VandenBroeck and Keller, Equation (23)

Percentage Accurate: 75.6% → 99.6%

Time: 17.4s

Alternatives: 21

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: 75.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2 \cdot 10^{+41}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1000:\\ \;\;\;\;F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -2e+41)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1000.0)
       (- (* 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 <= -2e+41) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1000.0) {
		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 <= -2e+41)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1000.0)
		tmp = Float64(Float64(F * Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) / sin(B))) - t_0);
	else
		tmp = Float64(Float64(1.0 - 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, -2e+41], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1000.0], N[(N[(F * N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 - N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.00000000000000001e41

   1. Initial program 53.4%

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

   3. Simplified 71.8%

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

   5. Taylor expanded in F around -inf 99.8%

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

   if -2.00000000000000001e41 < F < 1e3

   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 99.6%

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

   if 1e3 < F

   1. Initial program 56.2%

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

   3. Simplified 69.8%

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

      1. fma-define 69.8%

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

      2. fma-undefine 69.8%

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

      3. *-commutative 69.8%

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

      4. +-commutative 69.8%

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

      5. add-sqr-sqrt 69.8%

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

      6. metadata-eval 69.8%

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

      7. metadata-eval 69.8%

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

      8. unpow-prod-down 69.6%

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

      9. +-commutative 69.6%

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

      10. fma-define 69.6%

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

      11. fma-define 69.6%

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

      12. metadata-eval 69.6%

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

      13. metadata-eval 69.6%

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

      14. +-commutative 69.6%

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

      15. fma-define 69.6%

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

      16. fma-define 69.6%

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

      17. metadata-eval 69.6%

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

   6. Applied egg-rr 69.6%

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

      1. pow-sqr 69.8%

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

      2. metadata-eval 69.8%

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

      3. unpow-1 69.8%

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

      4. fma-undefine 69.8%

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

      5. *-commutative 69.8%

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

      6. fma-undefine 69.8%

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

      7. unpow2 69.8%

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

      8. +-commutative 69.8%

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

      9. fma-define 69.8%

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

      10. +-commutative 69.8%

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

      11. unpow2 69.8%

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

      12. fma-undefine 69.8%

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

   8. Simplified 69.8%

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

   9. Taylor expanded in F around inf 99.8%

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

       1. div-sub 99.8%

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

   11. Simplified 99.8%

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

Alternative 2: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.25 \cdot 10^{+73}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 10000000000:\\ \;\;\;\;F \cdot \frac{\frac{1}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.25e+73)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 10000000000.0)
       (- (* F (/ (/ 1.0 (sqrt (fma F F 2.0))) (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 <= -1.25e+73) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 10000000000.0) {
		tmp = (F * ((1.0 / sqrt(fma(F, F, 2.0))) / 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 <= -1.25e+73)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 10000000000.0)
		tmp = Float64(Float64(F * Float64(Float64(1.0 / sqrt(fma(F, F, 2.0))) / 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, -1.25e+73], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 10000000000.0], N[(N[(F * N[(N[(1.0 / N[Sqrt[N[(F * F + 2.0), $MachinePrecision]], $MachinePrecision]), $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 < -1.24999999999999994e73

   1. Initial program 50.6%

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

   3. Simplified 68.8%

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

   5. Taylor expanded in F around -inf 99.8%

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

   if -1.24999999999999994e73 < F < 1e10

   1. Initial program 98.6%

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

   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. fma-define 99.6%

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

      2. fma-undefine 99.6%

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

      3. *-commutative 99.6%

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

      4. +-commutative 99.6%

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

      5. add-sqr-sqrt 99.5%

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

      6. metadata-eval 99.5%

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

      7. metadata-eval 99.5%

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

      8. unpow-prod-down 99.6%

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

      9. +-commutative 99.6%

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

      10. fma-define 99.6%

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

      11. fma-define 99.6%

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

      12. metadata-eval 99.6%

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

      13. metadata-eval 99.6%

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

      14. +-commutative 99.6%

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

      15. fma-define 99.6%

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

      16. fma-define 99.6%

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

      17. metadata-eval 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. pow-sqr 99.5%

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

      2. metadata-eval 99.5%

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

      3. unpow-1 99.6%

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

      4. fma-undefine 99.6%

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

      5. *-commutative 99.6%

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

      6. fma-undefine 99.6%

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

      7. unpow2 99.6%

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

      8. +-commutative 99.6%

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

      9. fma-define 99.6%

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

      10. +-commutative 99.6%

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

      11. unpow2 99.6%

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

      12. fma-undefine 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in x around 0 99.6%

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

       1. +-commutative 99.6%

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

       2. unpow2 99.6%

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

       3. fma-undefine 99.6%

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

   11. Simplified 99.6%

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

   if 1e10 < F

   1. Initial program 56.2%

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

   3. Simplified 69.8%

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

      1. fma-define 69.8%

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

      2. fma-undefine 69.8%

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

      3. *-commutative 69.8%

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

      4. +-commutative 69.8%

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

      5. add-sqr-sqrt 69.8%

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

      6. metadata-eval 69.8%

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

      7. metadata-eval 69.8%

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

      8. unpow-prod-down 69.6%

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

      9. +-commutative 69.6%

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

      10. fma-define 69.6%

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

      11. fma-define 69.6%

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

      12. metadata-eval 69.6%

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

      13. metadata-eval 69.6%

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

      14. +-commutative 69.6%

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

      15. fma-define 69.6%

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

      16. fma-define 69.6%

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

      17. metadata-eval 69.6%

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

   6. Applied egg-rr 69.6%

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

      1. pow-sqr 69.8%

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

      2. metadata-eval 69.8%

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

      3. unpow-1 69.8%

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

      4. fma-undefine 69.8%

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

      5. *-commutative 69.8%

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

      6. fma-undefine 69.8%

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

      7. unpow2 69.8%

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

      8. +-commutative 69.8%

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

      9. fma-define 69.8%

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

      10. +-commutative 69.8%

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

      11. unpow2 69.8%

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

      12. fma-undefine 69.8%

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

   8. Simplified 69.8%

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

   9. Taylor expanded in F around inf 99.8%

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

       1. div-sub 99.8%

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

   11. Simplified 99.8%

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

Alternative 3: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{+23}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 10^{+16}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2e+23)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 1e+16)
     (+
      (* x (/ -1.0 (tan B)))
      (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)))
     (/ (- 1.0 (* x (cos B))) (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2e+23) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 1e+16) {
		tmp = (x * (-1.0 / tan(B))) + ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -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 <= (-2d+23)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 1d+16) then
        tmp = (x * ((-1.0d0) / tan(b))) + ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-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 <= -2e+23) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 1e+16) {
		tmp = (x * (-1.0 / Math.tan(B))) + ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5));
	} else {
		tmp = (1.0 - (x * Math.cos(B))) / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2e+23:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 1e+16:
		tmp = (x * (-1.0 / math.tan(B))) + ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5))
	else:
		tmp = (1.0 - (x * math.cos(B))) / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2e+23)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 1e+16)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)));
	else
		tmp = Float64(Float64(1.0 - Float64(x * cos(B))) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2e+23)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 1e+16)
		tmp = (x * (-1.0 / tan(B))) + ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5));
	else
		tmp = (1.0 - (x * cos(B))) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2e+23], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1e+16], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.9999999999999998e23

   1. Initial program 57.5%

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

   3. Simplified 74.2%

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

   5. Taylor expanded in F around -inf 99.8%

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

   if -1.9999999999999998e23 < F < 1e16

   1. Initial program 99.4%

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

      1. metadata-eval 99.4%

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

      2. metadata-eval 99.4%

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

   4. Applied egg-rr 99.4%

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

   if 1e16 < F

   1. Initial program 55.1%

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

   3. Simplified 69.1%

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

      1. fma-define 69.1%

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

      2. fma-undefine 69.1%

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

      3. *-commutative 69.1%

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

      4. +-commutative 69.1%

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

      5. add-sqr-sqrt 69.1%

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

      6. metadata-eval 69.1%

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

      7. metadata-eval 69.1%

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

      8. unpow-prod-down 68.9%

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

      9. +-commutative 68.9%

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

      10. fma-define 68.9%

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

      11. fma-define 68.9%

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

      12. metadata-eval 68.9%

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

      13. metadata-eval 68.9%

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

      14. +-commutative 68.9%

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

      15. fma-define 68.9%

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

      16. fma-define 68.9%

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

      17. metadata-eval 68.9%

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

   6. Applied egg-rr 68.9%

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

      1. pow-sqr 69.1%

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

      2. metadata-eval 69.1%

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

      3. unpow-1 69.1%

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

      4. fma-undefine 69.1%

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

      5. *-commutative 69.1%

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

      6. fma-undefine 69.1%

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

      7. unpow2 69.1%

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

      8. +-commutative 69.1%

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

      9. fma-define 69.1%

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

      10. +-commutative 69.1%

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

      11. unpow2 69.1%

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

      12. fma-undefine 69.1%

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

   8. Simplified 69.1%

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

   9. Taylor expanded in F around inf 99.8%

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

       1. div-sub 99.8%

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

   11. 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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{+23}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 10^{+16}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.45:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.45)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.4)
       (- (* 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 <= -1.45) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.4) {
		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 <= (-1.45d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.4d0) 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 <= -1.45) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.4) {
		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 <= -1.45:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.4:
		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 <= -1.45)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.4)
		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 <= -1.45)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.4)
		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, -1.45], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.4], 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 < -1.44999999999999996

   1. Initial program 58.9%

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

   3. Simplified 75.1%

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

   5. Taylor expanded in F around -inf 98.5%

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

   if -1.44999999999999996 < F < 1.3999999999999999

   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 99.6%

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

      1. fma-define 99.6%

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

      2. fma-undefine 99.6%

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

      3. *-commutative 99.6%

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

      4. +-commutative 99.6%

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

      5. add-sqr-sqrt 99.5%

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

      6. metadata-eval 99.5%

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

      7. metadata-eval 99.5%

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

      8. unpow-prod-down 99.5%

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

      9. +-commutative 99.5%

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

      10. fma-define 99.5%

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

      11. fma-define 99.5%

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

      12. metadata-eval 99.5%

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

      13. metadata-eval 99.5%

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

      14. +-commutative 99.5%

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

      15. fma-define 99.5%

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

      16. fma-define 99.5%

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

      17. metadata-eval 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. pow-sqr 99.5%

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

      2. metadata-eval 99.5%

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

      3. unpow-1 99.5%

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

      4. fma-undefine 99.5%

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

      5. *-commutative 99.5%

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

      6. fma-undefine 99.5%

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

      7. unpow2 99.5%

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

      8. +-commutative 99.5%

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

      9. fma-define 99.5%

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

      10. +-commutative 99.5%

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

      11. unpow2 99.5%

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

      12. fma-undefine 99.5%

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

   8. Simplified 99.5%

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

   9. Taylor expanded in x around 0 99.5%

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

       1. +-commutative 99.5%

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

       2. unpow2 99.5%

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

       3. fma-undefine 99.5%

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

   11. Simplified 99.5%

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

   12. Taylor expanded in F around 0 98.9%

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

       1. associate-/l* 99.0%

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

   14. Simplified 99.0%

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

   if 1.3999999999999999 < F

   1. Initial program 57.2%

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

   3. Simplified 70.5%

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

      1. fma-define 70.5%

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

      2. fma-undefine 70.5%

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

      3. *-commutative 70.5%

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

      4. +-commutative 70.5%

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

      5. add-sqr-sqrt 70.5%

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

      6. metadata-eval 70.5%

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

      7. metadata-eval 70.5%

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

      8. unpow-prod-down 70.3%

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

      9. +-commutative 70.3%

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

      10. fma-define 70.3%

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

      11. fma-define 70.3%

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

      12. metadata-eval 70.3%

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

      13. metadata-eval 70.3%

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

      14. +-commutative 70.3%

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

      15. fma-define 70.3%

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

      16. fma-define 70.3%

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

      17. metadata-eval 70.3%

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

   6. Applied egg-rr 70.3%

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

      1. pow-sqr 70.4%

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

      2. metadata-eval 70.4%

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

      3. unpow-1 70.4%

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

      4. fma-undefine 70.4%

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

      5. *-commutative 70.4%

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

      6. fma-undefine 70.4%

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

      7. unpow2 70.4%

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

      8. +-commutative 70.4%

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

      9. fma-define 70.4%

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

      10. +-commutative 70.4%

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

      11. unpow2 70.4%

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

      12. fma-undefine 70.4%

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

   8. Simplified 70.4%

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

   9. Taylor expanded in F around inf 98.4%

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

       1. div-sub 98.4%

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

   11. Simplified 98.4%

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

Alternative 5: 87.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos B\\ t_1 := {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{if}\;F \leq -2.8 \cdot 10^{-50}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -4 \cdot 10^{-229}:\\ \;\;\;\;t\_1 \cdot \left(F \cdot \frac{1}{\sin B}\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{-80}:\\ \;\;\;\;\frac{t\_0}{-\sin B}\\ \mathbf{elif}\;F \leq 800000:\\ \;\;\;\;t\_1 \cdot \frac{1}{\frac{\sin B}{F}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (cos B))) (t_1 (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)))
   (if (<= F -2.8e-50)
     (- (/ -1.0 (sin B)) (/ x (tan B)))
     (if (<= F -4e-229)
       (- (* t_1 (* F (/ 1.0 (sin B)))) (/ x B))
       (if (<= F 4.5e-80)
         (/ t_0 (- (sin B)))
         (if (<= F 800000.0)
           (- (* t_1 (/ 1.0 (/ (sin B) F))) (/ x B))
           (/ (- 1.0 t_0) (sin B))))))))

C

double code(double F, double B, double x) {
	double t_0 = x * cos(B);
	double t_1 = pow(((2.0 + (F * F)) + (x * 2.0)), -0.5);
	double tmp;
	if (F <= -2.8e-50) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= -4e-229) {
		tmp = (t_1 * (F * (1.0 / sin(B)))) - (x / B);
	} else if (F <= 4.5e-80) {
		tmp = t_0 / -sin(B);
	} else if (F <= 800000.0) {
		tmp = (t_1 * (1.0 / (sin(B) / F))) - (x / 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) :: t_1
    real(8) :: tmp
    t_0 = x * cos(b)
    t_1 = ((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)
    if (f <= (-2.8d-50)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= (-4d-229)) then
        tmp = (t_1 * (f * (1.0d0 / sin(b)))) - (x / b)
    else if (f <= 4.5d-80) then
        tmp = t_0 / -sin(b)
    else if (f <= 800000.0d0) then
        tmp = (t_1 * (1.0d0 / (sin(b) / f))) - (x / 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 t_1 = Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5);
	double tmp;
	if (F <= -2.8e-50) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= -4e-229) {
		tmp = (t_1 * (F * (1.0 / Math.sin(B)))) - (x / B);
	} else if (F <= 4.5e-80) {
		tmp = t_0 / -Math.sin(B);
	} else if (F <= 800000.0) {
		tmp = (t_1 * (1.0 / (Math.sin(B) / F))) - (x / B);
	} else {
		tmp = (1.0 - t_0) / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * math.cos(B)
	t_1 = math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)
	tmp = 0
	if F <= -2.8e-50:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= -4e-229:
		tmp = (t_1 * (F * (1.0 / math.sin(B)))) - (x / B)
	elif F <= 4.5e-80:
		tmp = t_0 / -math.sin(B)
	elif F <= 800000.0:
		tmp = (t_1 * (1.0 / (math.sin(B) / F))) - (x / B)
	else:
		tmp = (1.0 - t_0) / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * cos(B))
	t_1 = Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5
	tmp = 0.0
	if (F <= -2.8e-50)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= -4e-229)
		tmp = Float64(Float64(t_1 * Float64(F * Float64(1.0 / sin(B)))) - Float64(x / B));
	elseif (F <= 4.5e-80)
		tmp = Float64(t_0 / Float64(-sin(B)));
	elseif (F <= 800000.0)
		tmp = Float64(Float64(t_1 * Float64(1.0 / Float64(sin(B) / F))) - Float64(x / 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);
	t_1 = ((2.0 + (F * F)) + (x * 2.0)) ^ -0.5;
	tmp = 0.0;
	if (F <= -2.8e-50)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= -4e-229)
		tmp = (t_1 * (F * (1.0 / sin(B)))) - (x / B);
	elseif (F <= 4.5e-80)
		tmp = t_0 / -sin(B);
	elseif (F <= 800000.0)
		tmp = (t_1 * (1.0 / (sin(B) / F))) - (x / 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]}, Block[{t$95$1 = N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]}, If[LessEqual[F, -2.8e-50], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -4e-229], N[(N[(t$95$1 * N[(F * N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.5e-80], N[(t$95$0 / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 800000.0], N[(N[(t$95$1 * N[(1.0 / N[(N[Sin[B], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -2.7999999999999998e-50

   1. Initial program 64.3%

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

   3. Simplified 78.3%

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

   5. Taylor expanded in F around -inf 94.6%

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

   if -2.7999999999999998e-50 < F < -4.00000000000000028e-229

   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 90.4%

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

      1. associate-*r/ 90.4%

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

      2. neg-mul-1 90.4%

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

   5. Simplified 90.4%

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

      1. div-inv 90.4%

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

   7. Applied egg-rr 90.4%

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

   if -4.00000000000000028e-229 < F < 4.5000000000000003e-80

   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. Step-by-step derivation

      1. fma-define 99.7%

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

      2. fma-undefine 99.7%

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

      3. *-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. add-sqr-sqrt 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. unpow-prod-down 99.6%

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

      9. +-commutative 99.6%

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

      10. fma-define 99.6%

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

      11. fma-define 99.6%

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

      12. metadata-eval 99.6%

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

      13. metadata-eval 99.6%

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

      14. +-commutative 99.6%

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

      15. fma-define 99.6%

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

      16. fma-define 99.6%

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

      17. metadata-eval 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. pow-sqr 99.6%

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

      2. metadata-eval 99.6%

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

      3. unpow-1 99.6%

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

      4. fma-undefine 99.6%

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

      5. *-commutative 99.6%

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

      6. fma-undefine 99.6%

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

      7. unpow2 99.6%

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

      8. +-commutative 99.6%

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

      9. fma-define 99.6%

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

      10. +-commutative 99.6%

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

      11. unpow2 99.6%

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

      12. fma-undefine 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in F around 0 79.8%

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

       1. associate-*r/ 79.8%

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

       2. neg-mul-1 79.8%

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

       3. distribute-rgt-neg-in 79.8%

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

   11. Simplified 79.8%

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

   if 4.5000000000000003e-80 < F < 8e5

   1. Initial program 98.9%

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

   3. Taylor expanded in B around 0 89.3%

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

      1. associate-*r/ 89.3%

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

      2. neg-mul-1 89.3%

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

   5. Simplified 89.3%

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

      1. clear-num 89.5%

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

      2. inv-pow 89.5%

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

   7. Applied egg-rr 89.5%

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

      1. unpow-1 89.5%

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

   9. Simplified 89.5%

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

       1. metadata-eval 98.9%

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

       2. metadata-eval 98.9%

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

   11. Applied egg-rr 89.5%

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

   if 8e5 < F

   1. Initial program 56.2%

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

   3. Simplified 69.8%

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

      1. fma-define 69.8%

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

      2. fma-undefine 69.8%

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

      3. *-commutative 69.8%

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

      4. +-commutative 69.8%

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

      5. add-sqr-sqrt 69.8%

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

      6. metadata-eval 69.8%

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

      7. metadata-eval 69.8%

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

      8. unpow-prod-down 69.6%

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

      9. +-commutative 69.6%

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

      10. fma-define 69.6%

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

      11. fma-define 69.6%

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

      12. metadata-eval 69.6%

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

      13. metadata-eval 69.6%

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

      14. +-commutative 69.6%

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

      15. fma-define 69.6%

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

      16. fma-define 69.6%

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

      17. metadata-eval 69.6%

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

   6. Applied egg-rr 69.6%

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

      1. pow-sqr 69.8%

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

      2. metadata-eval 69.8%

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

      3. unpow-1 69.8%

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

      4. fma-undefine 69.8%

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

      5. *-commutative 69.8%

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

      6. fma-undefine 69.8%

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

      7. unpow2 69.8%

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

      8. +-commutative 69.8%

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

      9. fma-define 69.8%

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

      10. +-commutative 69.8%

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

      11. unpow2 69.8%

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

      12. fma-undefine 69.8%

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

   8. Simplified 69.8%

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

   9. Taylor expanded in F around inf 99.8%

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

       1. div-sub 99.8%

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

   11. Simplified 99.8%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.8 \cdot 10^{-50}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -4 \cdot 10^{-229}:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \left(F \cdot \frac{1}{\sin B}\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{-80}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{elif}\;F \leq 800000:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{1}{\frac{\sin B}{F}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos B\\ \mathbf{if}\;F \leq -2.8 \cdot 10^{-50}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.3 \cdot 10^{-228}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.05 \cdot 10^{-80}:\\ \;\;\;\;\frac{t\_0}{-\sin B}\\ \mathbf{elif}\;F \leq 250000:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{1}{\frac{\sin B}{F}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (cos B))))
   (if (<= F -2.8e-50)
     (- (/ -1.0 (sin B)) (/ x (tan B)))
     (if (<= F -1.3e-228)
       (- (* (/ F (sin B)) (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))) (/ x B))
       (if (<= F 3.05e-80)
         (/ t_0 (- (sin B)))
         (if (<= F 250000.0)
           (-
            (* (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5) (/ 1.0 (/ (sin B) F)))
            (/ x 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 <= -2.8e-50) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= -1.3e-228) {
		tmp = ((F / sin(B)) * sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B);
	} else if (F <= 3.05e-80) {
		tmp = t_0 / -sin(B);
	} else if (F <= 250000.0) {
		tmp = (pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (1.0 / (sin(B) / F))) - (x / 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 <= (-2.8d-50)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= (-1.3d-228)) then
        tmp = ((f / sin(b)) * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))) - (x / b)
    else if (f <= 3.05d-80) then
        tmp = t_0 / -sin(b)
    else if (f <= 250000.0d0) then
        tmp = ((((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)) * (1.0d0 / (sin(b) / f))) - (x / 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 <= -2.8e-50) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= -1.3e-228) {
		tmp = ((F / Math.sin(B)) * Math.sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B);
	} else if (F <= 3.05e-80) {
		tmp = t_0 / -Math.sin(B);
	} else if (F <= 250000.0) {
		tmp = (Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (1.0 / (Math.sin(B) / F))) - (x / 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 <= -2.8e-50:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= -1.3e-228:
		tmp = ((F / math.sin(B)) * math.sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B)
	elif F <= 3.05e-80:
		tmp = t_0 / -math.sin(B)
	elif F <= 250000.0:
		tmp = (math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (1.0 / (math.sin(B) / F))) - (x / 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 <= -2.8e-50)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= -1.3e-228)
		tmp = Float64(Float64(Float64(F / sin(B)) * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))) - Float64(x / B));
	elseif (F <= 3.05e-80)
		tmp = Float64(t_0 / Float64(-sin(B)));
	elseif (F <= 250000.0)
		tmp = Float64(Float64((Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5) * Float64(1.0 / Float64(sin(B) / F))) - Float64(x / 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 <= -2.8e-50)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= -1.3e-228)
		tmp = ((F / sin(B)) * sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B);
	elseif (F <= 3.05e-80)
		tmp = t_0 / -sin(B);
	elseif (F <= 250000.0)
		tmp = ((((2.0 + (F * F)) + (x * 2.0)) ^ -0.5) * (1.0 / (sin(B) / F))) - (x / 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, -2.8e-50], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.3e-228], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.05e-80], N[(t$95$0 / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 250000.0], N[(N[(N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(1.0 / N[(N[Sin[B], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -2.7999999999999998e-50

   1. Initial program 64.3%

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

   3. Simplified 78.3%

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

   5. Taylor expanded in F around -inf 94.6%

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

   if -2.7999999999999998e-50 < F < -1.3e-228

   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 90.4%

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

      1. associate-*r/ 90.4%

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

      2. neg-mul-1 90.4%

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

   5. Simplified 90.4%

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

   6. Taylor expanded in F around 0 90.4%

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

      1. neg-mul-1 90.4%

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

      2. distribute-frac-neg 90.4%

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

      3. +-commutative 90.4%

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

      4. distribute-frac-neg 90.4%

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

      5. unsub-neg 90.4%

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

      6. *-commutative 90.4%

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

   8. Simplified 90.4%

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

   if -1.3e-228 < F < 3.0500000000000001e-80

   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. Step-by-step derivation

      1. fma-define 99.7%

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

      2. fma-undefine 99.7%

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

      3. *-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. add-sqr-sqrt 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. unpow-prod-down 99.6%

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

      9. +-commutative 99.6%

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

      10. fma-define 99.6%

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

      11. fma-define 99.6%

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

      12. metadata-eval 99.6%

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

      13. metadata-eval 99.6%

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

      14. +-commutative 99.6%

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

      15. fma-define 99.6%

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

      16. fma-define 99.6%

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

      17. metadata-eval 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. pow-sqr 99.6%

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

      2. metadata-eval 99.6%

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

      3. unpow-1 99.6%

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

      4. fma-undefine 99.6%

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

      5. *-commutative 99.6%

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

      6. fma-undefine 99.6%

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

      7. unpow2 99.6%

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

      8. +-commutative 99.6%

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

      9. fma-define 99.6%

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

      10. +-commutative 99.6%

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

      11. unpow2 99.6%

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

      12. fma-undefine 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in F around 0 79.8%

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

       1. associate-*r/ 79.8%

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

       2. neg-mul-1 79.8%

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

       3. distribute-rgt-neg-in 79.8%

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

   11. Simplified 79.8%

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

   if 3.0500000000000001e-80 < F < 2.5e5

   1. Initial program 98.9%

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

   3. Taylor expanded in B around 0 89.3%

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

      1. associate-*r/ 89.3%

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

      2. neg-mul-1 89.3%

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

   5. Simplified 89.3%

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

      1. clear-num 89.5%

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

      2. inv-pow 89.5%

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

   7. Applied egg-rr 89.5%

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

      1. unpow-1 89.5%

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

   9. Simplified 89.5%

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

       1. metadata-eval 98.9%

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

       2. metadata-eval 98.9%

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

   11. Applied egg-rr 89.5%

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

   if 2.5e5 < F

   1. Initial program 56.2%

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

   3. Simplified 69.8%

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

      1. fma-define 69.8%

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

      2. fma-undefine 69.8%

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

      3. *-commutative 69.8%

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

      4. +-commutative 69.8%

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

      5. add-sqr-sqrt 69.8%

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

      6. metadata-eval 69.8%

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

      7. metadata-eval 69.8%

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

      8. unpow-prod-down 69.6%

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

      9. +-commutative 69.6%

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

      10. fma-define 69.6%

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

      11. fma-define 69.6%

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

      12. metadata-eval 69.6%

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

      13. metadata-eval 69.6%

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

      14. +-commutative 69.6%

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

      15. fma-define 69.6%

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

      16. fma-define 69.6%

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

      17. metadata-eval 69.6%

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

   6. Applied egg-rr 69.6%

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

      1. pow-sqr 69.8%

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

      2. metadata-eval 69.8%

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

      3. unpow-1 69.8%

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

      4. fma-undefine 69.8%

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

      5. *-commutative 69.8%

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

      6. fma-undefine 69.8%

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

      7. unpow2 69.8%

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

      8. +-commutative 69.8%

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

      9. fma-define 69.8%

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

      10. +-commutative 69.8%

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

      11. unpow2 69.8%

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

      12. fma-undefine 69.8%

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

   8. Simplified 69.8%

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

   9. Taylor expanded in F around inf 99.8%

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

       1. div-sub 99.8%

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

   11. Simplified 99.8%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.8 \cdot 10^{-50}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.3 \cdot 10^{-228}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.05 \cdot 10^{-80}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{elif}\;F \leq 250000:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{1}{\frac{\sin B}{F}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos B\\ t_1 := \frac{F}{\sin B}\\ \mathbf{if}\;F \leq -2.8 \cdot 10^{-50}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.16 \cdot 10^{-228}:\\ \;\;\;\;t\_1 \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.05 \cdot 10^{-80}:\\ \;\;\;\;\frac{t\_0}{-\sin B}\\ \mathbf{elif}\;F \leq 94000:\\ \;\;\;\;t\_1 \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (cos B))) (t_1 (/ F (sin B))))
   (if (<= F -2.8e-50)
     (- (/ -1.0 (sin B)) (/ x (tan B)))
     (if (<= F -1.16e-228)
       (- (* t_1 (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))) (/ x B))
       (if (<= F 3.05e-80)
         (/ t_0 (- (sin B)))
         (if (<= F 94000.0)
           (- (* t_1 (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)) (/ x B))
           (/ (- 1.0 t_0) (sin B))))))))

C

double code(double F, double B, double x) {
	double t_0 = x * cos(B);
	double t_1 = F / sin(B);
	double tmp;
	if (F <= -2.8e-50) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= -1.16e-228) {
		tmp = (t_1 * sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B);
	} else if (F <= 3.05e-80) {
		tmp = t_0 / -sin(B);
	} else if (F <= 94000.0) {
		tmp = (t_1 * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / 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) :: t_1
    real(8) :: tmp
    t_0 = x * cos(b)
    t_1 = f / sin(b)
    if (f <= (-2.8d-50)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= (-1.16d-228)) then
        tmp = (t_1 * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))) - (x / b)
    else if (f <= 3.05d-80) then
        tmp = t_0 / -sin(b)
    else if (f <= 94000.0d0) then
        tmp = (t_1 * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0))) - (x / 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 t_1 = F / Math.sin(B);
	double tmp;
	if (F <= -2.8e-50) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= -1.16e-228) {
		tmp = (t_1 * Math.sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B);
	} else if (F <= 3.05e-80) {
		tmp = t_0 / -Math.sin(B);
	} else if (F <= 94000.0) {
		tmp = (t_1 * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B);
	} else {
		tmp = (1.0 - t_0) / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * math.cos(B)
	t_1 = F / math.sin(B)
	tmp = 0
	if F <= -2.8e-50:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= -1.16e-228:
		tmp = (t_1 * math.sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B)
	elif F <= 3.05e-80:
		tmp = t_0 / -math.sin(B)
	elif F <= 94000.0:
		tmp = (t_1 * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B)
	else:
		tmp = (1.0 - t_0) / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * cos(B))
	t_1 = Float64(F / sin(B))
	tmp = 0.0
	if (F <= -2.8e-50)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= -1.16e-228)
		tmp = Float64(Float64(t_1 * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))) - Float64(x / B));
	elseif (F <= 3.05e-80)
		tmp = Float64(t_0 / Float64(-sin(B)));
	elseif (F <= 94000.0)
		tmp = Float64(Float64(t_1 * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)) - Float64(x / 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);
	t_1 = F / sin(B);
	tmp = 0.0;
	if (F <= -2.8e-50)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= -1.16e-228)
		tmp = (t_1 * sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B);
	elseif (F <= 3.05e-80)
		tmp = t_0 / -sin(B);
	elseif (F <= 94000.0)
		tmp = (t_1 * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5)) - (x / 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]}, Block[{t$95$1 = N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.8e-50], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.16e-228], N[(N[(t$95$1 * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.05e-80], N[(t$95$0 / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 94000.0], N[(N[(t$95$1 * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -2.7999999999999998e-50

   1. Initial program 64.3%

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

   3. Simplified 78.3%

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

   5. Taylor expanded in F around -inf 94.6%

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

   if -2.7999999999999998e-50 < F < -1.16e-228

   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 90.4%

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

      1. associate-*r/ 90.4%

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

      2. neg-mul-1 90.4%

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

   5. Simplified 90.4%

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

   6. Taylor expanded in F around 0 90.4%

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

      1. neg-mul-1 90.4%

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

      2. distribute-frac-neg 90.4%

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

      3. +-commutative 90.4%

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

      4. distribute-frac-neg 90.4%

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

      5. unsub-neg 90.4%

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

      6. *-commutative 90.4%

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

   8. Simplified 90.4%

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

   if -1.16e-228 < F < 3.0500000000000001e-80

   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. Step-by-step derivation

      1. fma-define 99.7%

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

      2. fma-undefine 99.7%

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

      3. *-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. add-sqr-sqrt 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. unpow-prod-down 99.6%

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

      9. +-commutative 99.6%

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

      10. fma-define 99.6%

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

      11. fma-define 99.6%

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

      12. metadata-eval 99.6%

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

      13. metadata-eval 99.6%

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

      14. +-commutative 99.6%

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

      15. fma-define 99.6%

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

      16. fma-define 99.6%

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

      17. metadata-eval 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. pow-sqr 99.6%

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

      2. metadata-eval 99.6%

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

      3. unpow-1 99.6%

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

      4. fma-undefine 99.6%

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

      5. *-commutative 99.6%

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

      6. fma-undefine 99.6%

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

      7. unpow2 99.6%

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

      8. +-commutative 99.6%

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

      9. fma-define 99.6%

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

      10. +-commutative 99.6%

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

      11. unpow2 99.6%

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

      12. fma-undefine 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in F around 0 79.8%

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

       1. associate-*r/ 79.8%

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

       2. neg-mul-1 79.8%

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

       3. distribute-rgt-neg-in 79.8%

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

   11. Simplified 79.8%

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

   if 3.0500000000000001e-80 < F < 94000

   1. Initial program 98.9%

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

   3. Taylor expanded in B around 0 89.3%

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

      1. associate-*r/ 89.3%

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

      2. neg-mul-1 89.3%

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

   5. Simplified 89.3%

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

      1. metadata-eval 98.9%

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

      2. metadata-eval 98.9%

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

   7. Applied egg-rr 89.3%

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

   if 94000 < F

   1. Initial program 56.2%

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

   3. Simplified 69.8%

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

      1. fma-define 69.8%

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

      2. fma-undefine 69.8%

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

      3. *-commutative 69.8%

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

      4. +-commutative 69.8%

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

      5. add-sqr-sqrt 69.8%

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

      6. metadata-eval 69.8%

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

      7. metadata-eval 69.8%

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

      8. unpow-prod-down 69.6%

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

      9. +-commutative 69.6%

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

      10. fma-define 69.6%

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

      11. fma-define 69.6%

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

      12. metadata-eval 69.6%

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

      13. metadata-eval 69.6%

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

      14. +-commutative 69.6%

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

      15. fma-define 69.6%

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

      16. fma-define 69.6%

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

      17. metadata-eval 69.6%

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

   6. Applied egg-rr 69.6%

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

      1. pow-sqr 69.8%

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

      2. metadata-eval 69.8%

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

      3. unpow-1 69.8%

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

      4. fma-undefine 69.8%

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

      5. *-commutative 69.8%

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

      6. fma-undefine 69.8%

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

      7. unpow2 69.8%

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

      8. +-commutative 69.8%

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

      9. fma-define 69.8%

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

      10. +-commutative 69.8%

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

      11. unpow2 69.8%

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

      12. fma-undefine 69.8%

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

   8. Simplified 69.8%

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

   9. Taylor expanded in F around inf 99.8%

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

       1. div-sub 99.8%

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

   11. Simplified 99.8%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.8 \cdot 10^{-50}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.16 \cdot 10^{-228}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.05 \cdot 10^{-80}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{elif}\;F \leq 94000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 87.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos B\\ t_1 := \frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{if}\;F \leq -2.8 \cdot 10^{-50}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -3.4 \cdot 10^{-229}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;F \leq 3.05 \cdot 10^{-80}:\\ \;\;\;\;\frac{t\_0}{-\sin B}\\ \mathbf{elif}\;F \leq 0.5:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (cos B)))
        (t_1 (- (* (/ F (sin B)) (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))) (/ x B))))
   (if (<= F -2.8e-50)
     (- (/ -1.0 (sin B)) (/ x (tan B)))
     (if (<= F -3.4e-229)
       t_1
       (if (<= F 3.05e-80)
         (/ t_0 (- (sin B)))
         (if (<= F 0.5) t_1 (/ (- 1.0 t_0) (sin B))))))))

C

double code(double F, double B, double x) {
	double t_0 = x * cos(B);
	double t_1 = ((F / sin(B)) * sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B);
	double tmp;
	if (F <= -2.8e-50) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= -3.4e-229) {
		tmp = t_1;
	} else if (F <= 3.05e-80) {
		tmp = t_0 / -sin(B);
	} else if (F <= 0.5) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = x * cos(b)
    t_1 = ((f / sin(b)) * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))) - (x / b)
    if (f <= (-2.8d-50)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= (-3.4d-229)) then
        tmp = t_1
    else if (f <= 3.05d-80) then
        tmp = t_0 / -sin(b)
    else if (f <= 0.5d0) then
        tmp = t_1
    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 t_1 = ((F / Math.sin(B)) * Math.sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B);
	double tmp;
	if (F <= -2.8e-50) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= -3.4e-229) {
		tmp = t_1;
	} else if (F <= 3.05e-80) {
		tmp = t_0 / -Math.sin(B);
	} else if (F <= 0.5) {
		tmp = t_1;
	} else {
		tmp = (1.0 - t_0) / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * math.cos(B)
	t_1 = ((F / math.sin(B)) * math.sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B)
	tmp = 0
	if F <= -2.8e-50:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= -3.4e-229:
		tmp = t_1
	elif F <= 3.05e-80:
		tmp = t_0 / -math.sin(B)
	elif F <= 0.5:
		tmp = t_1
	else:
		tmp = (1.0 - t_0) / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * cos(B))
	t_1 = Float64(Float64(Float64(F / sin(B)) * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))) - Float64(x / B))
	tmp = 0.0
	if (F <= -2.8e-50)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= -3.4e-229)
		tmp = t_1;
	elseif (F <= 3.05e-80)
		tmp = Float64(t_0 / Float64(-sin(B)));
	elseif (F <= 0.5)
		tmp = t_1;
	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);
	t_1 = ((F / sin(B)) * sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B);
	tmp = 0.0;
	if (F <= -2.8e-50)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= -3.4e-229)
		tmp = t_1;
	elseif (F <= 3.05e-80)
		tmp = t_0 / -sin(B);
	elseif (F <= 0.5)
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.8e-50], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -3.4e-229], t$95$1, If[LessEqual[F, 3.05e-80], N[(t$95$0 / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 0.5], t$95$1, N[(N[(1.0 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.7999999999999998e-50

   1. Initial program 64.3%

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

   3. Simplified 78.3%

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

   5. Taylor expanded in F around -inf 94.6%

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

   if -2.7999999999999998e-50 < F < -3.3999999999999999e-229 or 3.0500000000000001e-80 < F < 0.5

   1. Initial program 99.3%

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

   3. Taylor expanded in B around 0 89.6%

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

      1. associate-*r/ 89.6%

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

      2. neg-mul-1 89.6%

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

   5. Simplified 89.6%

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

   6. Taylor expanded in F around 0 87.9%

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

      1. neg-mul-1 87.9%

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

      2. distribute-frac-neg 87.9%

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

      3. +-commutative 87.9%

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

      4. distribute-frac-neg 87.9%

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

      5. unsub-neg 87.9%

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

      6. *-commutative 87.9%

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

   8. Simplified 87.9%

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

   if -3.3999999999999999e-229 < F < 3.0500000000000001e-80

   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. Step-by-step derivation

      1. fma-define 99.7%

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

      2. fma-undefine 99.7%

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

      3. *-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. add-sqr-sqrt 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. unpow-prod-down 99.6%

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

      9. +-commutative 99.6%

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

      10. fma-define 99.6%

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

      11. fma-define 99.6%

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

      12. metadata-eval 99.6%

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

      13. metadata-eval 99.6%

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

      14. +-commutative 99.6%

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

      15. fma-define 99.6%

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

      16. fma-define 99.6%

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

      17. metadata-eval 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. pow-sqr 99.6%

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

      2. metadata-eval 99.6%

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

      3. unpow-1 99.6%

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

      4. fma-undefine 99.6%

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

      5. *-commutative 99.6%

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

      6. fma-undefine 99.6%

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

      7. unpow2 99.6%

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

      8. +-commutative 99.6%

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

      9. fma-define 99.6%

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

      10. +-commutative 99.6%

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

      11. unpow2 99.6%

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

      12. fma-undefine 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in F around 0 79.8%

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

       1. associate-*r/ 79.8%

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

       2. neg-mul-1 79.8%

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

       3. distribute-rgt-neg-in 79.8%

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

   11. Simplified 79.8%

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

   if 0.5 < F

   1. Initial program 57.2%

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

   3. Simplified 70.5%

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

      1. fma-define 70.5%

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

      2. fma-undefine 70.5%

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

      3. *-commutative 70.5%

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

      4. +-commutative 70.5%

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

      5. add-sqr-sqrt 70.5%

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

      6. metadata-eval 70.5%

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

      7. metadata-eval 70.5%

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

      8. unpow-prod-down 70.3%

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

      9. +-commutative 70.3%

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

      10. fma-define 70.3%

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

      11. fma-define 70.3%

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

      12. metadata-eval 70.3%

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

      13. metadata-eval 70.3%

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

      14. +-commutative 70.3%

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

      15. fma-define 70.3%

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

      16. fma-define 70.3%

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

      17. metadata-eval 70.3%

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

   6. Applied egg-rr 70.3%

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

      1. pow-sqr 70.4%

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

      2. metadata-eval 70.4%

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

      3. unpow-1 70.4%

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

      4. fma-undefine 70.4%

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

      5. *-commutative 70.4%

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

      6. fma-undefine 70.4%

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

      7. unpow2 70.4%

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

      8. +-commutative 70.4%

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

      9. fma-define 70.4%

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

      10. +-commutative 70.4%

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

      11. unpow2 70.4%

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

      12. fma-undefine 70.4%

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

   8. Simplified 70.4%

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

   9. Taylor expanded in F around inf 98.4%

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

       1. div-sub 98.4%

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

   11. Simplified 98.4%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.8 \cdot 10^{-50}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -3.4 \cdot 10^{-229}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.05 \cdot 10^{-80}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{elif}\;F \leq 0.5:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \cos B\\ \mathbf{if}\;F \leq -4.4 \cdot 10^{-88}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 6.4 \cdot 10^{-70}:\\ \;\;\;\;\frac{t\_0}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (cos B))))
   (if (<= F -4.4e-88)
     (- (/ -1.0 (sin B)) (/ x (tan B)))
     (if (<= F 6.4e-70) (/ 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 <= -4.4e-88) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 6.4e-70) {
		tmp = 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 <= (-4.4d-88)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 6.4d-70) then
        tmp = 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 <= -4.4e-88) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 6.4e-70) {
		tmp = 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 <= -4.4e-88:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 6.4e-70:
		tmp = 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 <= -4.4e-88)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 6.4e-70)
		tmp = Float64(t_0 / Float64(-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 <= -4.4e-88)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 6.4e-70)
		tmp = 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, -4.4e-88], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 6.4e-70], N[(t$95$0 / (-N[Sin[B], $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 < -4.4000000000000001e-88

   1. Initial program 68.4%

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

   3. Simplified 80.8%

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

   5. Taylor expanded in F around -inf 87.9%

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

   if -4.4000000000000001e-88 < F < 6.3999999999999995e-70

   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. Step-by-step derivation

      1. fma-define 99.7%

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

      2. fma-undefine 99.7%

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

      3. *-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. add-sqr-sqrt 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. unpow-prod-down 99.6%

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

      9. +-commutative 99.6%

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

      10. fma-define 99.6%

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

      11. fma-define 99.6%

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

      12. metadata-eval 99.6%

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

      13. metadata-eval 99.6%

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

      14. +-commutative 99.6%

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

      15. fma-define 99.6%

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

      16. fma-define 99.6%

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

      17. metadata-eval 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. pow-sqr 99.6%

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

      2. metadata-eval 99.6%

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

      3. unpow-1 99.6%

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

      4. fma-undefine 99.6%

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

      5. *-commutative 99.6%

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

      6. fma-undefine 99.6%

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

      7. unpow2 99.6%

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

      8. +-commutative 99.6%

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

      9. fma-define 99.6%

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

      10. +-commutative 99.6%

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

      11. unpow2 99.6%

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

      12. fma-undefine 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in F around 0 79.1%

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

       1. associate-*r/ 79.1%

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

       2. neg-mul-1 79.1%

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

       3. distribute-rgt-neg-in 79.1%

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

   11. Simplified 79.1%

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

   if 6.3999999999999995e-70 < F

   1. Initial program 62.7%

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

   3. Simplified 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. Step-by-step derivation

      1. fma-define 74.3%

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

      2. fma-undefine 74.3%

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

      3. *-commutative 74.3%

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

      4. +-commutative 74.3%

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

      5. add-sqr-sqrt 74.2%

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

      6. metadata-eval 74.2%

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

      7. metadata-eval 74.2%

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

      8. unpow-prod-down 74.1%

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

      9. +-commutative 74.1%

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

      10. fma-define 74.1%

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

      11. fma-define 74.1%

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

      12. metadata-eval 74.1%

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

      13. metadata-eval 74.1%

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

      14. +-commutative 74.1%

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

      15. fma-define 74.1%

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

      16. fma-define 74.1%

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

      17. metadata-eval 74.1%

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

   6. Applied egg-rr 74.1%

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

      1. pow-sqr 74.2%

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

      2. metadata-eval 74.2%

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

      3. unpow-1 74.2%

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

      4. fma-undefine 74.2%

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

      5. *-commutative 74.2%

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

      6. fma-undefine 74.2%

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

      7. unpow2 74.2%

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

      8. +-commutative 74.2%

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

      9. fma-define 74.2%

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

      10. +-commutative 74.2%

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

      11. unpow2 74.2%

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

      12. fma-undefine 74.2%

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

   8. Simplified 74.2%

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

   9. Taylor expanded in F around inf 90.3%

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

       1. div-sub 90.3%

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

   11. Simplified 90.3%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.4 \cdot 10^{-88}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 6.4 \cdot 10^{-70}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 83.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -4.4 \cdot 10^{-88}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 5.4 \cdot 10^{-70}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -4.4e-88)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 5.4e-70)
       (/ (* x (cos B)) (- (sin B)))
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -4.4e-88) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 5.4e-70) {
		tmp = (x * cos(B)) / -sin(B);
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-4.4d-88)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 5.4d-70) then
        tmp = (x * cos(b)) / -sin(b)
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -4.4e-88) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 5.4e-70) {
		tmp = (x * Math.cos(B)) / -Math.sin(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -4.4e-88:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 5.4e-70:
		tmp = (x * math.cos(B)) / -math.sin(B)
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -4.4e-88)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 5.4e-70)
		tmp = Float64(Float64(x * cos(B)) / Float64(-sin(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -4.4e-88)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 5.4e-70)
		tmp = (x * cos(B)) / -sin(B);
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -4.4e-88], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 5.4e-70], N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.4000000000000001e-88

   1. Initial program 68.4%

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

   3. Simplified 80.8%

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

   5. Taylor expanded in F around -inf 87.9%

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

   if -4.4000000000000001e-88 < F < 5.4000000000000003e-70

   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. Step-by-step derivation

      1. fma-define 99.7%

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

      2. fma-undefine 99.7%

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

      3. *-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. add-sqr-sqrt 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. unpow-prod-down 99.6%

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

      9. +-commutative 99.6%

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

      10. fma-define 99.6%

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

      11. fma-define 99.6%

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

      12. metadata-eval 99.6%

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

      13. metadata-eval 99.6%

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

      14. +-commutative 99.6%

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

      15. fma-define 99.6%

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

      16. fma-define 99.6%

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

      17. metadata-eval 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. pow-sqr 99.6%

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

      2. metadata-eval 99.6%

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

      3. unpow-1 99.6%

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

      4. fma-undefine 99.6%

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

      5. *-commutative 99.6%

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

      6. fma-undefine 99.6%

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

      7. unpow2 99.6%

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

      8. +-commutative 99.6%

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

      9. fma-define 99.6%

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

      10. +-commutative 99.6%

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

      11. unpow2 99.6%

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

      12. fma-undefine 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in F around 0 79.1%

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

       1. associate-*r/ 79.1%

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

       2. neg-mul-1 79.1%

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

       3. distribute-rgt-neg-in 79.1%

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

   11. Simplified 79.1%

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

   if 5.4000000000000003e-70 < F

   1. Initial program 62.7%

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

   3. Simplified 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 90.3%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.4 \cdot 10^{-88}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 5.4 \cdot 10^{-70}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 77.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -4.2 \cdot 10^{-88}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{-32}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4.2e-88)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 2.3e-32)
     (/ (* x (cos B)) (- (sin B)))
     (- (/ 1.0 (sin B)) (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.2e-88) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 2.3e-32) {
		tmp = (x * cos(B)) / -sin(B);
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-4.2d-88)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 2.3d-32) then
        tmp = (x * cos(b)) / -sin(b)
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.2e-88) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 2.3e-32) {
		tmp = (x * Math.cos(B)) / -Math.sin(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -4.2e-88:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 2.3e-32:
		tmp = (x * math.cos(B)) / -math.sin(B)
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -4.2e-88)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 2.3e-32)
		tmp = Float64(Float64(x * cos(B)) / Float64(-sin(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -4.2e-88)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 2.3e-32)
		tmp = (x * cos(B)) / -sin(B);
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -4.2e-88], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.3e-32], N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.1999999999999999e-88

   1. Initial program 68.4%

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

   3. Simplified 80.8%

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

   5. Taylor expanded in F around -inf 87.9%

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

   if -4.1999999999999999e-88 < F < 2.3000000000000001e-32

   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. Step-by-step derivation

      1. fma-define 99.7%

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

      2. fma-undefine 99.7%

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

      3. *-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. add-sqr-sqrt 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. unpow-prod-down 99.6%

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

      9. +-commutative 99.6%

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

      10. fma-define 99.6%

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

      11. fma-define 99.6%

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

      12. metadata-eval 99.6%

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

      13. metadata-eval 99.6%

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

      14. +-commutative 99.6%

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

      15. fma-define 99.6%

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

      16. fma-define 99.6%

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

      17. metadata-eval 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. pow-sqr 99.6%

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

      2. metadata-eval 99.6%

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

      3. unpow-1 99.6%

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

      4. fma-undefine 99.6%

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

      5. *-commutative 99.6%

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

      6. fma-undefine 99.6%

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

      7. unpow2 99.6%

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

      8. +-commutative 99.6%

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

      9. fma-define 99.6%

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

      10. +-commutative 99.6%

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

      11. unpow2 99.6%

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

      12. fma-undefine 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in F around 0 75.2%

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

       1. associate-*r/ 75.2%

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

       2. neg-mul-1 75.2%

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

       3. distribute-rgt-neg-in 75.2%

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

   11. Simplified 75.2%

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

   if 2.3000000000000001e-32 < F

   1. Initial program 59.9%

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

   3. Simplified 72.3%

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

   5. Taylor expanded in F around inf 94.5%

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

   6. Taylor expanded in B around 0 76.8%

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

   7. Taylor expanded in F around 0 76.9%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.2 \cdot 10^{-88}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{-32}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 71.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.45:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{-32}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.45)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 2.3e-32)
     (/ (* x (cos B)) (- (sin B)))
     (- (/ 1.0 (sin B)) (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.45) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 2.3e-32) {
		tmp = (x * cos(B)) / -sin(B);
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.45d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 2.3d-32) then
        tmp = (x * cos(b)) / -sin(b)
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.45) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 2.3e-32) {
		tmp = (x * Math.cos(B)) / -Math.sin(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.45:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 2.3e-32:
		tmp = (x * math.cos(B)) / -math.sin(B)
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.45)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 2.3e-32)
		tmp = Float64(Float64(x * cos(B)) / Float64(-sin(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.45)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 2.3e-32)
		tmp = (x * cos(B)) / -sin(B);
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.45], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.3e-32], N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.44999999999999996

   1. Initial program 58.9%

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

   3. Simplified 75.1%

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

   5. Taylor expanded in F around -inf 98.5%

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

   6. Taylor expanded in B around 0 79.0%

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

   if -1.44999999999999996 < F < 2.3000000000000001e-32

   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. Step-by-step derivation

      1. fma-define 99.7%

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

      2. fma-undefine 99.7%

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

      3. *-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. add-sqr-sqrt 99.5%

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

      6. metadata-eval 99.5%

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

      7. metadata-eval 99.5%

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

      8. unpow-prod-down 99.5%

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

      9. +-commutative 99.5%

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

      10. fma-define 99.5%

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

      11. fma-define 99.5%

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

      12. metadata-eval 99.5%

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

      13. metadata-eval 99.5%

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

      14. +-commutative 99.5%

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

      15. fma-define 99.5%

          \[\leadsto F \cdot \frac{{\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{-0.5} \cdot {\left(\sqrt{\color{blue}{\mathsf{fma}\left(2, x, F \cdot F + 2\right)}}\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} - \frac{x}{\tan B} \]

      16. fma-define 99.5%

          \[\leadsto F \cdot \frac{{\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{-0.5} \cdot {\left(\sqrt{\mathsf{fma}\left(2, x, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} - \frac{x}{\tan B} \]

      17. metadata-eval 99.5%

          \[\leadsto F \cdot \frac{{\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{-0.5} \cdot {\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{\left(-\color{blue}{0.5}\right)}}{\sin B} - \frac{x}{\tan B} \]

   6. Applied egg-rr 99.5%

      \[\leadsto F \cdot \frac{\color{blue}{{\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{-0.5} \cdot {\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{-0.5}}}{\sin B} - \frac{x}{\tan B} \]
   7. Step-by-step derivation

      1. pow-sqr 99.5%

         \[\leadsto F \cdot \frac{\color{blue}{{\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{\left(2 \cdot -0.5\right)}}}{\sin B} - \frac{x}{\tan B} \]

      2. metadata-eval 99.5%

         \[\leadsto F \cdot \frac{{\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{\color{blue}{-1}}}{\sin B} - \frac{x}{\tan B} \]

      3. unpow-1 99.5%

         \[\leadsto F \cdot \frac{\color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}}{\sin B} - \frac{x}{\tan B} \]

      4. fma-undefine 99.5%

         \[\leadsto F \cdot \frac{\frac{1}{\sqrt{\color{blue}{2 \cdot x + \mathsf{fma}\left(F, F, 2\right)}}}}{\sin B} - \frac{x}{\tan B} \]

      5. *-commutative 99.5%

         \[\leadsto F \cdot \frac{\frac{1}{\sqrt{\color{blue}{x \cdot 2} + \mathsf{fma}\left(F, F, 2\right)}}}{\sin B} - \frac{x}{\tan B} \]

      6. fma-undefine 99.5%

         \[\leadsto F \cdot \frac{\frac{1}{\sqrt{x \cdot 2 + \color{blue}{\left(F \cdot F + 2\right)}}}}{\sin B} - \frac{x}{\tan B} \]

      7. unpow2 99.5%

         \[\leadsto F \cdot \frac{\frac{1}{\sqrt{x \cdot 2 + \left(\color{blue}{{F}^{2}} + 2\right)}}}{\sin B} - \frac{x}{\tan B} \]

      8. +-commutative 99.5%

         \[\leadsto F \cdot \frac{\frac{1}{\sqrt{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}}{\sin B} - \frac{x}{\tan B} \]

      9. fma-define 99.5%

         \[\leadsto F \cdot \frac{\frac{1}{\sqrt{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}}{\sin B} - \frac{x}{\tan B} \]

      10. +-commutative 99.5%

          \[\leadsto F \cdot \frac{\frac{1}{\sqrt{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}}{\sin B} - \frac{x}{\tan B} \]

      11. unpow2 99.5%

          \[\leadsto F \cdot \frac{\frac{1}{\sqrt{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}}{\sin B} - \frac{x}{\tan B} \]

      12. fma-undefine 99.5%

          \[\leadsto F \cdot \frac{\frac{1}{\sqrt{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}}{\sin B} - \frac{x}{\tan B} \]

   8. Simplified 99.5%

      \[\leadsto F \cdot \frac{\color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}}{\sin B} - \frac{x}{\tan B} \]

   9. Taylor expanded in F around 0 71.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   10. Step-by-step derivation

       1. associate-*r/ 71.2%

          \[\leadsto \color{blue}{\frac{-1 \cdot \left(x \cdot \cos B\right)}{\sin B}} \]

       2. neg-mul-1 71.2%

          \[\leadsto \frac{\color{blue}{-x \cdot \cos B}}{\sin B} \]

       3. distribute-rgt-neg-in 71.2%

          \[\leadsto \frac{\color{blue}{x \cdot \left(-\cos B\right)}}{\sin B} \]

   11. Simplified 71.2%

       \[\leadsto \color{blue}{\frac{x \cdot \left(-\cos B\right)}{\sin B}} \]

   if 2.3000000000000001e-32 < F

   1. Initial program 59.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Simplified 72.3%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in F around inf 94.5%

      \[\leadsto F \cdot \frac{\color{blue}{\frac{1}{F}}}{\sin B} - \frac{x}{\tan B} \]

   6. Taylor expanded in B around 0 76.8%

      \[\leadsto F \cdot \frac{\frac{1}{F}}{\sin B} - \color{blue}{\frac{x}{B}} \]

   7. Taylor expanded in F around 0 76.9%

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.45:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{-32}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 71.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.25 \cdot 10^{-9}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 9.2 \cdot 10^{-34}:\\ \;\;\;\;x \cdot \frac{\cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.25e-9)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 9.2e-34)
     (* x (/ (cos B) (- (sin B))))
     (- (/ 1.0 (sin B)) (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.25e-9) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 9.2e-34) {
		tmp = x * (cos(B) / -sin(B));
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.25d-9)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 9.2d-34) then
        tmp = x * (cos(b) / -sin(b))
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.25e-9) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 9.2e-34) {
		tmp = x * (Math.cos(B) / -Math.sin(B));
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.25e-9:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 9.2e-34:
		tmp = x * (math.cos(B) / -math.sin(B))
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.25e-9)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 9.2e-34)
		tmp = Float64(x * Float64(cos(B) / Float64(-sin(B))));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.25e-9)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 9.2e-34)
		tmp = x * (cos(B) / -sin(B));
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.25e-9], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 9.2e-34], N[(x * N[(N[Cos[B], $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.25e-9

   1. Initial program 59.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Simplified 75.5%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in F around -inf 98.5%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} - \frac{x}{\tan B} \]

   6. Taylor expanded in B around 0 79.4%

      \[\leadsto \frac{-1}{\sin B} - \color{blue}{\frac{x}{B}} \]

   if -1.25e-9 < F < 9.20000000000000045e-34

   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. Step-by-step derivation

      1. fma-define 99.7%

         \[\leadsto F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F + 2}\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B} \]

      2. fma-undefine 99.7%

         \[\leadsto F \cdot \frac{{\color{blue}{\left(x \cdot 2 + \left(F \cdot F + 2\right)\right)}}^{-0.5}}{\sin B} - \frac{x}{\tan B} \]

      3. *-commutative 99.7%

         \[\leadsto F \cdot \frac{{\left(\color{blue}{2 \cdot x} + \left(F \cdot F + 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B} \]

      4. +-commutative 99.7%

         \[\leadsto F \cdot \frac{{\color{blue}{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}}^{-0.5}}{\sin B} - \frac{x}{\tan B} \]

      5. add-sqr-sqrt 99.5%

         \[\leadsto F \cdot \frac{{\color{blue}{\left(\sqrt{\left(F \cdot F + 2\right) + 2 \cdot x} \cdot \sqrt{\left(F \cdot F + 2\right) + 2 \cdot x}\right)}}^{-0.5}}{\sin B} - \frac{x}{\tan B} \]

      6. metadata-eval 99.5%

         \[\leadsto F \cdot \frac{{\left(\sqrt{\left(F \cdot F + 2\right) + 2 \cdot x} \cdot \sqrt{\left(F \cdot F + 2\right) + 2 \cdot x}\right)}^{\color{blue}{\left(-0.5\right)}}}{\sin B} - \frac{x}{\tan B} \]

      7. metadata-eval 99.5%

         \[\leadsto F \cdot \frac{{\left(\sqrt{\left(F \cdot F + 2\right) + 2 \cdot x} \cdot \sqrt{\left(F \cdot F + 2\right) + 2 \cdot x}\right)}^{\left(-\color{blue}{\frac{1}{2}}\right)}}{\sin B} - \frac{x}{\tan B} \]

      8. unpow-prod-down 99.5%

         \[\leadsto F \cdot \frac{\color{blue}{{\left(\sqrt{\left(F \cdot F + 2\right) + 2 \cdot x}\right)}^{\left(-\frac{1}{2}\right)} \cdot {\left(\sqrt{\left(F \cdot F + 2\right) + 2 \cdot x}\right)}^{\left(-\frac{1}{2}\right)}}}{\sin B} - \frac{x}{\tan B} \]

      9. +-commutative 99.5%

         \[\leadsto F \cdot \frac{{\left(\sqrt{\color{blue}{2 \cdot x + \left(F \cdot F + 2\right)}}\right)}^{\left(-\frac{1}{2}\right)} \cdot {\left(\sqrt{\left(F \cdot F + 2\right) + 2 \cdot x}\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} - \frac{x}{\tan B} \]

      10. fma-define 99.5%

          \[\leadsto F \cdot \frac{{\left(\sqrt{\color{blue}{\mathsf{fma}\left(2, x, F \cdot F + 2\right)}}\right)}^{\left(-\frac{1}{2}\right)} \cdot {\left(\sqrt{\left(F \cdot F + 2\right) + 2 \cdot x}\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} - \frac{x}{\tan B} \]

      11. fma-define 99.5%

          \[\leadsto F \cdot \frac{{\left(\sqrt{\mathsf{fma}\left(2, x, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}\right)}^{\left(-\frac{1}{2}\right)} \cdot {\left(\sqrt{\left(F \cdot F + 2\right) + 2 \cdot x}\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} - \frac{x}{\tan B} \]

      12. metadata-eval 99.5%

          \[\leadsto F \cdot \frac{{\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{\left(-\color{blue}{0.5}\right)} \cdot {\left(\sqrt{\left(F \cdot F + 2\right) + 2 \cdot x}\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} - \frac{x}{\tan B} \]

      13. metadata-eval 99.5%

          \[\leadsto F \cdot \frac{{\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{\color{blue}{-0.5}} \cdot {\left(\sqrt{\left(F \cdot F + 2\right) + 2 \cdot x}\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} - \frac{x}{\tan B} \]

      14. +-commutative 99.5%

          \[\leadsto F \cdot \frac{{\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{-0.5} \cdot {\left(\sqrt{\color{blue}{2 \cdot x + \left(F \cdot F + 2\right)}}\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} - \frac{x}{\tan B} \]

      15. fma-define 99.5%

          \[\leadsto F \cdot \frac{{\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{-0.5} \cdot {\left(\sqrt{\color{blue}{\mathsf{fma}\left(2, x, F \cdot F + 2\right)}}\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} - \frac{x}{\tan B} \]

      16. fma-define 99.5%

          \[\leadsto F \cdot \frac{{\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{-0.5} \cdot {\left(\sqrt{\mathsf{fma}\left(2, x, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} - \frac{x}{\tan B} \]

      17. metadata-eval 99.5%

          \[\leadsto F \cdot \frac{{\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{-0.5} \cdot {\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{\left(-\color{blue}{0.5}\right)}}{\sin B} - \frac{x}{\tan B} \]

   6. Applied egg-rr 99.5%

      \[\leadsto F \cdot \frac{\color{blue}{{\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{-0.5} \cdot {\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{-0.5}}}{\sin B} - \frac{x}{\tan B} \]
   7. Step-by-step derivation

      1. pow-sqr 99.5%

         \[\leadsto F \cdot \frac{\color{blue}{{\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{\left(2 \cdot -0.5\right)}}}{\sin B} - \frac{x}{\tan B} \]

      2. metadata-eval 99.5%

         \[\leadsto F \cdot \frac{{\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{\color{blue}{-1}}}{\sin B} - \frac{x}{\tan B} \]

      3. unpow-1 99.5%

         \[\leadsto F \cdot \frac{\color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}}{\sin B} - \frac{x}{\tan B} \]

      4. fma-undefine 99.5%

         \[\leadsto F \cdot \frac{\frac{1}{\sqrt{\color{blue}{2 \cdot x + \mathsf{fma}\left(F, F, 2\right)}}}}{\sin B} - \frac{x}{\tan B} \]

      5. *-commutative 99.5%

         \[\leadsto F \cdot \frac{\frac{1}{\sqrt{\color{blue}{x \cdot 2} + \mathsf{fma}\left(F, F, 2\right)}}}{\sin B} - \frac{x}{\tan B} \]

      6. fma-undefine 99.5%

         \[\leadsto F \cdot \frac{\frac{1}{\sqrt{x \cdot 2 + \color{blue}{\left(F \cdot F + 2\right)}}}}{\sin B} - \frac{x}{\tan B} \]

      7. unpow2 99.5%

         \[\leadsto F \cdot \frac{\frac{1}{\sqrt{x \cdot 2 + \left(\color{blue}{{F}^{2}} + 2\right)}}}{\sin B} - \frac{x}{\tan B} \]

      8. +-commutative 99.5%

         \[\leadsto F \cdot \frac{\frac{1}{\sqrt{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}}{\sin B} - \frac{x}{\tan B} \]

      9. fma-define 99.5%

         \[\leadsto F \cdot \frac{\frac{1}{\sqrt{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}}{\sin B} - \frac{x}{\tan B} \]

      10. +-commutative 99.5%

          \[\leadsto F \cdot \frac{\frac{1}{\sqrt{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}}{\sin B} - \frac{x}{\tan B} \]

      11. unpow2 99.5%

          \[\leadsto F \cdot \frac{\frac{1}{\sqrt{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}}{\sin B} - \frac{x}{\tan B} \]

      12. fma-undefine 99.5%

          \[\leadsto F \cdot \frac{\frac{1}{\sqrt{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}}{\sin B} - \frac{x}{\tan B} \]

   8. Simplified 99.5%

      \[\leadsto F \cdot \frac{\color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}}{\sin B} - \frac{x}{\tan B} \]

   9. Taylor expanded in F around 0 71.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   10. Step-by-step derivation

       1. mul-1-neg 71.0%

          \[\leadsto \color{blue}{-\frac{x \cdot \cos B}{\sin B}} \]

       2. associate-/l* 70.9%

          \[\leadsto -\color{blue}{x \cdot \frac{\cos B}{\sin B}} \]

       3. distribute-rgt-neg-in 70.9%

          \[\leadsto \color{blue}{x \cdot \left(-\frac{\cos B}{\sin B}\right)} \]

   11. Simplified 70.9%

       \[\leadsto \color{blue}{x \cdot \left(-\frac{\cos B}{\sin B}\right)} \]

   if 9.20000000000000045e-34 < F

   1. Initial program 59.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Simplified 72.3%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in F around inf 94.5%

      \[\leadsto F \cdot \frac{\color{blue}{\frac{1}{F}}}{\sin B} - \frac{x}{\tan B} \]

   6. Taylor expanded in B around 0 76.8%

      \[\leadsto F \cdot \frac{\frac{1}{F}}{\sin B} - \color{blue}{\frac{x}{B}} \]

   7. Taylor expanded in F around 0 76.9%

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.25 \cdot 10^{-9}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 9.2 \cdot 10^{-34}:\\ \;\;\;\;x \cdot \frac{\cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 64.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -90000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -7 \cdot 10^{-230}:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.35 \cdot 10^{+74}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -90000000.0)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F -7e-230)
     (- (* (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5) (/ F B)) (/ x B))
     (if (<= F 1.35e+74)
       (- (/ 1.0 B) (/ x (tan B)))
       (- (/ 1.0 (sin B)) (/ x B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -90000000.0) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -7e-230) {
		tmp = (pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else if (F <= 1.35e+74) {
		tmp = (1.0 / B) - (x / tan(B));
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-90000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-7d-230)) then
        tmp = ((((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)) * (f / b)) - (x / b)
    else if (f <= 1.35d+74) then
        tmp = (1.0d0 / b) - (x / tan(b))
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -90000000.0) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -7e-230) {
		tmp = (Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else if (F <= 1.35e+74) {
		tmp = (1.0 / B) - (x / Math.tan(B));
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -90000000.0:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -7e-230:
		tmp = (math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B)
	elif F <= 1.35e+74:
		tmp = (1.0 / B) - (x / math.tan(B))
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -90000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -7e-230)
		tmp = Float64(Float64((Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5) * Float64(F / B)) - Float64(x / B));
	elseif (F <= 1.35e+74)
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -90000000.0)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -7e-230)
		tmp = ((((2.0 + (F * F)) + (x * 2.0)) ^ -0.5) * (F / B)) - (x / B);
	elseif (F <= 1.35e+74)
		tmp = (1.0 / B) - (x / tan(B));
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -90000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -7e-230], N[(N[(N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.35e+74], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -9e7

   1. Initial program 57.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Simplified 74.2%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in F around -inf 99.8%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} - \frac{x}{\tan B} \]

   6. Taylor expanded in B around 0 79.7%

      \[\leadsto \frac{-1}{\sin B} - \color{blue}{\frac{x}{B}} \]

   if -9e7 < F < -6.99999999999999975e-230

   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 83.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ 83.9%

         \[\leadsto \color{blue}{\frac{-1 \cdot x}{B}} + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. neg-mul-1 83.9%

         \[\leadsto \frac{\color{blue}{-x}}{B} + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   5. Simplified 83.9%

      \[\leadsto \color{blue}{\frac{-x}{B}} + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   6. Taylor expanded in B around 0 70.3%

      \[\leadsto \frac{-x}{B} + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   if -6.99999999999999975e-230 < F < 1.3499999999999999e74

   1. Initial program 99.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in F around inf 37.6%

      \[\leadsto F \cdot \frac{\color{blue}{\frac{1}{F}}}{\sin B} - \frac{x}{\tan B} \]

   6. Taylor expanded in B around 0 56.1%

      \[\leadsto \color{blue}{\frac{1}{B}} - \frac{x}{\tan B} \]

   if 1.3499999999999999e74 < F

   1. Initial program 47.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Simplified 63.9%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in F around inf 99.7%

      \[\leadsto F \cdot \frac{\color{blue}{\frac{1}{F}}}{\sin B} - \frac{x}{\tan B} \]

   6. Taylor expanded in B around 0 81.8%

      \[\leadsto F \cdot \frac{\frac{1}{F}}{\sin B} - \color{blue}{\frac{x}{B}} \]

   7. Taylor expanded in F around 0 81.9%

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{B}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -90000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -7 \cdot 10^{-230}:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.35 \cdot 10^{+74}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 60.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -5.2 \cdot 10^{-150}:\\ \;\;\;\;\frac{-1}{B} - t\_0\\ \mathbf{elif}\;F \leq -3.8 \cdot 10^{-230}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;F \leq 3 \cdot 10^{+74}:\\ \;\;\;\;\frac{1}{B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -5.2e-150)
     (- (/ -1.0 B) t_0)
     (if (<= F -3.8e-230)
       (/ x (- B))
       (if (<= F 3e+74) (- (/ 1.0 B) t_0) (- (/ 1.0 (sin B)) (/ x B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -5.2e-150) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= -3.8e-230) {
		tmp = x / -B;
	} else if (F <= 3e+74) {
		tmp = (1.0 / B) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-5.2d-150)) then
        tmp = ((-1.0d0) / b) - t_0
    else if (f <= (-3.8d-230)) then
        tmp = x / -b
    else if (f <= 3d+74) then
        tmp = (1.0d0 / b) - t_0
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -5.2e-150) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= -3.8e-230) {
		tmp = x / -B;
	} else if (F <= 3e+74) {
		tmp = (1.0 / B) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -5.2e-150:
		tmp = (-1.0 / B) - t_0
	elif F <= -3.8e-230:
		tmp = x / -B
	elif F <= 3e+74:
		tmp = (1.0 / B) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -5.2e-150)
		tmp = Float64(Float64(-1.0 / B) - t_0);
	elseif (F <= -3.8e-230)
		tmp = Float64(x / Float64(-B));
	elseif (F <= 3e+74)
		tmp = Float64(Float64(1.0 / B) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -5.2e-150)
		tmp = (-1.0 / B) - t_0;
	elseif (F <= -3.8e-230)
		tmp = x / -B;
	elseif (F <= 3e+74)
		tmp = (1.0 / B) - t_0;
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -5.2e-150], N[(N[(-1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -3.8e-230], N[(x / (-B)), $MachinePrecision], If[LessEqual[F, 3e+74], N[(N[(1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -5.1999999999999995e-150

   1. Initial program 71.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Simplified 82.7%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in F around -inf 83.6%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} - \frac{x}{\tan B} \]

   6. Taylor expanded in B around 0 68.6%

      \[\leadsto \color{blue}{\frac{-1}{B}} - \frac{x}{\tan B} \]

   if -5.1999999999999995e-150 < F < -3.7999999999999998e-230

   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 100.0%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in F around inf 2.4%

      \[\leadsto F \cdot \frac{\color{blue}{\frac{1}{F}}}{\sin B} - \frac{x}{\tan B} \]

   6. Taylor expanded in B around 0 26.8%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

   7. Taylor expanded in x around inf 85.3%

      \[\leadsto \frac{\color{blue}{-1 \cdot x}}{B} \]
   8. Step-by-step derivation

      1. neg-mul-1 85.3%

         \[\leadsto \frac{\color{blue}{-x}}{B} \]

   9. Simplified 85.3%

      \[\leadsto \frac{\color{blue}{-x}}{B} \]

   if -3.7999999999999998e-230 < F < 3e74

   1. Initial program 99.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in F around inf 37.6%

      \[\leadsto F \cdot \frac{\color{blue}{\frac{1}{F}}}{\sin B} - \frac{x}{\tan B} \]

   6. Taylor expanded in B around 0 56.1%

      \[\leadsto \color{blue}{\frac{1}{B}} - \frac{x}{\tan B} \]

   if 3e74 < F

   1. Initial program 47.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Simplified 63.9%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in F around inf 99.7%

      \[\leadsto F \cdot \frac{\color{blue}{\frac{1}{F}}}{\sin B} - \frac{x}{\tan B} \]

   6. Taylor expanded in B around 0 81.8%

      \[\leadsto F \cdot \frac{\frac{1}{F}}{\sin B} - \color{blue}{\frac{x}{B}} \]

   7. Taylor expanded in F around 0 81.9%

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{B}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5.2 \cdot 10^{-150}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -3.8 \cdot 10^{-230}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;F \leq 3 \cdot 10^{+74}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 53.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;F \leq 2.1 \cdot 10^{-32}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 2.5 \cdot 10^{+182}:\\ \;\;\;\;\frac{1 - x}{B}\\ \mathbf{elif}\;F \leq 4.2 \cdot 10^{+259}:\\ \;\;\;\;x \cdot \frac{\frac{1}{x}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ -1.0 B) (/ x (tan B)))))
   (if (<= F 2.1e-32)
     t_0
     (if (<= F 2.5e+182)
       (/ (- 1.0 x) B)
       (if (<= F 4.2e+259) (* x (/ (/ 1.0 x) (sin B))) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 / B) - (x / tan(B));
	double tmp;
	if (F <= 2.1e-32) {
		tmp = t_0;
	} else if (F <= 2.5e+182) {
		tmp = (1.0 - x) / B;
	} else if (F <= 4.2e+259) {
		tmp = x * ((1.0 / x) / sin(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 (f <= 2.1d-32) then
        tmp = t_0
    else if (f <= 2.5d+182) then
        tmp = (1.0d0 - x) / b
    else if (f <= 4.2d+259) then
        tmp = x * ((1.0d0 / x) / sin(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 (F <= 2.1e-32) {
		tmp = t_0;
	} else if (F <= 2.5e+182) {
		tmp = (1.0 - x) / B;
	} else if (F <= 4.2e+259) {
		tmp = x * ((1.0 / x) / Math.sin(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 F <= 2.1e-32:
		tmp = t_0
	elif F <= 2.5e+182:
		tmp = (1.0 - x) / B
	elif F <= 4.2e+259:
		tmp = x * ((1.0 / x) / math.sin(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 (F <= 2.1e-32)
		tmp = t_0;
	elseif (F <= 2.5e+182)
		tmp = Float64(Float64(1.0 - x) / B);
	elseif (F <= 4.2e+259)
		tmp = Float64(x * Float64(Float64(1.0 / x) / sin(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 (F <= 2.1e-32)
		tmp = t_0;
	elseif (F <= 2.5e+182)
		tmp = (1.0 - x) / B;
	elseif (F <= 4.2e+259)
		tmp = x * ((1.0 / x) / sin(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[F, 2.1e-32], t$95$0, If[LessEqual[F, 2.5e+182], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 4.2e+259], N[(x * N[(N[(1.0 / x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if F < 2.0999999999999999e-32 or 4.20000000000000011e259 < F

   1. Initial program 80.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Simplified 90.2%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in F around -inf 61.6%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} - \frac{x}{\tan B} \]

   6. Taylor expanded in B around 0 60.8%

      \[\leadsto \color{blue}{\frac{-1}{B}} - \frac{x}{\tan B} \]

   if 2.0999999999999999e-32 < F < 2.49999999999999987e182

   1. Initial program 83.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Simplified 88.7%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in F around inf 91.1%

      \[\leadsto F \cdot \frac{\color{blue}{\frac{1}{F}}}{\sin B} - \frac{x}{\tan B} \]

   6. Taylor expanded in B around 0 56.0%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

   if 2.49999999999999987e182 < F < 4.20000000000000011e259

   1. Initial program 12.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Simplified 17.1%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in F around inf 99.5%

      \[\leadsto F \cdot \frac{\color{blue}{\frac{1}{F}}}{\sin B} - \frac{x}{\tan B} \]

   6. Taylor expanded in B around 0 90.2%

      \[\leadsto F \cdot \frac{\frac{1}{F}}{\sin B} - \color{blue}{\frac{x}{B}} \]

   7. Taylor expanded in x around inf 76.1%

      \[\leadsto \color{blue}{x \cdot \left(\frac{1}{x \cdot \sin B} - \frac{1}{B}\right)} \]

   8. Taylor expanded in x around 0 67.5%

      \[\leadsto x \cdot \color{blue}{\frac{1}{x \cdot \sin B}} \]
   9. Step-by-step derivation

      1. associate-/r* 67.3%

         \[\leadsto x \cdot \color{blue}{\frac{\frac{1}{x}}{\sin B}} \]

   10. Simplified 67.3%

       \[\leadsto x \cdot \color{blue}{\frac{\frac{1}{x}}{\sin B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 60.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2 \cdot 10^{-150}:\\ \;\;\;\;\frac{-1}{B} - t\_0\\ \mathbf{elif}\;F \leq -5.4 \cdot 10^{-230}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -2e-150)
     (- (/ -1.0 B) t_0)
     (if (<= F -5.4e-230) (/ x (- B)) (- (/ 1.0 B) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -2e-150) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= -5.4e-230) {
		tmp = x / -B;
	} else {
		tmp = (1.0 / B) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-2d-150)) then
        tmp = ((-1.0d0) / b) - t_0
    else if (f <= (-5.4d-230)) then
        tmp = x / -b
    else
        tmp = (1.0d0 / b) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -2e-150) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= -5.4e-230) {
		tmp = x / -B;
	} else {
		tmp = (1.0 / B) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -2e-150:
		tmp = (-1.0 / B) - t_0
	elif F <= -5.4e-230:
		tmp = x / -B
	else:
		tmp = (1.0 / B) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -2e-150)
		tmp = Float64(Float64(-1.0 / B) - t_0);
	elseif (F <= -5.4e-230)
		tmp = Float64(x / Float64(-B));
	else
		tmp = Float64(Float64(1.0 / B) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -2e-150)
		tmp = (-1.0 / B) - t_0;
	elseif (F <= -5.4e-230)
		tmp = x / -B;
	else
		tmp = (1.0 / B) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2e-150], N[(N[(-1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -5.4e-230], N[(x / (-B)), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.00000000000000001e-150

   1. Initial program 71.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Simplified 82.7%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in F around -inf 83.6%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} - \frac{x}{\tan B} \]

   6. Taylor expanded in B around 0 68.6%

      \[\leadsto \color{blue}{\frac{-1}{B}} - \frac{x}{\tan B} \]

   if -2.00000000000000001e-150 < F < -5.40000000000000023e-230

   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 100.0%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in F around inf 2.4%

      \[\leadsto F \cdot \frac{\color{blue}{\frac{1}{F}}}{\sin B} - \frac{x}{\tan B} \]

   6. Taylor expanded in B around 0 26.8%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

   7. Taylor expanded in x around inf 85.3%

      \[\leadsto \frac{\color{blue}{-1 \cdot x}}{B} \]
   8. Step-by-step derivation

      1. neg-mul-1 85.3%

         \[\leadsto \frac{\color{blue}{-x}}{B} \]

   9. Simplified 85.3%

      \[\leadsto \frac{\color{blue}{-x}}{B} \]

   if -5.40000000000000023e-230 < F

   1. Initial program 76.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Simplified 83.7%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in F around inf 65.3%

      \[\leadsto F \cdot \frac{\color{blue}{\frac{1}{F}}}{\sin B} - \frac{x}{\tan B} \]

   6. Taylor expanded in B around 0 61.8%

      \[\leadsto \color{blue}{\frac{1}{B}} - \frac{x}{\tan B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{-150}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -5.4 \cdot 10^{-230}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 44.1% accurate, 21.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -8.5 \cdot 10^{-89}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 10^{-95}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -8.5e-89)
   (/ (- -1.0 x) B)
   (if (<= F 1e-95) (/ x (- B)) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -8.5e-89) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1e-95) {
		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 <= (-8.5d-89)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 1d-95) 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 <= -8.5e-89) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1e-95) {
		tmp = x / -B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -8.5e-89:
		tmp = (-1.0 - x) / B
	elif F <= 1e-95:
		tmp = x / -B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -8.5e-89)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 1e-95)
		tmp = Float64(x / Float64(-B));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -8.5e-89)
		tmp = (-1.0 - x) / B;
	elseif (F <= 1e-95)
		tmp = x / -B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -8.5e-89], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1e-95], N[(x / (-B)), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -8.49999999999999937e-89

   1. Initial program 68.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Simplified 80.8%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in F around -inf 87.9%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} - \frac{x}{\tan B} \]

   6. Taylor expanded in B around 0 52.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   7. Step-by-step derivation

      1. mul-1-neg 52.3%

         \[\leadsto \color{blue}{-\frac{1 + x}{B}} \]

      2. distribute-neg-frac2 52.3%

         \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

      3. +-commutative 52.3%

         \[\leadsto \frac{\color{blue}{x + 1}}{-B} \]

   8. Simplified 52.3%

      \[\leadsto \color{blue}{\frac{x + 1}{-B}} \]

   if -8.49999999999999937e-89 < F < 9.99999999999999989e-96

   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 inf 21.0%

      \[\leadsto F \cdot \frac{\color{blue}{\frac{1}{F}}}{\sin B} - \frac{x}{\tan B} \]

   6. Taylor expanded in B around 0 24.6%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

   7. Taylor expanded in x around inf 46.7%

      \[\leadsto \frac{\color{blue}{-1 \cdot x}}{B} \]
   8. Step-by-step derivation

      1. neg-mul-1 46.7%

         \[\leadsto \frac{\color{blue}{-x}}{B} \]

   9. Simplified 46.7%

      \[\leadsto \frac{\color{blue}{-x}}{B} \]

   if 9.99999999999999989e-96 < F

   1. Initial program 64.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Simplified 75.7%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in F around inf 86.1%

      \[\leadsto F \cdot \frac{\color{blue}{\frac{1}{F}}}{\sin B} - \frac{x}{\tan B} \]

   6. Taylor expanded in B around 0 47.2%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -8.5 \cdot 10^{-89}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 10^{-95}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 30.1% accurate, 23.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 500 \lor \neg \left(F \leq 1.2 \cdot 10^{+256}\right):\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (or (<= F 500.0) (not (<= F 1.2e+256))) (/ x (- B)) (/ 1.0 B)))

C

double code(double F, double B, double x) {
	double tmp;
	if ((F <= 500.0) || !(F <= 1.2e+256)) {
		tmp = x / -B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((f <= 500.0d0) .or. (.not. (f <= 1.2d+256))) then
        tmp = x / -b
    else
        tmp = 1.0d0 / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if ((F <= 500.0) || !(F <= 1.2e+256)) {
		tmp = x / -B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if (F <= 500.0) or not (F <= 1.2e+256):
		tmp = x / -B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if ((F <= 500.0) || !(F <= 1.2e+256))
		tmp = Float64(x / Float64(-B));
	else
		tmp = Float64(1.0 / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if ((F <= 500.0) || ~((F <= 1.2e+256)))
		tmp = x / -B;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[Or[LessEqual[F, 500.0], N[Not[LessEqual[F, 1.2e+256]], $MachinePrecision]], N[(x / (-B)), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 500 or 1.20000000000000007e256 < F

   1. Initial program 80.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Simplified 90.1%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in F around inf 37.8%

      \[\leadsto F \cdot \frac{\color{blue}{\frac{1}{F}}}{\sin B} - \frac{x}{\tan B} \]

   6. Taylor expanded in B around 0 27.4%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

   7. Taylor expanded in x around inf 35.9%

      \[\leadsto \frac{\color{blue}{-1 \cdot x}}{B} \]
   8. Step-by-step derivation

      1. neg-mul-1 35.9%

         \[\leadsto \frac{\color{blue}{-x}}{B} \]

   9. Simplified 35.9%

      \[\leadsto \frac{\color{blue}{-x}}{B} \]

   if 500 < F < 1.20000000000000007e256

   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 67.8%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in F around inf 98.8%

      \[\leadsto F \cdot \frac{\color{blue}{\frac{1}{F}}}{\sin B} - \frac{x}{\tan B} \]

   6. Taylor expanded in B around 0 53.6%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

   7. Taylor expanded in x around 0 38.7%

      \[\leadsto \color{blue}{\frac{1}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 500 \lor \neg \left(F \leq 1.2 \cdot 10^{+256}\right):\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 36.5% accurate, 32.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 9.6 \cdot 10^{-96}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F 9.6e-96) (/ x (- B)) (/ (- 1.0 x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= 9.6e-96) {
		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 <= 9.6d-96) 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 <= 9.6e-96) {
		tmp = x / -B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= 9.6e-96:
		tmp = x / -B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= 9.6e-96)
		tmp = Float64(x / Float64(-B));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= 9.6e-96)
		tmp = x / -B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, 9.6e-96], N[(x / (-B)), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 9.60000000000000076e-96

   1. Initial program 83.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Simplified 90.0%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in F around inf 31.7%

      \[\leadsto F \cdot \frac{\color{blue}{\frac{1}{F}}}{\sin B} - \frac{x}{\tan B} \]

   6. Taylor expanded in B around 0 25.4%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

   7. Taylor expanded in x around inf 36.7%

      \[\leadsto \frac{\color{blue}{-1 \cdot x}}{B} \]
   8. Step-by-step derivation

      1. neg-mul-1 36.7%

         \[\leadsto \frac{\color{blue}{-x}}{B} \]

   9. Simplified 36.7%

      \[\leadsto \frac{\color{blue}{-x}}{B} \]

   if 9.60000000000000076e-96 < F

   1. Initial program 64.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Simplified 75.7%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in F around inf 86.1%

      \[\leadsto F \cdot \frac{\color{blue}{\frac{1}{F}}}{\sin B} - \frac{x}{\tan B} \]

   6. Taylor expanded in B around 0 47.2%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 9.6 \cdot 10^{-96}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 10.0% 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 75.9%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

3. Simplified 84.2%

   \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
4. Add Preprocessing

5. Taylor expanded in F around inf 54.0%

   \[\leadsto F \cdot \frac{\color{blue}{\frac{1}{F}}}{\sin B} - \frac{x}{\tan B} \]

6. Taylor expanded in B around 0 34.3%

   \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

7. Taylor expanded in x around 0 12.5%

   \[\leadsto \color{blue}{\frac{1}{B}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024146 
(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))))))