1.0 / (2.0 + (x * 2.0))))) + ((B * x) * 0.3333333333333333)) - (x / B)
	else:
		tmp = (1.0 - x) / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.8e+88)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -9e-280)
		tmp = Float64(Float64(-1.0 / B) - Float64(x * Float64(1.0 / tan(B))));
	elseif (F <= 0.125)
		tmp = Float64(Float64(Float64(Float64(F / B) * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))) + Float64(Float64(B * x) * 0.3333333333333333)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 - x) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -3.8e+88)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -9e-280)
		tmp = (-1.0 / B) - (x * (1.0 / tan(B)));
	elseif (F <= 0.125)
		tmp = (((F / B) * sqrt((1.0 / (2.0 + (x * 2.0))))) + ((B * x) * 0.3333333333333333)) - (x / B);
	else
		tmp = (1.0 - x) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.8e+88], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -9e-280], N[(N[(-1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.125], N[(N[(N[(N[(F / B), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(B * x), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.7999999999999997e88

   1. Initial program 52.4%

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

   3. Taylor expanded in F around -inf 99.7%

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

   4. Taylor expanded in B around 0 79.9%

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

   if -3.7999999999999997e88 < F < -8.9999999999999991e-280

   1. Initial program 99.5%

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

   3. Taylor expanded in F around -inf 52.9%

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

   4. Taylor expanded in B around 0 60.7%

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

   if -8.9999999999999991e-280 < F < 0.125

   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 99.7%

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

   5. Taylor expanded in F around 0 98.9%

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

   6. Taylor expanded in B around 0 85.9%

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

   7. Taylor expanded in B around 0 57.5%

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

   if 0.125 < F

   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 75.4%

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

   5. Taylor expanded in F around inf 99.6%

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

      1. associate-/r* 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in F around 0 99.8%

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

      1. div-sub 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in B around 0 81.9%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.8 \cdot 10^{+88}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -9 \cdot 10^{-280}:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq 0.125:\\ \;\;\;\;\left(\frac{F}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} + \left(B \cdot x\right) \cdot 0.3333333333333333\right) - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 64.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.2 \cdot 10^{+88}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 10^{-276}:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq 0.2:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.2e+88)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 1e-276)
     (- (/ -1.0 B) (* x (/ 1.0 (tan B))))
     (if (<= F 0.2)
       (/ (- (* F (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))) x) B)
       (/ (- 1.0 x) (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.2e+88) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 1e-276) {
		tmp = (-1.0 / B) - (x * (1.0 / tan(B)));
	} else if (F <= 0.2) {
		tmp = ((F * sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B;
	} else {
		tmp = (1.0 - x) / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-2.2d+88)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 1d-276) then
        tmp = ((-1.0d0) / b) - (x * (1.0d0 / tan(b)))
    else if (f <= 0.2d0) then
        tmp = ((f * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))) - x) / b
    else
        tmp = (1.0d0 - x) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.2e+88) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 1e-276) {
		tmp = (-1.0 / B) - (x * (1.0 / Math.tan(B)));
	} else if (F <= 0.2) {
		tmp = ((F * Math.sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B;
	} else {
		tmp = (1.0 - x) / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.2e+88:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 1e-276:
		tmp = (-1.0 / B) - (x * (1.0 / math.tan(B)))
	elif F <= 0.2:
		tmp = ((F * math.sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B
	else:
		tmp = (1.0 - x) / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.2e+88)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 1e-276)
		tmp = Float64(Float64(-1.0 / B) - Float64(x * Float64(1.0 / tan(B))));
	elseif (F <= 0.2)
		tmp = Float64(Float64(Float64(F * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))) - x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.2e+88)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 1e-276)
		tmp = (-1.0 / B) - (x * (1.0 / tan(B)));
	elseif (F <= 0.2)
		tmp = ((F * sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B;
	else
		tmp = (1.0 - x) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.2e+88], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1e-276], N[(N[(-1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.2], 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], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.20000000000000009e88

   1. Initial program 52.4%

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

   3. Taylor expanded in F around -inf 99.7%

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

   4. Taylor expanded in B around 0 79.9%

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

   if -2.20000000000000009e88 < F < 1e-276

   1. Initial program 99.5%

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

   3. Taylor expanded in F around -inf 48.0%

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

   4. Taylor expanded in B around 0 58.9%

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

   if 1e-276 < F < 0.20000000000000001

   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 99.7%

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

   5. Taylor expanded in F around 0 98.6%

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

   6. Taylor expanded in B around 0 58.4%

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

   if 0.20000000000000001 < F

   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 75.4%

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

   5. Taylor expanded in F around inf 99.6%

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

      1. associate-/r* 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in F around 0 99.8%

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

      1. div-sub 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in B around 0 81.9%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.2 \cdot 10^{+88}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 10^{-276}:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq 0.2:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 61.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2.3 \cdot 10^{-18}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.2 \cdot 10^{-276}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 10^{-192}:\\ \;\;\;\;x \cdot \left(B \cdot 0.3333333333333333 + \frac{-1}{B}\right)\\ \mathbf{elif}\;F \leq 1.7 \cdot 10^{+96}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 B) (/ x (tan B)))))
   (if (<= F -2.3e-18)
     (- (/ -1.0 (sin B)) (/ x B))
     (if (<= F 1.2e-276)
       t_0
       (if (<= F 1e-192)
         (* x (+ (* B 0.3333333333333333) (/ -1.0 B)))
         (if (<= F 1.7e+96) t_0 (/ (- 1.0 x) (sin B))))))))

C

double code(double F, double B, double x) {
	double t_0 = (1.0 / B) - (x / tan(B));
	double tmp;
	if (F <= -2.3e-18) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 1.2e-276) {
		tmp = t_0;
	} else if (F <= 1e-192) {
		tmp = x * ((B * 0.3333333333333333) + (-1.0 / B));
	} else if (F <= 1.7e+96) {
		tmp = t_0;
	} else {
		tmp = (1.0 - x) / 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 = (1.0d0 / b) - (x / tan(b))
    if (f <= (-2.3d-18)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 1.2d-276) then
        tmp = t_0
    else if (f <= 1d-192) then
        tmp = x * ((b * 0.3333333333333333d0) + ((-1.0d0) / b))
    else if (f <= 1.7d+96) then
        tmp = t_0
    else
        tmp = (1.0d0 - x) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (1.0 / B) - (x / Math.tan(B));
	double tmp;
	if (F <= -2.3e-18) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 1.2e-276) {
		tmp = t_0;
	} else if (F <= 1e-192) {
		tmp = x * ((B * 0.3333333333333333) + (-1.0 / B));
	} else if (F <= 1.7e+96) {
		tmp = t_0;
	} else {
		tmp = (1.0 - x) / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (1.0 / B) - (x / math.tan(B))
	tmp = 0
	if F <= -2.3e-18:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 1.2e-276:
		tmp = t_0
	elif F <= 1e-192:
		tmp = x * ((B * 0.3333333333333333) + (-1.0 / B))
	elif F <= 1.7e+96:
		tmp = t_0
	else:
		tmp = (1.0 - x) / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(1.0 / B) - Float64(x / tan(B)))
	tmp = 0.0
	if (F <= -2.3e-18)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 1.2e-276)
		tmp = t_0;
	elseif (F <= 1e-192)
		tmp = Float64(x * Float64(Float64(B * 0.3333333333333333) + Float64(-1.0 / B)));
	elseif (F <= 1.7e+96)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 - x) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (1.0 / B) - (x / tan(B));
	tmp = 0.0;
	if (F <= -2.3e-18)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 1.2e-276)
		tmp = t_0;
	elseif (F <= 1e-192)
		tmp = x * ((B * 0.3333333333333333) + (-1.0 / B));
	elseif (F <= 1.7e+96)
		tmp = t_0;
	else
		tmp = (1.0 - x) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.3e-18], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.2e-276], t$95$0, If[LessEqual[F, 1e-192], N[(x * N[(N[(B * 0.3333333333333333), $MachinePrecision] + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.7e+96], t$95$0, N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.3000000000000001e-18

   1. Initial program 64.0%

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

   3. Taylor expanded in F around -inf 94.2%

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

   4. Taylor expanded in B around 0 72.4%

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

   if -2.3000000000000001e-18 < F < 1.19999999999999991e-276 or 1.0000000000000001e-192 < F < 1.7e96

   1. Initial program 97.8%

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

   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 inf 43.8%

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

      1. associate-/r* 43.8%

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

   7. Simplified 43.8%

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

   8. Taylor expanded in B around 0 56.5%

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

   if 1.19999999999999991e-276 < F < 1.0000000000000001e-192

   1. Initial program 99.6%

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

   3. Taylor expanded in B around 0 70.4%

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

   4. Taylor expanded in F around inf 9.6%

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

   5. Taylor expanded in x around -inf 64.7%

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(\frac{1}{B} - 0.3333333333333333 \cdot B\right)\right)} \]

   if 1.7e96 < F

   1. Initial program 40.7%

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

   3. Simplified 65.9%

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

   5. Taylor expanded in F around inf 99.6%

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

      1. associate-/r* 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in F around 0 99.8%

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

      1. div-sub 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in B around 0 85.9%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.3 \cdot 10^{-18}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.2 \cdot 10^{-276}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 10^{-192}:\\ \;\;\;\;x \cdot \left(B \cdot 0.3333333333333333 + \frac{-1}{B}\right)\\ \mathbf{elif}\;F \leq 1.7 \cdot 10^{+96}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 61.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.8 \cdot 10^{+88}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -3 \cdot 10^{-279}:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq 6 \cdot 10^{-193}:\\ \;\;\;\;x \cdot \left(B \cdot 0.3333333333333333 + \frac{-1}{B}\right)\\ \mathbf{elif}\;F \leq 1.7 \cdot 10^{+96}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.8e+88)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F -3e-279)
     (- (/ -1.0 B) (* x (/ 1.0 (tan B))))
     (if (<= F 6e-193)
       (* x (+ (* B 0.3333333333333333) (/ -1.0 B)))
       (if (<= F 1.7e+96)
         (- (/ 1.0 B) (/ x (tan B)))
         (/ (- 1.0 x) (sin B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.8e+88) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -3e-279) {
		tmp = (-1.0 / B) - (x * (1.0 / tan(B)));
	} else if (F <= 6e-193) {
		tmp = x * ((B * 0.3333333333333333) + (-1.0 / B));
	} else if (F <= 1.7e+96) {
		tmp = (1.0 / B) - (x / tan(B));
	} else {
		tmp = (1.0 - x) / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-3.8d+88)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-3d-279)) then
        tmp = ((-1.0d0) / b) - (x * (1.0d0 / tan(b)))
    else if (f <= 6d-193) then
        tmp = x * ((b * 0.3333333333333333d0) + ((-1.0d0) / b))
    else if (f <= 1.7d+96) then
        tmp = (1.0d0 / b) - (x / tan(b))
    else
        tmp = (1.0d0 - x) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.8e+88) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -3e-279) {
		tmp = (-1.0 / B) - (x * (1.0 / Math.tan(B)));
	} else if (F <= 6e-193) {
		tmp = x * ((B * 0.3333333333333333) + (-1.0 / B));
	} else if (F <= 1.7e+96) {
		tmp = (1.0 / B) - (x / Math.tan(B));
	} else {
		tmp = (1.0 - x) / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.8e+88:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -3e-279:
		tmp = (-1.0 / B) - (x * (1.0 / math.tan(B)))
	elif F <= 6e-193:
		tmp = x * ((B * 0.3333333333333333) + (-1.0 / B))
	elif F <= 1.7e+96:
		tmp = (1.0 / B) - (x / math.tan(B))
	else:
		tmp = (1.0 - x) / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.8e+88)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -3e-279)
		tmp = Float64(Float64(-1.0 / B) - Float64(x * Float64(1.0 / tan(B))));
	elseif (F <= 6e-193)
		tmp = Float64(x * Float64(Float64(B * 0.3333333333333333) + Float64(-1.0 / B)));
	elseif (F <= 1.7e+96)
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(1.0 - x) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -3.8e+88)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -3e-279)
		tmp = (-1.0 / B) - (x * (1.0 / tan(B)));
	elseif (F <= 6e-193)
		tmp = x * ((B * 0.3333333333333333) + (-1.0 / B));
	elseif (F <= 1.7e+96)
		tmp = (1.0 / B) - (x / tan(B));
	else
		tmp = (1.0 - x) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.8e+88], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -3e-279], N[(N[(-1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 6e-193], N[(x * N[(N[(B * 0.3333333333333333), $MachinePrecision] + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.7e+96], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -3.7999999999999997e88

   1. Initial program 52.4%

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

   3. Taylor expanded in F around -inf 99.7%

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

   4. Taylor expanded in B around 0 79.9%

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

   if -3.7999999999999997e88 < F < -3e-279

   1. Initial program 99.5%

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

   3. Taylor expanded in F around -inf 52.9%

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

   4. Taylor expanded in B around 0 60.7%

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

   if -3e-279 < F < 5.9999999999999998e-193

   1. Initial program 99.6%

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

   3. Taylor expanded in B around 0 61.1%

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

   4. Taylor expanded in F around inf 12.2%

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

   5. Taylor expanded in x around -inf 58.1%

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(\frac{1}{B} - 0.3333333333333333 \cdot B\right)\right)} \]

   if 5.9999999999999998e-193 < F < 1.7e96

   1. Initial program 95.8%

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

   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 60.4%

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

      1. associate-/r* 60.4%

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

   7. Simplified 60.4%

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

   8. Taylor expanded in B around 0 56.4%

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

   if 1.7e96 < F

   1. Initial program 40.7%

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

   3. Simplified 65.9%

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

   5. Taylor expanded in F around inf 99.6%

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

      1. associate-/r* 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in F around 0 99.8%

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

      1. div-sub 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in B around 0 85.9%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.8 \cdot 10^{+88}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -3 \cdot 10^{-279}:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq 6 \cdot 10^{-193}:\\ \;\;\;\;x \cdot \left(B \cdot 0.3333333333333333 + \frac{-1}{B}\right)\\ \mathbf{elif}\;F \leq 1.7 \cdot 10^{+96}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 56.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{-132}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-149}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{elif}\;x \leq 0.00025:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 B) (/ x (tan B)))))
   (if (<= x -1.35e-132)
     t_0
     (if (<= x 4.8e-149)
       (/ 1.0 (sin B))
       (if (<= x 0.00025) (/ (- 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 <= -1.35e-132) {
		tmp = t_0;
	} else if (x <= 4.8e-149) {
		tmp = 1.0 / sin(B);
	} else if (x <= 0.00025) {
		tmp = -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 <= (-1.35d-132)) then
        tmp = t_0
    else if (x <= 4.8d-149) then
        tmp = 1.0d0 / sin(b)
    else if (x <= 0.00025d0) then
        tmp = -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 <= -1.35e-132) {
		tmp = t_0;
	} else if (x <= 4.8e-149) {
		tmp = 1.0 / Math.sin(B);
	} else if (x <= 0.00025) {
		tmp = -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 <= -1.35e-132:
		tmp = t_0
	elif x <= 4.8e-149:
		tmp = 1.0 / math.sin(B)
	elif x <= 0.00025:
		tmp = -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 <= -1.35e-132)
		tmp = t_0;
	elseif (x <= 4.8e-149)
		tmp = Float64(1.0 / sin(B));
	elseif (x <= 0.00025)
		tmp = Float64(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 <= -1.35e-132)
		tmp = t_0;
	elseif (x <= 4.8e-149)
		tmp = 1.0 / sin(B);
	elseif (x <= 0.00025)
		tmp = -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, -1.35e-132], t$95$0, If[LessEqual[x, 4.8e-149], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.00025], N[((-x) / B), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.34999999999999995e-132 or 2.5000000000000001e-4 < x

   1. Initial program 80.0%

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

   3. Simplified 94.1%

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

   5. Taylor expanded in F around inf 74.3%

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

      1. associate-/r* 74.3%

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

   7. Simplified 74.3%

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

   8. Taylor expanded in B around 0 83.5%

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

   if -1.34999999999999995e-132 < x < 4.8000000000000002e-149

   1. Initial program 71.6%

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

   3. Taylor expanded in B around 0 52.7%

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

   4. Taylor expanded in F around inf 30.1%

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

   5. Taylor expanded in x around 0 35.6%

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

   if 4.8000000000000002e-149 < x < 2.5000000000000001e-4

   1. Initial program 81.0%

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

   3. Simplified 81.6%

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

   5. Taylor expanded in F around inf 12.7%

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

      1. associate-/r* 12.5%

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

   7. Simplified 12.5%

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

   8. Taylor expanded in B around 0 7.7%

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

   9. Taylor expanded in x around inf 40.0%

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

       1. neg-mul-1 40.0%

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

       2. distribute-neg-frac 40.0%

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

   11. Simplified 40.0%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-132}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-149}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{elif}\;x \leq 0.00025:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 52.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.4 \cdot 10^{-57}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 7.4 \cdot 10^{-75}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.4e-57)
   (/ (- -1.0 x) B)
   (if (<= F 7.4e-75) (/ (- x) B) (/ (- 1.0 x) (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.4e-57) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 7.4e-75) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.4d-57)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 7.4d-75) then
        tmp = -x / b
    else
        tmp = (1.0d0 - x) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.4e-57) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 7.4e-75) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.4e-57:
		tmp = (-1.0 - x) / B
	elif F <= 7.4e-75:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.4e-57)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 7.4e-75)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.4e-57)
		tmp = (-1.0 - x) / B;
	elseif (F <= 7.4e-75)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.4e-57], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 7.4e-75], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.4e-57

   1. Initial program 66.8%

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

   3. Taylor expanded in F around -inf 92.2%

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

   4. Taylor expanded in B around 0 43.3%

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

      1. associate-*r/ 43.3%

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

      2. mul-1-neg 43.3%

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

   6. Simplified 43.3%

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

   if -1.4e-57 < F < 7.40000000000000047e-75

   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 inf 24.3%

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

      1. associate-/r* 24.3%

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

   7. Simplified 24.3%

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

   8. Taylor expanded in B around 0 18.6%

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

   9. Taylor expanded in x around inf 41.1%

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

       1. neg-mul-1 41.1%

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

       2. distribute-neg-frac 41.1%

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

   11. Simplified 41.1%

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

   if 7.40000000000000047e-75 < F

   1. Initial program 61.9%

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

   3. Simplified 79.4%

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

   5. Taylor expanded in F around inf 91.1%

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

      1. associate-/r* 91.0%

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

   7. Simplified 91.0%

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

   8. Taylor expanded in F around 0 91.2%

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

      1. div-sub 91.2%

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

   10. Simplified 91.2%

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

   11. Taylor expanded in B around 0 73.8%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.4 \cdot 10^{-57}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 7.4 \cdot 10^{-75}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 44.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.8 \cdot 10^{-109}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 11500:\\ \;\;\;\;x \cdot \left(B \cdot 0.3333333333333333 + \frac{-1}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.8e-109)
   (/ (- -1.0 x) B)
   (if (<= F 11500.0)
     (* x (+ (* B 0.3333333333333333) (/ -1.0 B)))
     (/ 1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.8e-109) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 11500.0) {
		tmp = x * ((B * 0.3333333333333333) + (-1.0 / B));
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-2.8d-109)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 11500.0d0) then
        tmp = x * ((b * 0.3333333333333333d0) + ((-1.0d0) / b))
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.8e-109) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 11500.0) {
		tmp = x * ((B * 0.3333333333333333) + (-1.0 / B));
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.8e-109:
		tmp = (-1.0 - x) / B
	elif F <= 11500.0:
		tmp = x * ((B * 0.3333333333333333) + (-1.0 / B))
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.8e-109)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 11500.0)
		tmp = Float64(x * Float64(Float64(B * 0.3333333333333333) + Float64(-1.0 / B)));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.8e-109)
		tmp = (-1.0 - x) / B;
	elseif (F <= 11500.0)
		tmp = x * ((B * 0.3333333333333333) + (-1.0 / B));
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.8e-109], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 11500.0], N[(x * N[(N[(B * 0.3333333333333333), $MachinePrecision] + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.79999999999999979e-109

   1. Initial program 70.0%

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

   3. Taylor expanded in F around -inf 85.5%

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

   4. Taylor expanded in B around 0 40.7%

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

      1. associate-*r/ 40.7%

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

      2. mul-1-neg 40.7%

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

   6. Simplified 40.7%

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

   if -2.79999999999999979e-109 < F < 11500

   1. Initial program 99.5%

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

   3. Taylor expanded in B around 0 60.1%

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

   4. Taylor expanded in F around inf 21.4%

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

   5. Taylor expanded in x around -inf 42.2%

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(\frac{1}{B} - 0.3333333333333333 \cdot B\right)\right)} \]

   if 11500 < F

   1. Initial program 53.9%

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

   3. Taylor expanded in B around 0 28.2%

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

   4. Taylor expanded in F around inf 64.3%

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

   5. Taylor expanded in x around 0 58.4%

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

4. Final simplification 46.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.8 \cdot 10^{-109}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 11500:\\ \;\;\;\;x \cdot \left(B \cdot 0.3333333333333333 + \frac{-1}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 44.1% accurate, 17.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.8 \cdot 10^{-109}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{-15}:\\ \;\;\;\;x \cdot \left(B \cdot 0.3333333333333333 + \frac{-1}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.8e-109)
   (/ (- -1.0 x) B)
   (if (<= F 1.4e-15)
     (* x (+ (* B 0.3333333333333333) (/ -1.0 B)))
     (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.8e-109) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.4e-15) {
		tmp = x * ((B * 0.3333333333333333) + (-1.0 / 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 <= (-2.8d-109)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 1.4d-15) then
        tmp = x * ((b * 0.3333333333333333d0) + ((-1.0d0) / 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 <= -2.8e-109) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.4e-15) {
		tmp = x * ((B * 0.3333333333333333) + (-1.0 / B));
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.8e-109:
		tmp = (-1.0 - x) / B
	elif F <= 1.4e-15:
		tmp = x * ((B * 0.3333333333333333) + (-1.0 / B))
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.8e-109)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 1.4e-15)
		tmp = Float64(x * Float64(Float64(B * 0.3333333333333333) + Float64(-1.0 / 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 <= -2.8e-109)
		tmp = (-1.0 - x) / B;
	elseif (F <= 1.4e-15)
		tmp = x * ((B * 0.3333333333333333) + (-1.0 / B));
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.8e-109], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.4e-15], N[(x * N[(N[(B * 0.3333333333333333), $MachinePrecision] + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.79999999999999979e-109

   1. Initial program 70.0%

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

   3. Taylor expanded in F around -inf 85.5%

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

   4. Taylor expanded in B around 0 40.7%

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

      1. associate-*r/ 40.7%

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

      2. mul-1-neg 40.7%

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

   6. Simplified 40.7%

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

   if -2.79999999999999979e-109 < F < 1.40000000000000007e-15

   1. Initial program 99.5%

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

   3. Taylor expanded in B around 0 59.5%

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

   4. Taylor expanded in F around inf 20.1%

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

   5. Taylor expanded in x around -inf 41.9%

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(\frac{1}{B} - 0.3333333333333333 \cdot B\right)\right)} \]

   if 1.40000000000000007e-15 < F

   1. Initial program 56.3%

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

   3. Simplified 76.4%

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

   5. Taylor expanded in F around inf 98.4%

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

      1. associate-/r* 98.4%

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

   7. Simplified 98.4%

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

   8. Taylor expanded in B around 0 46.6%

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

4. Final simplification 42.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.8 \cdot 10^{-109}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{-15}:\\ \;\;\;\;x \cdot \left(B \cdot 0.3333333333333333 + \frac{-1}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 44.3% accurate, 21.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.9 \cdot 10^{-58}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.1 \cdot 10^{-62}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.9e-58)
   (/ (- -1.0 x) B)
   (if (<= F 1.1e-62) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.9e-58) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.1e-62) {
		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 <= (-3.9d-58)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 1.1d-62) 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 <= -3.9e-58) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.1e-62) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.9e-58:
		tmp = (-1.0 - x) / B
	elif F <= 1.1e-62:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.9e-58)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 1.1e-62)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -3.9e-58)
		tmp = (-1.0 - x) / B;
	elseif (F <= 1.1e-62)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.9e-58], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.1e-62], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.89999999999999992e-58

   1. Initial program 66.8%

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

   3. Taylor expanded in F around -inf 92.2%

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

   4. Taylor expanded in B around 0 43.3%

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

      1. associate-*r/ 43.3%

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

      2. mul-1-neg 43.3%

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

   6. Simplified 43.3%

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

   if -3.89999999999999992e-58 < F < 1.10000000000000009e-62

   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 inf 24.9%

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

      1. associate-/r* 24.9%

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

   7. Simplified 24.9%

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

   8. Taylor expanded in B around 0 18.3%

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

   9. Taylor expanded in x around inf 40.3%

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

       1. neg-mul-1 40.3%

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

       2. distribute-neg-frac 40.3%

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

   11. Simplified 40.3%

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

   if 1.10000000000000009e-62 < F

   1. Initial program 61.0%

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

   3. Simplified 78.9%

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

   5. Taylor expanded in F around inf 92.0%

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

      1. associate-/r* 91.9%

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

   7. Simplified 91.9%

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

   8. Taylor expanded in B around 0 45.2%

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

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.9 \cdot 10^{-58}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.1 \cdot 10^{-62}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 37.5% accurate, 32.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 3.1 \cdot 10^{-63}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F 3.1e-63) (/ (- x) B) (/ (- 1.0 x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= 3.1e-63) {
		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 <= 3.1d-63) 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 <= 3.1e-63) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= 3.1e-63:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= 3.1e-63)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= 3.1e-63)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, 3.1e-63], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 3.09999999999999984e-63

   1. Initial program 85.3%

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

   3. Simplified 89.6%

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

   5. Taylor expanded in F around inf 34.0%

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

      1. associate-/r* 34.0%

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

   7. Simplified 34.0%

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

   8. Taylor expanded in B around 0 19.9%

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

   9. Taylor expanded in x around inf 32.6%

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

       1. neg-mul-1 32.6%

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

       2. distribute-neg-frac 32.6%

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

   11. Simplified 32.6%

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

   if 3.09999999999999984e-63 < F

   1. Initial program 61.0%

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

   3. Simplified 78.9%

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

   5. Taylor expanded in F around inf 92.0%

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

      1. associate-/r* 91.9%

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

   7. Simplified 91.9%

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

   8. Taylor expanded in B around 0 45.2%

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

4. Final simplification 36.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 3.1 \cdot 10^{-63}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 30.5% accurate, 81.0× speedup

Math

\[\begin{array}{l} \\ \frac{-x}{B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (/ (- x) B))

C

double code(double F, double B, double x) {
	return -x / B;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = -x / b
end function

Java

public static double code(double F, double B, double x) {
	return -x / B;
}

Python

def code(F, B, x):
	return -x / B

Julia

function code(F, B, x)
	return Float64(Float64(-x) / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = -x / B;
end

Wolfram

code[F_, B_, x_] := N[((-x) / B), $MachinePrecision]

Derivation

1. Initial program 77.4%

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

3. Simplified 86.1%

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

5. Taylor expanded in F around inf 52.8%

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

   1. associate-/r* 52.8%

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

7. Simplified 52.8%

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

8. Taylor expanded in B around 0 28.1%

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

9. Taylor expanded in x around inf 30.9%

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

    1. neg-mul-1 30.9%

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

    2. distribute-neg-frac 30.9%

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

11. Simplified 30.9%

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

12. Final simplification 30.9%

    \[\leadsto \frac{-x}{B} \]
13. Add Preprocessing

Alternative 26: 9.7% accurate, 108.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (/ 1.0 B))

C

double code(double F, double B, double x) {
	return 1.0 / B;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = 1.0d0 / b
end function

Java

public static double code(double F, double B, double x) {
	return 1.0 / B;
}

Python

def code(F, B, x):
	return 1.0 / B

Julia

function code(F, B, x)
	return Float64(1.0 / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = 1.0 / B;
end

Wolfram

code[F_, B_, x_] := N[(1.0 / B), $MachinePrecision]

Derivation

1. Initial program 77.4%

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

3. Simplified 86.1%

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

5. Taylor expanded in F around inf 52.8%

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

   1. associate-/r* 52.8%

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

7. Simplified 52.8%

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

8. Taylor expanded in B around 0 28.1%

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

9. Taylor expanded in x around 0 8.7%

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

10. Final simplification 8.7%

    \[\leadsto \frac{1}{B} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024011 
(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))))))