VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.6% → 99.7%

Time: 16.7s

Alternatives: 23

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.6% 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: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -145000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x, \cos B, -1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -145000000.0)
   (/ (fma (- x) (cos B) -1.0) (sin B))
   (if (<= F 5e+38)
     (fma F (/ (pow (fma 2.0 x (fma F F 2.0)) -0.5) (sin B)) (/ (- x) (tan B)))
     (/ (- 1.0 (* x (cos B))) (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -145000000.0) {
		tmp = fma(-x, cos(B), -1.0) / sin(B);
	} else if (F <= 5e+38) {
		tmp = fma(F, (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) / sin(B)), (-x / tan(B)));
	} else {
		tmp = (1.0 - (x * cos(B))) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -145000000.0)
		tmp = Float64(fma(Float64(-x), cos(B), -1.0) / sin(B));
	elseif (F <= 5e+38)
		tmp = fma(F, Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) / sin(B)), Float64(Float64(-x) / tan(B)));
	else
		tmp = Float64(Float64(1.0 - Float64(x * cos(B))) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -145000000.0], N[(N[((-x) * N[Cos[B], $MachinePrecision] + -1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5e+38], N[(F * N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.45e8

   1. Initial program 58.7%

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

   3. Taylor expanded in F around -inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. div-add-rev N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*r* N/A

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-sin.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if -1.45e8 < F < 4.9999999999999997e38

   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. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 99.5%

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

   if 4.9999999999999997e38 < F

   1. Initial program 50.5%

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

   3. Taylor expanded in F around inf

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

      1. associate-/l* N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. distribute-lft-neg-in N/A

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

      4. associate-/l* N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. associate-*r* N/A

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

      11. mul-1-neg N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-sin.f64 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -145000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x, \cos B, -1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{\tan B}\\ t_1 := \frac{-x}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B}\\ t_2 := \mathsf{fma}\left(F, \frac{{\left(\sqrt{\mathsf{fma}\left(F, F, 2\right)}\right)}^{-1}}{\sin B}, t\_1\right)\\ \mathbf{if}\;F \leq -2.4 \cdot 10^{+111}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{-1}{\mathsf{fma}\left(\left(B \cdot B\right) \cdot F, -0.16666666666666666, F\right) \cdot B}, t\_0\right)\\ \mathbf{elif}\;F \leq -6.1 \cdot 10^{-121}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;F \leq 9.5 \cdot 10^{-223}:\\ \;\;\;\;\mathsf{fma}\left(F, \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot {B}^{-1}, t\_0\right)\\ \mathbf{elif}\;F \leq 2 \cdot 10^{+125}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1 + {\sin B}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B)))
        (t_1 (/ (- x) (* (fma 0.3333333333333333 (* B B) 1.0) B)))
        (t_2 (fma F (/ (pow (sqrt (fma F F 2.0)) -1.0) (sin B)) t_1)))
   (if (<= F -2.4e+111)
     (fma F (/ -1.0 (* (fma (* (* B B) F) -0.16666666666666666 F) B)) t_0)
     (if (<= F -6.1e-121)
       t_2
       (if (<= F 9.5e-223)
         (fma
          F
          (* (sqrt (pow (+ (fma 2.0 x (* F F)) 2.0) -1.0)) (pow B -1.0))
          t_0)
         (if (<= F 2e+125) t_2 (+ t_1 (pow (sin B) -1.0))))))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double t_1 = -x / (fma(0.3333333333333333, (B * B), 1.0) * B);
	double t_2 = fma(F, (pow(sqrt(fma(F, F, 2.0)), -1.0) / sin(B)), t_1);
	double tmp;
	if (F <= -2.4e+111) {
		tmp = fma(F, (-1.0 / (fma(((B * B) * F), -0.16666666666666666, F) * B)), t_0);
	} else if (F <= -6.1e-121) {
		tmp = t_2;
	} else if (F <= 9.5e-223) {
		tmp = fma(F, (sqrt(pow((fma(2.0, x, (F * F)) + 2.0), -1.0)) * pow(B, -1.0)), t_0);
	} else if (F <= 2e+125) {
		tmp = t_2;
	} else {
		tmp = t_1 + pow(sin(B), -1.0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	t_1 = Float64(Float64(-x) / Float64(fma(0.3333333333333333, Float64(B * B), 1.0) * B))
	t_2 = fma(F, Float64((sqrt(fma(F, F, 2.0)) ^ -1.0) / sin(B)), t_1)
	tmp = 0.0
	if (F <= -2.4e+111)
		tmp = fma(F, Float64(-1.0 / Float64(fma(Float64(Float64(B * B) * F), -0.16666666666666666, F) * B)), t_0);
	elseif (F <= -6.1e-121)
		tmp = t_2;
	elseif (F <= 9.5e-223)
		tmp = fma(F, Float64(sqrt((Float64(fma(2.0, x, Float64(F * F)) + 2.0) ^ -1.0)) * (B ^ -1.0)), t_0);
	elseif (F <= 2e+125)
		tmp = t_2;
	else
		tmp = Float64(t_1 + (sin(B) ^ -1.0));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-x) / N[(N[(0.3333333333333333 * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(F * N[(N[Power[N[Sqrt[N[(F * F + 2.0), $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[F, -2.4e+111], N[(F * N[(-1.0 / N[(N[(N[(N[(B * B), $MachinePrecision] * F), $MachinePrecision] * -0.16666666666666666 + F), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[F, -6.1e-121], t$95$2, If[LessEqual[F, 9.5e-223], N[(F * N[(N[Sqrt[N[Power[N[(N[(2.0 * x + N[(F * F), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[B, -1.0], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[F, 2e+125], t$95$2, N[(t$95$1 + N[Power[N[Sin[B], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.40000000000000006e111

   1. Initial program 46.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 53.4%

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

   5. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 99.6

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in B around 0

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

   10. Applied rewrites 67.6%

       \[\leadsto \mathsf{fma}\left(F, \frac{-1}{\mathsf{fma}\left(\left(B \cdot B\right) \cdot F, -0.16666666666666666, F\right) \cdot \color{blue}{B}}, \frac{-x}{\tan B}\right) \]

   if -2.40000000000000006e111 < F < -6.09999999999999978e-121 or 9.49999999999999992e-223 < F < 1.9999999999999998e125

   1. Initial program 95.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 99.5

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

   7. Applied rewrites 99.5%

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

   9. Applied rewrites 99.5%

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

   10. Taylor expanded in B around 0

       \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}, \frac{-x}{\color{blue}{B \cdot \left(1 + \frac{1}{3} \cdot {B}^{2}\right)}}\right) \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}, \frac{-x}{\color{blue}{\left(1 + \frac{1}{3} \cdot {B}^{2}\right) \cdot B}}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}, \frac{-x}{\color{blue}{\left(1 + \frac{1}{3} \cdot {B}^{2}\right) \cdot B}}\right) \]

       3. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}, \frac{-x}{\color{blue}{\left(\frac{1}{3} \cdot {B}^{2} + 1\right)} \cdot B}\right) \]

       4. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}, \frac{-x}{\color{blue}{\mathsf{fma}\left(\frac{1}{3}, {B}^{2}, 1\right)} \cdot B}\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}, \frac{-x}{\mathsf{fma}\left(\frac{1}{3}, \color{blue}{B \cdot B}, 1\right) \cdot B}\right) \]

       6. lower-*.f64 84.2

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

   12. Applied rewrites 84.2%

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

   if -6.09999999999999978e-121 < F < 9.49999999999999992e-223

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 93.5

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

   7. Applied rewrites 93.5%

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

   if 1.9999999999999998e125 < F

   1. Initial program 36.3%

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

   3. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 99.7

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

   5. Applied rewrites 99.7%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. distribute-neg-frac N/A

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

      6. *-rgt-identity N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-/.f64 99.8

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in B around 0

      \[\leadsto \frac{-x}{\color{blue}{B \cdot \left(1 + \frac{1}{3} \cdot {B}^{2}\right)}} + \frac{1}{\sin B} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{-x}{\color{blue}{\left(1 + \frac{1}{3} \cdot {B}^{2}\right) \cdot B}} + \frac{1}{\sin B} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-x}{\color{blue}{\left(1 + \frac{1}{3} \cdot {B}^{2}\right) \cdot B}} + \frac{1}{\sin B} \]

      3. +-commutative N/A

         \[\leadsto \frac{-x}{\color{blue}{\left(\frac{1}{3} \cdot {B}^{2} + 1\right)} \cdot B} + \frac{1}{\sin B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{-x}{\color{blue}{\mathsf{fma}\left(\frac{1}{3}, {B}^{2}, 1\right)} \cdot B} + \frac{1}{\sin B} \]

      5. unpow2 N/A

         \[\leadsto \frac{-x}{\mathsf{fma}\left(\frac{1}{3}, \color{blue}{B \cdot B}, 1\right) \cdot B} + \frac{1}{\sin B} \]

      6. lower-*.f64 76.0

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

   10. Applied rewrites 76.0%

       \[\leadsto \frac{-x}{\color{blue}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B}} + \frac{1}{\sin B} \]
3. Recombined 4 regimes into one program.

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.4 \cdot 10^{+111}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{-1}{\mathsf{fma}\left(\left(B \cdot B\right) \cdot F, -0.16666666666666666, F\right) \cdot B}, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;F \leq -6.1 \cdot 10^{-121}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\sqrt{\mathsf{fma}\left(F, F, 2\right)}\right)}^{-1}}{\sin B}, \frac{-x}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B}\right)\\ \mathbf{elif}\;F \leq 9.5 \cdot 10^{-223}:\\ \;\;\;\;\mathsf{fma}\left(F, \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot {B}^{-1}, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;F \leq 2 \cdot 10^{+125}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\sqrt{\mathsf{fma}\left(F, F, 2\right)}\right)}^{-1}}{\sin B}, \frac{-x}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B} + {\sin B}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -7 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x, \cos B, -1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 9.5 \cdot 10^{-223}:\\ \;\;\;\;\mathsf{fma}\left(F, \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot {B}^{-1}, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;F \leq 35000:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\sqrt{\mathsf{fma}\left(F, F, 2\right)}\right)}^{-1}}{\sin B}, \frac{-x}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -7e-5)
   (/ (fma (- x) (cos B) -1.0) (sin B))
   (if (<= F 9.5e-223)
     (fma
      F
      (* (sqrt (pow (+ (fma 2.0 x (* F F)) 2.0) -1.0)) (pow B -1.0))
      (/ (- x) (tan B)))
     (if (<= F 35000.0)
       (fma
        F
        (/ (pow (sqrt (fma F F 2.0)) -1.0) (sin B))
        (/ (- x) (* (fma 0.3333333333333333 (* B B) 1.0) B)))
       (/ (- 1.0 (* x (cos B))) (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -7e-5) {
		tmp = fma(-x, cos(B), -1.0) / sin(B);
	} else if (F <= 9.5e-223) {
		tmp = fma(F, (sqrt(pow((fma(2.0, x, (F * F)) + 2.0), -1.0)) * pow(B, -1.0)), (-x / tan(B)));
	} else if (F <= 35000.0) {
		tmp = fma(F, (pow(sqrt(fma(F, F, 2.0)), -1.0) / sin(B)), (-x / (fma(0.3333333333333333, (B * B), 1.0) * B)));
	} else {
		tmp = (1.0 - (x * cos(B))) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -7e-5)
		tmp = Float64(fma(Float64(-x), cos(B), -1.0) / sin(B));
	elseif (F <= 9.5e-223)
		tmp = fma(F, Float64(sqrt((Float64(fma(2.0, x, Float64(F * F)) + 2.0) ^ -1.0)) * (B ^ -1.0)), Float64(Float64(-x) / tan(B)));
	elseif (F <= 35000.0)
		tmp = fma(F, Float64((sqrt(fma(F, F, 2.0)) ^ -1.0) / sin(B)), Float64(Float64(-x) / Float64(fma(0.3333333333333333, Float64(B * B), 1.0) * B)));
	else
		tmp = Float64(Float64(1.0 - Float64(x * cos(B))) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -7e-5], N[(N[((-x) * N[Cos[B], $MachinePrecision] + -1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 9.5e-223], N[(F * N[(N[Sqrt[N[Power[N[(N[(2.0 * x + N[(F * F), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[B, -1.0], $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 35000.0], N[(F * N[(N[Power[N[Sqrt[N[(F * F + 2.0), $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[((-x) / N[(N[(0.3333333333333333 * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -6.9999999999999994e-5

   1. Initial program 59.8%

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

   3. Taylor expanded in F around -inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. div-add-rev N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*r* N/A

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-sin.f64 98.4

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

   5. Applied rewrites 98.4%

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

   if -6.9999999999999994e-5 < F < 9.49999999999999992e-223

   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. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 88.2

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

   7. Applied rewrites 88.2%

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

   if 9.49999999999999992e-223 < F < 35000

   1. Initial program 99.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in x around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 99.3

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

   7. Applied rewrites 99.3%

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

   9. Applied rewrites 99.5%

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

   10. Taylor expanded in B around 0

       \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}, \frac{-x}{\color{blue}{B \cdot \left(1 + \frac{1}{3} \cdot {B}^{2}\right)}}\right) \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}, \frac{-x}{\color{blue}{\left(1 + \frac{1}{3} \cdot {B}^{2}\right) \cdot B}}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}, \frac{-x}{\color{blue}{\left(1 + \frac{1}{3} \cdot {B}^{2}\right) \cdot B}}\right) \]

       3. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}, \frac{-x}{\color{blue}{\left(\frac{1}{3} \cdot {B}^{2} + 1\right)} \cdot B}\right) \]

       4. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}, \frac{-x}{\color{blue}{\mathsf{fma}\left(\frac{1}{3}, {B}^{2}, 1\right)} \cdot B}\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}, \frac{-x}{\mathsf{fma}\left(\frac{1}{3}, \color{blue}{B \cdot B}, 1\right) \cdot B}\right) \]

       6. lower-*.f64 89.9

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

   12. Applied rewrites 89.9%

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

   if 35000 < F

   1. Initial program 54.8%

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

   3. Taylor expanded in F around inf

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

      1. associate-/l* N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. distribute-lft-neg-in N/A

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

      4. associate-/l* N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. associate-*r* N/A

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

      11. mul-1-neg N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-sin.f64 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x, \cos B, -1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 9.5 \cdot 10^{-223}:\\ \;\;\;\;\mathsf{fma}\left(F, \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot {B}^{-1}, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;F \leq 35000:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\sqrt{\mathsf{fma}\left(F, F, 2\right)}\right)}^{-1}}{\sin B}, \frac{-x}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B}\\ \mathbf{if}\;F \leq -7 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x, \cos B, -1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 9.5 \cdot 10^{-223}:\\ \;\;\;\;\mathsf{fma}\left(F, \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot {B}^{-1}, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;F \leq 2 \cdot 10^{+125}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\sqrt{\mathsf{fma}\left(F, F, 2\right)}\right)}^{-1}}{\sin B}, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + {\sin B}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (* (fma 0.3333333333333333 (* B B) 1.0) B))))
   (if (<= F -7e-5)
     (/ (fma (- x) (cos B) -1.0) (sin B))
     (if (<= F 9.5e-223)
       (fma
        F
        (* (sqrt (pow (+ (fma 2.0 x (* F F)) 2.0) -1.0)) (pow B -1.0))
        (/ (- x) (tan B)))
       (if (<= F 2e+125)
         (fma F (/ (pow (sqrt (fma F F 2.0)) -1.0) (sin B)) t_0)
         (+ t_0 (pow (sin B) -1.0)))))))

C

double code(double F, double B, double x) {
	double t_0 = -x / (fma(0.3333333333333333, (B * B), 1.0) * B);
	double tmp;
	if (F <= -7e-5) {
		tmp = fma(-x, cos(B), -1.0) / sin(B);
	} else if (F <= 9.5e-223) {
		tmp = fma(F, (sqrt(pow((fma(2.0, x, (F * F)) + 2.0), -1.0)) * pow(B, -1.0)), (-x / tan(B)));
	} else if (F <= 2e+125) {
		tmp = fma(F, (pow(sqrt(fma(F, F, 2.0)), -1.0) / sin(B)), t_0);
	} else {
		tmp = t_0 + pow(sin(B), -1.0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / Float64(fma(0.3333333333333333, Float64(B * B), 1.0) * B))
	tmp = 0.0
	if (F <= -7e-5)
		tmp = Float64(fma(Float64(-x), cos(B), -1.0) / sin(B));
	elseif (F <= 9.5e-223)
		tmp = fma(F, Float64(sqrt((Float64(fma(2.0, x, Float64(F * F)) + 2.0) ^ -1.0)) * (B ^ -1.0)), Float64(Float64(-x) / tan(B)));
	elseif (F <= 2e+125)
		tmp = fma(F, Float64((sqrt(fma(F, F, 2.0)) ^ -1.0) / sin(B)), t_0);
	else
		tmp = Float64(t_0 + (sin(B) ^ -1.0));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-x) / N[(N[(0.3333333333333333 * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -7e-5], N[(N[((-x) * N[Cos[B], $MachinePrecision] + -1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 9.5e-223], N[(F * N[(N[Sqrt[N[Power[N[(N[(2.0 * x + N[(F * F), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[B, -1.0], $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2e+125], N[(F * N[(N[Power[N[Sqrt[N[(F * F + 2.0), $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], N[(t$95$0 + N[Power[N[Sin[B], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -6.9999999999999994e-5

   1. Initial program 59.8%

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

   3. Taylor expanded in F around -inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. div-add-rev N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*r* N/A

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-sin.f64 98.4

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

   5. Applied rewrites 98.4%

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

   if -6.9999999999999994e-5 < F < 9.49999999999999992e-223

   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. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 88.2

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

   7. Applied rewrites 88.2%

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

   if 9.49999999999999992e-223 < F < 1.9999999999999998e125

   1. Initial program 97.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 99.5

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

   7. Applied rewrites 99.5%

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

   9. Applied rewrites 99.6%

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

   10. Taylor expanded in B around 0

       \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}, \frac{-x}{\color{blue}{B \cdot \left(1 + \frac{1}{3} \cdot {B}^{2}\right)}}\right) \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}, \frac{-x}{\color{blue}{\left(1 + \frac{1}{3} \cdot {B}^{2}\right) \cdot B}}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}, \frac{-x}{\color{blue}{\left(1 + \frac{1}{3} \cdot {B}^{2}\right) \cdot B}}\right) \]

       3. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}, \frac{-x}{\color{blue}{\left(\frac{1}{3} \cdot {B}^{2} + 1\right)} \cdot B}\right) \]

       4. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}, \frac{-x}{\color{blue}{\mathsf{fma}\left(\frac{1}{3}, {B}^{2}, 1\right)} \cdot B}\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}, \frac{-x}{\mathsf{fma}\left(\frac{1}{3}, \color{blue}{B \cdot B}, 1\right) \cdot B}\right) \]

       6. lower-*.f64 84.9

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

   12. Applied rewrites 84.9%

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

   if 1.9999999999999998e125 < F

   1. Initial program 36.3%

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

   3. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 99.7

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

   5. Applied rewrites 99.7%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. distribute-neg-frac N/A

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

      6. *-rgt-identity N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-/.f64 99.8

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in B around 0

      \[\leadsto \frac{-x}{\color{blue}{B \cdot \left(1 + \frac{1}{3} \cdot {B}^{2}\right)}} + \frac{1}{\sin B} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{-x}{\color{blue}{\left(1 + \frac{1}{3} \cdot {B}^{2}\right) \cdot B}} + \frac{1}{\sin B} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-x}{\color{blue}{\left(1 + \frac{1}{3} \cdot {B}^{2}\right) \cdot B}} + \frac{1}{\sin B} \]

      3. +-commutative N/A

         \[\leadsto \frac{-x}{\color{blue}{\left(\frac{1}{3} \cdot {B}^{2} + 1\right)} \cdot B} + \frac{1}{\sin B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{-x}{\color{blue}{\mathsf{fma}\left(\frac{1}{3}, {B}^{2}, 1\right)} \cdot B} + \frac{1}{\sin B} \]

      5. unpow2 N/A

         \[\leadsto \frac{-x}{\mathsf{fma}\left(\frac{1}{3}, \color{blue}{B \cdot B}, 1\right) \cdot B} + \frac{1}{\sin B} \]

      6. lower-*.f64 76.0

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

   10. Applied rewrites 76.0%

       \[\leadsto \frac{-x}{\color{blue}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B}} + \frac{1}{\sin B} \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x, \cos B, -1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 9.5 \cdot 10^{-223}:\\ \;\;\;\;\mathsf{fma}\left(F, \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot {B}^{-1}, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;F \leq 2 \cdot 10^{+125}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\sqrt{\mathsf{fma}\left(F, F, 2\right)}\right)}^{-1}}{\sin B}, \frac{-x}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B} + {\sin B}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -7.8 \cdot 10^{+136}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{-1}{\mathsf{fma}\left(\left(B \cdot B\right) \cdot F, -0.16666666666666666, F\right) \cdot B}, t\_0\right)\\ \mathbf{elif}\;F \leq 3.7 \cdot 10^{-129}:\\ \;\;\;\;\mathsf{fma}\left(F, \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot {B}^{-1}, t\_0\right)\\ \mathbf{elif}\;F \leq 5800000:\\ \;\;\;\;\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot \frac{F}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B} + {\sin B}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))))
   (if (<= F -7.8e+136)
     (fma F (/ -1.0 (* (fma (* (* B B) F) -0.16666666666666666 F) B)) t_0)
     (if (<= F 3.7e-129)
       (fma
        F
        (* (sqrt (pow (+ (fma 2.0 x (* F F)) 2.0) -1.0)) (pow B -1.0))
        t_0)
       (if (<= F 5800000.0)
         (* (sqrt (pow (fma F F 2.0) -1.0)) (/ F (sin B)))
         (+
          (/ (- x) (* (fma 0.3333333333333333 (* B B) 1.0) B))
          (pow (sin B) -1.0)))))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double tmp;
	if (F <= -7.8e+136) {
		tmp = fma(F, (-1.0 / (fma(((B * B) * F), -0.16666666666666666, F) * B)), t_0);
	} else if (F <= 3.7e-129) {
		tmp = fma(F, (sqrt(pow((fma(2.0, x, (F * F)) + 2.0), -1.0)) * pow(B, -1.0)), t_0);
	} else if (F <= 5800000.0) {
		tmp = sqrt(pow(fma(F, F, 2.0), -1.0)) * (F / sin(B));
	} else {
		tmp = (-x / (fma(0.3333333333333333, (B * B), 1.0) * B)) + pow(sin(B), -1.0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (F <= -7.8e+136)
		tmp = fma(F, Float64(-1.0 / Float64(fma(Float64(Float64(B * B) * F), -0.16666666666666666, F) * B)), t_0);
	elseif (F <= 3.7e-129)
		tmp = fma(F, Float64(sqrt((Float64(fma(2.0, x, Float64(F * F)) + 2.0) ^ -1.0)) * (B ^ -1.0)), t_0);
	elseif (F <= 5800000.0)
		tmp = Float64(sqrt((fma(F, F, 2.0) ^ -1.0)) * Float64(F / sin(B)));
	else
		tmp = Float64(Float64(Float64(-x) / Float64(fma(0.3333333333333333, Float64(B * B), 1.0) * B)) + (sin(B) ^ -1.0));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -7.8e+136], N[(F * N[(-1.0 / N[(N[(N[(N[(B * B), $MachinePrecision] * F), $MachinePrecision] * -0.16666666666666666 + F), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[F, 3.7e-129], N[(F * N[(N[Sqrt[N[Power[N[(N[(2.0 * x + N[(F * F), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[B, -1.0], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[F, 5800000.0], N[(N[Sqrt[N[Power[N[(F * F + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-x) / N[(N[(0.3333333333333333 * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[B], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -7.80000000000000038e136

   1. Initial program 40.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 45.8%

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

   5. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 99.6

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in B around 0

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

   10. Applied rewrites 62.3%

       \[\leadsto \mathsf{fma}\left(F, \frac{-1}{\mathsf{fma}\left(\left(B \cdot B\right) \cdot F, -0.16666666666666666, F\right) \cdot \color{blue}{B}}, \frac{-x}{\tan B}\right) \]

   if -7.80000000000000038e136 < F < 3.7000000000000002e-129

   1. Initial program 96.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 80.6

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

   7. Applied rewrites 80.6%

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

   if 3.7000000000000002e-129 < F < 5.8e6

   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 x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sin.f64 79.8

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

   5. Applied rewrites 79.8%

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

   if 5.8e6 < F

   1. Initial program 54.8%

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

   3. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 99.7

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

   5. Applied rewrites 99.7%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. distribute-neg-frac N/A

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

      6. *-rgt-identity N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-/.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in B around 0

      \[\leadsto \frac{-x}{\color{blue}{B \cdot \left(1 + \frac{1}{3} \cdot {B}^{2}\right)}} + \frac{1}{\sin B} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{-x}{\color{blue}{\left(1 + \frac{1}{3} \cdot {B}^{2}\right) \cdot B}} + \frac{1}{\sin B} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-x}{\color{blue}{\left(1 + \frac{1}{3} \cdot {B}^{2}\right) \cdot B}} + \frac{1}{\sin B} \]

      3. +-commutative N/A

         \[\leadsto \frac{-x}{\color{blue}{\left(\frac{1}{3} \cdot {B}^{2} + 1\right)} \cdot B} + \frac{1}{\sin B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{-x}{\color{blue}{\mathsf{fma}\left(\frac{1}{3}, {B}^{2}, 1\right)} \cdot B} + \frac{1}{\sin B} \]

      5. unpow2 N/A

         \[\leadsto \frac{-x}{\mathsf{fma}\left(\frac{1}{3}, \color{blue}{B \cdot B}, 1\right) \cdot B} + \frac{1}{\sin B} \]

      6. lower-*.f64 76.3

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

   10. Applied rewrites 76.3%

       \[\leadsto \frac{-x}{\color{blue}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B}} + \frac{1}{\sin B} \]
3. Recombined 4 regimes into one program.

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7.8 \cdot 10^{+136}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{-1}{\mathsf{fma}\left(\left(B \cdot B\right) \cdot F, -0.16666666666666666, F\right) \cdot B}, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;F \leq 3.7 \cdot 10^{-129}:\\ \;\;\;\;\mathsf{fma}\left(F, \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot {B}^{-1}, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;F \leq 5800000:\\ \;\;\;\;\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot \frac{F}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B} + {\sin B}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -215000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x, \cos B, -1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}}}{\sin B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -215000000.0)
   (/ (fma (- x) (cos B) -1.0) (sin B))
   (if (<= F 5e+38)
     (fma F (/ (sqrt (pow (fma F F 2.0) -1.0)) (sin B)) (/ (- x) (tan B)))
     (/ (- 1.0 (* x (cos B))) (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -215000000.0) {
		tmp = fma(-x, cos(B), -1.0) / sin(B);
	} else if (F <= 5e+38) {
		tmp = fma(F, (sqrt(pow(fma(F, F, 2.0), -1.0)) / sin(B)), (-x / tan(B)));
	} else {
		tmp = (1.0 - (x * cos(B))) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -215000000.0)
		tmp = Float64(fma(Float64(-x), cos(B), -1.0) / sin(B));
	elseif (F <= 5e+38)
		tmp = fma(F, Float64(sqrt((fma(F, F, 2.0) ^ -1.0)) / sin(B)), Float64(Float64(-x) / tan(B)));
	else
		tmp = Float64(Float64(1.0 - Float64(x * cos(B))) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -215000000.0], N[(N[((-x) * N[Cos[B], $MachinePrecision] + -1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5e+38], N[(F * N[(N[Sqrt[N[Power[N[(F * F + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.15e8

   1. Initial program 58.7%

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

   3. Taylor expanded in F around -inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. div-add-rev N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*r* N/A

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-sin.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if -2.15e8 < F < 4.9999999999999997e38

   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. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 99.5

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

   7. Applied rewrites 99.5%

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

   if 4.9999999999999997e38 < F

   1. Initial program 50.5%

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

   3. Taylor expanded in F around inf

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

      1. associate-/l* N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. distribute-lft-neg-in N/A

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

      4. associate-/l* N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. associate-*r* N/A

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

      11. mul-1-neg N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-sin.f64 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -215000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x, \cos B, -1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}}}{\sin B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{+56}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x, \cos B, -1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 5000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}}, F, \left(-x\right) \cdot \cos B\right)}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1e+56)
   (/ (fma (- x) (cos B) -1.0) (sin B))
   (if (<= F 5000000.0)
     (/
      (fma (sqrt (pow (fma F F (fma 2.0 x 2.0)) -1.0)) F (* (- x) (cos B)))
      (sin B))
     (/ (- 1.0 (* x (cos B))) (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1e+56) {
		tmp = fma(-x, cos(B), -1.0) / sin(B);
	} else if (F <= 5000000.0) {
		tmp = fma(sqrt(pow(fma(F, F, fma(2.0, x, 2.0)), -1.0)), F, (-x * cos(B))) / sin(B);
	} else {
		tmp = (1.0 - (x * cos(B))) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1e+56)
		tmp = Float64(fma(Float64(-x), cos(B), -1.0) / sin(B));
	elseif (F <= 5000000.0)
		tmp = Float64(fma(sqrt((fma(F, F, fma(2.0, x, 2.0)) ^ -1.0)), F, Float64(Float64(-x) * cos(B))) / sin(B));
	else
		tmp = Float64(Float64(1.0 - Float64(x * cos(B))) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1e+56], N[(N[((-x) * N[Cos[B], $MachinePrecision] + -1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5000000.0], N[(N[(N[Sqrt[N[Power[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * F + N[((-x) * N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $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 < -1.00000000000000009e56

   1. Initial program 52.6%

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

   3. Taylor expanded in F around -inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. div-add-rev N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*r* N/A

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-sin.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if -1.00000000000000009e56 < F < 5e6

   1. Initial program 98.6%

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

   3. Taylor expanded in B around inf

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

      1. associate-/l* N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. associate-*l/ N/A

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

      4. distribute-lft-neg-in N/A

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

      5. associate-/l* N/A

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

      6. mul-1-neg N/A

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

      7. associate-*r/ N/A

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

      8. div-add-rev N/A

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

      9. lower-/.f64 N/A

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

   5. Applied rewrites 99.3%

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

   if 5e6 < F

   1. Initial program 54.8%

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

   3. Taylor expanded in F around inf

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

      1. associate-/l* N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. distribute-lft-neg-in N/A

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

      4. associate-/l* N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. associate-*r* N/A

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

      11. mul-1-neg N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-sin.f64 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{+56}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x, \cos B, -1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 5000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}}, F, \left(-x\right) \cdot \cos B\right)}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}}\\ t_1 := -\frac{x}{B}\\ \mathbf{if}\;F \leq -9 \cdot 10^{+26}:\\ \;\;\;\;t\_1 + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B}\\ \mathbf{elif}\;F \leq 820:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, F, \mathsf{fma}\left(0.16666666666666666 \cdot F, t\_0, 0.3333333333333333 \cdot x\right) \cdot \left(B \cdot B\right) - x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_1 + {\sin B}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (sqrt (pow (+ (fma 2.0 x (* F F)) 2.0) -1.0))) (t_1 (- (/ x B))))
   (if (<= F -9e+26)
     (+ t_1 (/ (- (* -0.16666666666666666 (* B B)) 1.0) B))
     (if (<= F 820.0)
       (/
        (fma
         t_0
         F
         (-
          (*
           (fma (* 0.16666666666666666 F) t_0 (* 0.3333333333333333 x))
           (* B B))
          x))
        B)
       (+ t_1 (pow (sin B) -1.0))))))

C

double code(double F, double B, double x) {
	double t_0 = sqrt(pow((fma(2.0, x, (F * F)) + 2.0), -1.0));
	double t_1 = -(x / B);
	double tmp;
	if (F <= -9e+26) {
		tmp = t_1 + (((-0.16666666666666666 * (B * B)) - 1.0) / B);
	} else if (F <= 820.0) {
		tmp = fma(t_0, F, ((fma((0.16666666666666666 * F), t_0, (0.3333333333333333 * x)) * (B * B)) - x)) / B;
	} else {
		tmp = t_1 + pow(sin(B), -1.0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = sqrt((Float64(fma(2.0, x, Float64(F * F)) + 2.0) ^ -1.0))
	t_1 = Float64(-Float64(x / B))
	tmp = 0.0
	if (F <= -9e+26)
		tmp = Float64(t_1 + Float64(Float64(Float64(-0.16666666666666666 * Float64(B * B)) - 1.0) / B));
	elseif (F <= 820.0)
		tmp = Float64(fma(t_0, F, Float64(Float64(fma(Float64(0.16666666666666666 * F), t_0, Float64(0.3333333333333333 * x)) * Float64(B * B)) - x)) / B);
	else
		tmp = Float64(t_1 + (sin(B) ^ -1.0));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[Sqrt[N[Power[N[(N[(2.0 * x + N[(F * F), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = (-N[(x / B), $MachinePrecision])}, If[LessEqual[F, -9e+26], N[(t$95$1 + N[(N[(N[(-0.16666666666666666 * N[(B * B), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 820.0], N[(N[(t$95$0 * F + N[(N[(N[(N[(0.16666666666666666 * F), $MachinePrecision] * t$95$0 + N[(0.3333333333333333 * x), $MachinePrecision]), $MachinePrecision] * N[(B * B), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], N[(t$95$1 + N[Power[N[Sin[B], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -8.99999999999999957e26

   1. Initial program 56.9%

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

   3. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 30.6%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 51.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B} \]

   9. Taylor expanded in B around 0

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

       1. lower-/.f64 39.6

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

   11. Applied rewrites 39.6%

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

   if -8.99999999999999957e26 < F < 820

   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

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 50.1

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

   5. Applied rewrites 50.1%

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

   6. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \frac{1}{3} \cdot x\right)\right) - x}{B}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

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

   8. Applied rewrites 50.2%

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

   if 820 < F

   1. Initial program 54.8%

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

   3. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in B around 0

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

      1. lower-/.f64 76.2

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

   8. Applied rewrites 76.2%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -9 \cdot 10^{+26}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B}\\ \mathbf{elif}\;F \leq 820:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}}, F, \mathsf{fma}\left(0.16666666666666666 \cdot F, \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}}, 0.3333333333333333 \cdot x\right) \cdot \left(B \cdot B\right) - x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + {\sin B}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 51.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}}\\ \mathbf{if}\;F \leq -5 \cdot 10^{+18}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B}\\ \mathbf{elif}\;F \leq 2.05 \cdot 10^{+124}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, F, \mathsf{fma}\left(0.16666666666666666 \cdot F, t\_0, 0.3333333333333333 \cdot x\right) \cdot \left(B \cdot B\right) - x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (sqrt (pow (+ (fma 2.0 x (* F F)) 2.0) -1.0))))
   (if (<= F -5e+18)
     (+ (- (/ x B)) (/ (- (* -0.16666666666666666 (* B B)) 1.0) B))
     (if (<= F 2.05e+124)
       (/
        (fma
         t_0
         F
         (-
          (*
           (fma (* 0.16666666666666666 F) t_0 (* 0.3333333333333333 x))
           (* B B))
          x))
        B)
       (/ (- 1.0 x) B)))))

C

double code(double F, double B, double x) {
	double t_0 = sqrt(pow((fma(2.0, x, (F * F)) + 2.0), -1.0));
	double tmp;
	if (F <= -5e+18) {
		tmp = -(x / B) + (((-0.16666666666666666 * (B * B)) - 1.0) / B);
	} else if (F <= 2.05e+124) {
		tmp = fma(t_0, F, ((fma((0.16666666666666666 * F), t_0, (0.3333333333333333 * x)) * (B * B)) - x)) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = sqrt((Float64(fma(2.0, x, Float64(F * F)) + 2.0) ^ -1.0))
	tmp = 0.0
	if (F <= -5e+18)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(Float64(Float64(-0.16666666666666666 * Float64(B * B)) - 1.0) / B));
	elseif (F <= 2.05e+124)
		tmp = Float64(fma(t_0, F, Float64(Float64(fma(Float64(0.16666666666666666 * F), t_0, Float64(0.3333333333333333 * x)) * Float64(B * B)) - x)) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[Sqrt[N[Power[N[(N[(2.0 * x + N[(F * F), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[F, -5e+18], N[((-N[(x / B), $MachinePrecision]) + N[(N[(N[(-0.16666666666666666 * N[(B * B), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.05e+124], N[(N[(t$95$0 * F + N[(N[(N[(N[(0.16666666666666666 * F), $MachinePrecision] * t$95$0 + N[(0.3333333333333333 * x), $MachinePrecision]), $MachinePrecision] * N[(B * B), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -5e18

   1. Initial program 58.1%

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

   3. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 29.8%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 49.5%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B} \]

   9. Taylor expanded in B around 0

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

       1. lower-/.f64 38.5

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

   11. Applied rewrites 38.5%

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

   if -5e18 < F < 2.05000000000000001e124

   1. Initial program 98.6%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 49.5

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

   5. Applied rewrites 49.5%

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

   6. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \frac{1}{3} \cdot x\right)\right) - x}{B}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

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

   8. Applied rewrites 49.8%

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

   if 2.05000000000000001e124 < F

   1. Initial program 37.4%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 34.7

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

   5. Applied rewrites 34.7%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 46.5%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{+18}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B}\\ \mathbf{elif}\;F \leq 2.05 \cdot 10^{+124}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}}, F, \mathsf{fma}\left(0.16666666666666666 \cdot F, \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}}, 0.3333333333333333 \cdot x\right) \cdot \left(B \cdot B\right) - x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.95 \cdot 10^{+135}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{-1}{\mathsf{fma}\left(\left(B \cdot B\right) \cdot F, -0.16666666666666666, F\right) \cdot B}, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;F \leq 3.7 \cdot 10^{-129}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{\tan B}, x, \frac{\sqrt{{\left(\mathsf{fma}\left(x, 2, F \cdot F\right) + 2\right)}^{-1}} \cdot F}{B}\right)\\ \mathbf{elif}\;F \leq 5800000:\\ \;\;\;\;\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot \frac{F}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B} + {\sin B}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.95e+135)
   (fma
    F
    (/ -1.0 (* (fma (* (* B B) F) -0.16666666666666666 F) B))
    (/ (- x) (tan B)))
   (if (<= F 3.7e-129)
     (fma
      (/ -1.0 (tan B))
      x
      (/ (* (sqrt (pow (+ (fma x 2.0 (* F F)) 2.0) -1.0)) F) B))
     (if (<= F 5800000.0)
       (* (sqrt (pow (fma F F 2.0) -1.0)) (/ F (sin B)))
       (+
        (/ (- x) (* (fma 0.3333333333333333 (* B B) 1.0) B))
        (pow (sin B) -1.0))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.95e+135) {
		tmp = fma(F, (-1.0 / (fma(((B * B) * F), -0.16666666666666666, F) * B)), (-x / tan(B)));
	} else if (F <= 3.7e-129) {
		tmp = fma((-1.0 / tan(B)), x, ((sqrt(pow((fma(x, 2.0, (F * F)) + 2.0), -1.0)) * F) / B));
	} else if (F <= 5800000.0) {
		tmp = sqrt(pow(fma(F, F, 2.0), -1.0)) * (F / sin(B));
	} else {
		tmp = (-x / (fma(0.3333333333333333, (B * B), 1.0) * B)) + pow(sin(B), -1.0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.95e+135)
		tmp = fma(F, Float64(-1.0 / Float64(fma(Float64(Float64(B * B) * F), -0.16666666666666666, F) * B)), Float64(Float64(-x) / tan(B)));
	elseif (F <= 3.7e-129)
		tmp = fma(Float64(-1.0 / tan(B)), x, Float64(Float64(sqrt((Float64(fma(x, 2.0, Float64(F * F)) + 2.0) ^ -1.0)) * F) / B));
	elseif (F <= 5800000.0)
		tmp = Float64(sqrt((fma(F, F, 2.0) ^ -1.0)) * Float64(F / sin(B)));
	else
		tmp = Float64(Float64(Float64(-x) / Float64(fma(0.3333333333333333, Float64(B * B), 1.0) * B)) + (sin(B) ^ -1.0));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.95e+135], N[(F * N[(-1.0 / N[(N[(N[(N[(B * B), $MachinePrecision] * F), $MachinePrecision] * -0.16666666666666666 + F), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.7e-129], N[(N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision] * x + N[(N[(N[Sqrt[N[Power[N[(N[(x * 2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * F), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5800000.0], N[(N[Sqrt[N[Power[N[(F * F + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-x) / N[(N[(0.3333333333333333 * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[B], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.95000000000000016e135

   1. Initial program 41.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 47.0%

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

   5. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 99.6

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in B around 0

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

   10. Applied rewrites 63.1%

       \[\leadsto \mathsf{fma}\left(F, \frac{-1}{\mathsf{fma}\left(\left(B \cdot B\right) \cdot F, -0.16666666666666666, F\right) \cdot \color{blue}{B}}, \frac{-x}{\tan B}\right) \]

   if -1.95000000000000016e135 < F < 3.7000000000000002e-129

   1. Initial program 96.1%

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

   3. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 64.6%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 29.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B} \]
   9. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{\frac{-1}{6} \cdot \left(B \cdot B\right) - 1}{B}} \]

   10. Applied rewrites 29.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{\tan B}, x, \frac{\left(B \cdot B\right) \cdot -0.16666666666666666 - 1}{B}\right)} \]

   11. Taylor expanded in B around 0

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

   13. Applied rewrites 80.3%

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

   if 3.7000000000000002e-129 < F < 5.8e6

   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 x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sin.f64 79.8

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

   5. Applied rewrites 79.8%

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

   if 5.8e6 < F

   1. Initial program 54.8%

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

   3. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 99.7

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

   5. Applied rewrites 99.7%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. distribute-neg-frac N/A

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

      6. *-rgt-identity N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-/.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in B around 0

      \[\leadsto \frac{-x}{\color{blue}{B \cdot \left(1 + \frac{1}{3} \cdot {B}^{2}\right)}} + \frac{1}{\sin B} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{-x}{\color{blue}{\left(1 + \frac{1}{3} \cdot {B}^{2}\right) \cdot B}} + \frac{1}{\sin B} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-x}{\color{blue}{\left(1 + \frac{1}{3} \cdot {B}^{2}\right) \cdot B}} + \frac{1}{\sin B} \]

      3. +-commutative N/A

         \[\leadsto \frac{-x}{\color{blue}{\left(\frac{1}{3} \cdot {B}^{2} + 1\right)} \cdot B} + \frac{1}{\sin B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{-x}{\color{blue}{\mathsf{fma}\left(\frac{1}{3}, {B}^{2}, 1\right)} \cdot B} + \frac{1}{\sin B} \]

      5. unpow2 N/A

         \[\leadsto \frac{-x}{\mathsf{fma}\left(\frac{1}{3}, \color{blue}{B \cdot B}, 1\right) \cdot B} + \frac{1}{\sin B} \]

      6. lower-*.f64 76.3

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

   10. Applied rewrites 76.3%

       \[\leadsto \frac{-x}{\color{blue}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B}} + \frac{1}{\sin B} \]
3. Recombined 4 regimes into one program.

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.95 \cdot 10^{+135}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{-1}{\mathsf{fma}\left(\left(B \cdot B\right) \cdot F, -0.16666666666666666, F\right) \cdot B}, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;F \leq 3.7 \cdot 10^{-129}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{\tan B}, x, \frac{\sqrt{{\left(\mathsf{fma}\left(x, 2, F \cdot F\right) + 2\right)}^{-1}} \cdot F}{B}\right)\\ \mathbf{elif}\;F \leq 5800000:\\ \;\;\;\;\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot \frac{F}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B} + {\sin B}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 99.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -7:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x, \cos B, -1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\sqrt{0.5}}{\sin B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -7.0)
   (/ (fma (- x) (cos B) -1.0) (sin B))
   (if (<= F 1.4)
     (fma F (/ (sqrt 0.5) (sin B)) (/ (- x) (tan B)))
     (/ (- 1.0 (* x (cos B))) (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -7.0) {
		tmp = fma(-x, cos(B), -1.0) / sin(B);
	} else if (F <= 1.4) {
		tmp = fma(F, (sqrt(0.5) / sin(B)), (-x / tan(B)));
	} else {
		tmp = (1.0 - (x * cos(B))) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -7.0)
		tmp = Float64(fma(Float64(-x), cos(B), -1.0) / sin(B));
	elseif (F <= 1.4)
		tmp = fma(F, Float64(sqrt(0.5) / sin(B)), Float64(Float64(-x) / tan(B)));
	else
		tmp = Float64(Float64(1.0 - Float64(x * cos(B))) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -7.0], N[(N[((-x) * N[Cos[B], $MachinePrecision] + -1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.4], N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -7

   1. Initial program 58.7%

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

   3. Taylor expanded in F around -inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. div-add-rev N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*r* N/A

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-sin.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if -7 < F < 1.3999999999999999

   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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in F around 0

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

   10. Applied rewrites 98.9%

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

   if 1.3999999999999999 < F

   1. Initial program 54.8%

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

   3. Taylor expanded in F around inf

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

      1. associate-/l* N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. distribute-lft-neg-in N/A

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

      4. associate-/l* N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. associate-*r* N/A

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

      11. mul-1-neg N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-sin.f64 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x, \cos B, -1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\sqrt{0.5}}{\sin B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 68.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-78} \lor \neg \left(x \leq 7.2 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{-x}{\tan B} + {B}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot \frac{F}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (or (<= x -7.5e-78) (not (<= x 7.2e-12)))
   (+ (/ (- x) (tan B)) (pow B -1.0))
   (* (sqrt (pow (fma F F 2.0) -1.0)) (/ F (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if ((x <= -7.5e-78) || !(x <= 7.2e-12)) {
		tmp = (-x / tan(B)) + pow(B, -1.0);
	} else {
		tmp = sqrt(pow(fma(F, F, 2.0), -1.0)) * (F / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if ((x <= -7.5e-78) || !(x <= 7.2e-12))
		tmp = Float64(Float64(Float64(-x) / tan(B)) + (B ^ -1.0));
	else
		tmp = Float64(sqrt((fma(F, F, 2.0) ^ -1.0)) * Float64(F / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[Or[LessEqual[x, -7.5e-78], N[Not[LessEqual[x, 7.2e-12]], $MachinePrecision]], N[(N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision] + N[Power[B, -1.0], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[Power[N[(F * F + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.50000000000000041e-78 or 7.2e-12 < x

   1. Initial program 80.7%

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

   3. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 85.2

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

   5. Applied rewrites 85.2%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. distribute-neg-frac N/A

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

      6. *-rgt-identity N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-/.f64 85.3

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

   7. Applied rewrites 85.3%

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

   8. Taylor expanded in B around 0

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

   10. Applied rewrites 86.1%

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

   if -7.50000000000000041e-78 < x < 7.2e-12

   1. Initial program 68.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sin.f64 52.8

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

   5. Applied rewrites 52.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\sin B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-78} \lor \neg \left(x \leq 7.2 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{-x}{\tan B} + {B}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot \frac{F}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 55.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.14:\\ \;\;\;\;\left(-x\right) \cdot \frac{\mathsf{fma}\left(-0.022222222222222223 \cdot \left(B \cdot B\right) - 0.3333333333333333, B \cdot B, 1\right)}{B} + \frac{\sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}} \cdot \mathsf{fma}\left(0.16666666666666666 \cdot F, B \cdot B, F\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B} + {B}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 0.14)
   (+
    (*
     (- x)
     (/
      (fma
       (- (* -0.022222222222222223 (* B B)) 0.3333333333333333)
       (* B B)
       1.0)
      B))
    (/
     (*
      (sqrt (pow (fma F F (fma 2.0 x 2.0)) -1.0))
      (fma (* 0.16666666666666666 F) (* B B) F))
     B))
   (+ (/ (- x) (tan B)) (pow B -1.0))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 0.14) {
		tmp = (-x * (fma(((-0.022222222222222223 * (B * B)) - 0.3333333333333333), (B * B), 1.0) / B)) + ((sqrt(pow(fma(F, F, fma(2.0, x, 2.0)), -1.0)) * fma((0.16666666666666666 * F), (B * B), F)) / B);
	} else {
		tmp = (-x / tan(B)) + pow(B, -1.0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 0.14)
		tmp = Float64(Float64(Float64(-x) * Float64(fma(Float64(Float64(-0.022222222222222223 * Float64(B * B)) - 0.3333333333333333), Float64(B * B), 1.0) / B)) + Float64(Float64(sqrt((fma(F, F, fma(2.0, x, 2.0)) ^ -1.0)) * fma(Float64(0.16666666666666666 * F), Float64(B * B), F)) / B));
	else
		tmp = Float64(Float64(Float64(-x) / tan(B)) + (B ^ -1.0));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 0.14], N[(N[((-x) * N[(N[(N[(N[(-0.022222222222222223 * N[(B * B), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[N[Power[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * N[(N[(0.16666666666666666 * F), $MachinePrecision] * N[(B * B), $MachinePrecision] + F), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], N[(N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision] + N[Power[B, -1.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 0.14000000000000001

   1. Initial program 73.3%

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

   3. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 58.6%

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

   6. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \color{blue}{\frac{1 + {B}^{2} \cdot \left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right)}{B}}\right) + \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(\frac{1}{6} \cdot F, B \cdot B, F\right)}{B} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \color{blue}{\frac{1 + {B}^{2} \cdot \left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right)}{B}}\right) + \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(\frac{1}{6} \cdot F, B \cdot B, F\right)}{B} \]

      2. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{\color{blue}{{B}^{2} \cdot \left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right) + 1}}{B}\right) + \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(\frac{1}{6} \cdot F, B \cdot B, F\right)}{B} \]

      3. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{\color{blue}{\left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right) \cdot {B}^{2}} + 1}{B}\right) + \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(\frac{1}{6} \cdot F, B \cdot B, F\right)}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}, {B}^{2}, 1\right)}}{B}\right) + \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(\frac{1}{6} \cdot F, B \cdot B, F\right)}{B} \]

      5. lower--.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}}, {B}^{2}, 1\right)}{B}\right) + \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(\frac{1}{6} \cdot F, B \cdot B, F\right)}{B} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{45} \cdot {B}^{2}} - \frac{1}{3}, {B}^{2}, 1\right)}{B}\right) + \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(\frac{1}{6} \cdot F, B \cdot B, F\right)}{B} \]

      7. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\frac{-1}{45} \cdot \color{blue}{\left(B \cdot B\right)} - \frac{1}{3}, {B}^{2}, 1\right)}{B}\right) + \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(\frac{1}{6} \cdot F, B \cdot B, F\right)}{B} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\frac{-1}{45} \cdot \color{blue}{\left(B \cdot B\right)} - \frac{1}{3}, {B}^{2}, 1\right)}{B}\right) + \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(\frac{1}{6} \cdot F, B \cdot B, F\right)}{B} \]

      9. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\frac{-1}{45} \cdot \left(B \cdot B\right) - \frac{1}{3}, \color{blue}{B \cdot B}, 1\right)}{B}\right) + \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(\frac{1}{6} \cdot F, B \cdot B, F\right)}{B} \]

      10. lower-*.f64 53.5

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(-0.022222222222222223 \cdot \left(B \cdot B\right) - 0.3333333333333333, \color{blue}{B \cdot B}, 1\right)}{B}\right) + \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(0.16666666666666666 \cdot F, B \cdot B, F\right)}{B} \]

   8. Applied rewrites 53.5%

      \[\leadsto \left(-x \cdot \color{blue}{\frac{\mathsf{fma}\left(-0.022222222222222223 \cdot \left(B \cdot B\right) - 0.3333333333333333, B \cdot B, 1\right)}{B}}\right) + \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(0.16666666666666666 \cdot F, B \cdot B, F\right)}{B} \]

   if 0.14000000000000001 < B

   1. Initial program 76.9%

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

   3. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 56.8

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

   5. Applied rewrites 56.8%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. distribute-neg-frac N/A

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

      6. *-rgt-identity N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-/.f64 56.8

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

   7. Applied rewrites 56.8%

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

   8. Taylor expanded in B around 0

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

   10. Applied rewrites 44.4%

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

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 0.14:\\ \;\;\;\;\left(-x\right) \cdot \frac{\mathsf{fma}\left(-0.022222222222222223 \cdot \left(B \cdot B\right) - 0.3333333333333333, B \cdot B, 1\right)}{B} + \frac{\sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}} \cdot \mathsf{fma}\left(0.16666666666666666 \cdot F, B \cdot B, F\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B} + {B}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 51.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -7.6 \cdot 10^{+18}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B}\\ \mathbf{elif}\;F \leq 5.4 \cdot 10^{+100}:\\ \;\;\;\;\left(-\frac{\mathsf{fma}\left(\left(B \cdot B\right) \cdot x, -0.3333333333333333, x\right)}{B}\right) + \frac{\sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}} \cdot \mathsf{fma}\left(0.16666666666666666 \cdot F, B \cdot B, F\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -7.6e+18)
   (+ (- (/ x B)) (/ (- (* -0.16666666666666666 (* B B)) 1.0) B))
   (if (<= F 5.4e+100)
     (+
      (- (/ (fma (* (* B B) x) -0.3333333333333333 x) B))
      (/
       (*
        (sqrt (pow (fma F F (fma 2.0 x 2.0)) -1.0))
        (fma (* 0.16666666666666666 F) (* B B) F))
       B))
     (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -7.6e+18) {
		tmp = -(x / B) + (((-0.16666666666666666 * (B * B)) - 1.0) / B);
	} else if (F <= 5.4e+100) {
		tmp = -(fma(((B * B) * x), -0.3333333333333333, x) / B) + ((sqrt(pow(fma(F, F, fma(2.0, x, 2.0)), -1.0)) * fma((0.16666666666666666 * F), (B * B), F)) / B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -7.6e+18)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(Float64(Float64(-0.16666666666666666 * Float64(B * B)) - 1.0) / B));
	elseif (F <= 5.4e+100)
		tmp = Float64(Float64(-Float64(fma(Float64(Float64(B * B) * x), -0.3333333333333333, x) / B)) + Float64(Float64(sqrt((fma(F, F, fma(2.0, x, 2.0)) ^ -1.0)) * fma(Float64(0.16666666666666666 * F), Float64(B * B), F)) / B));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -7.6e+18], N[((-N[(x / B), $MachinePrecision]) + N[(N[(N[(-0.16666666666666666 * N[(B * B), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.4e+100], N[((-N[(N[(N[(N[(B * B), $MachinePrecision] * x), $MachinePrecision] * -0.3333333333333333 + x), $MachinePrecision] / B), $MachinePrecision]) + N[(N[(N[Sqrt[N[Power[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * N[(N[(0.16666666666666666 * F), $MachinePrecision] * N[(B * B), $MachinePrecision] + F), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -7.6e18

   1. Initial program 57.5%

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

   3. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 30.2%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 50.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B} \]

   9. Taylor expanded in B around 0

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

       1. lower-/.f64 39.1

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

   11. Applied rewrites 39.1%

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

   if -7.6e18 < F < 5.39999999999999997e100

   1. Initial program 98.6%

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

   3. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 61.6%

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

   6. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x + \frac{-1}{3} \cdot \left({B}^{2} \cdot x\right)}{B}}\right) + \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(\frac{1}{6} \cdot F, B \cdot B, F\right)}{B} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 50.4

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

   8. Applied rewrites 50.4%

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

   if 5.39999999999999997e100 < F

   1. Initial program 42.5%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 33.6

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

   5. Applied rewrites 33.6%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 44.5%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7.6 \cdot 10^{+18}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B}\\ \mathbf{elif}\;F \leq 5.4 \cdot 10^{+100}:\\ \;\;\;\;\left(-\frac{\mathsf{fma}\left(\left(B \cdot B\right) \cdot x, -0.3333333333333333, x\right)}{B}\right) + \frac{\sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}} \cdot \mathsf{fma}\left(0.16666666666666666 \cdot F, B \cdot B, F\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 51.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -7.6 \cdot 10^{+18}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B}\\ \mathbf{elif}\;F \leq 5.4 \cdot 10^{+100}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}}, \mathsf{fma}\left(0.16666666666666666 \cdot F, B \cdot B, F\right), \left(0.3333333333333333 \cdot \left(B \cdot B\right)\right) \cdot x\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -7.6e+18)
   (+ (- (/ x B)) (/ (- (* -0.16666666666666666 (* B B)) 1.0) B))
   (if (<= F 5.4e+100)
     (/
      (-
       (fma
        (sqrt (pow (fma F F (fma 2.0 x 2.0)) -1.0))
        (fma (* 0.16666666666666666 F) (* B B) F)
        (* (* 0.3333333333333333 (* B B)) x))
       x)
      B)
     (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -7.6e+18) {
		tmp = -(x / B) + (((-0.16666666666666666 * (B * B)) - 1.0) / B);
	} else if (F <= 5.4e+100) {
		tmp = (fma(sqrt(pow(fma(F, F, fma(2.0, x, 2.0)), -1.0)), fma((0.16666666666666666 * F), (B * B), F), ((0.3333333333333333 * (B * B)) * x)) - x) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -7.6e+18)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(Float64(Float64(-0.16666666666666666 * Float64(B * B)) - 1.0) / B));
	elseif (F <= 5.4e+100)
		tmp = Float64(Float64(fma(sqrt((fma(F, F, fma(2.0, x, 2.0)) ^ -1.0)), fma(Float64(0.16666666666666666 * F), Float64(B * B), F), Float64(Float64(0.3333333333333333 * Float64(B * B)) * x)) - x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -7.6e+18], N[((-N[(x / B), $MachinePrecision]) + N[(N[(N[(-0.16666666666666666 * N[(B * B), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.4e+100], N[(N[(N[(N[Sqrt[N[Power[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * N[(N[(0.16666666666666666 * F), $MachinePrecision] * N[(B * B), $MachinePrecision] + F), $MachinePrecision] + N[(N[(0.3333333333333333 * N[(B * B), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -7.6e18

   1. Initial program 57.5%

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

   3. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 30.2%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 50.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B} \]

   9. Taylor expanded in B around 0

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

       1. lower-/.f64 39.1

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

   11. Applied rewrites 39.1%

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

   if -7.6e18 < F < 5.39999999999999997e100

   1. Initial program 98.6%

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

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \frac{1}{3} \cdot x\right)\right) - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

   5. Applied rewrites 50.3%

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

   if 5.39999999999999997e100 < F

   1. Initial program 42.5%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 33.6

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

   5. Applied rewrites 33.6%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 44.5%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7.6 \cdot 10^{+18}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B}\\ \mathbf{elif}\;F \leq 5.4 \cdot 10^{+100}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}}, \mathsf{fma}\left(0.16666666666666666 \cdot F, B \cdot B, F\right), \left(0.3333333333333333 \cdot \left(B \cdot B\right)\right) \cdot x\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 42.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}}\\ \mathbf{if}\;F \leq -8.8 \cdot 10^{+18}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B}\\ \mathbf{elif}\;F \leq -8 \cdot 10^{-185}:\\ \;\;\;\;\frac{t\_0 \cdot F}{B}\\ \mathbf{elif}\;F \leq 2.4 \cdot 10^{-164}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 32000000000:\\ \;\;\;\;t\_0 \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (sqrt (pow (fma F F 2.0) -1.0))))
   (if (<= F -8.8e+18)
     (+ (- (/ x B)) (/ (- (* -0.16666666666666666 (* B B)) 1.0) B))
     (if (<= F -8e-185)
       (/ (* t_0 F) B)
       (if (<= F 2.4e-164)
         (/ (- x) B)
         (if (<= F 32000000000.0) (* t_0 (/ F B)) (/ (- 1.0 x) B)))))))

C

double code(double F, double B, double x) {
	double t_0 = sqrt(pow(fma(F, F, 2.0), -1.0));
	double tmp;
	if (F <= -8.8e+18) {
		tmp = -(x / B) + (((-0.16666666666666666 * (B * B)) - 1.0) / B);
	} else if (F <= -8e-185) {
		tmp = (t_0 * F) / B;
	} else if (F <= 2.4e-164) {
		tmp = -x / B;
	} else if (F <= 32000000000.0) {
		tmp = t_0 * (F / B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = sqrt((fma(F, F, 2.0) ^ -1.0))
	tmp = 0.0
	if (F <= -8.8e+18)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(Float64(Float64(-0.16666666666666666 * Float64(B * B)) - 1.0) / B));
	elseif (F <= -8e-185)
		tmp = Float64(Float64(t_0 * F) / B);
	elseif (F <= 2.4e-164)
		tmp = Float64(Float64(-x) / B);
	elseif (F <= 32000000000.0)
		tmp = Float64(t_0 * Float64(F / B));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[Sqrt[N[Power[N[(F * F + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[F, -8.8e+18], N[((-N[(x / B), $MachinePrecision]) + N[(N[(N[(-0.16666666666666666 * N[(B * B), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -8e-185], N[(N[(t$95$0 * F), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 2.4e-164], N[((-x) / B), $MachinePrecision], If[LessEqual[F, 32000000000.0], N[(t$95$0 * N[(F / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -8.8e18

   1. Initial program 56.9%

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

   3. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 30.6%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 51.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B} \]

   9. Taylor expanded in B around 0

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

       1. lower-/.f64 39.6

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

   11. Applied rewrites 39.6%

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

   if -8.8e18 < F < -7.9999999999999999e-185

   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

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 53.9

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

   5. Applied rewrites 53.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 38.2%

      \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot F}{B} \]

   if -7.9999999999999999e-185 < F < 2.39999999999999983e-164

   1. Initial program 99.4%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 47.3

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

   5. Applied rewrites 47.3%

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

   6. Taylor expanded in F around 0

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

   8. Applied rewrites 39.6%

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

   if 2.39999999999999983e-164 < F < 3.2e10

   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

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \cdot F - x}{B} \]

      12. lower-fma.f64 47.9

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 47.9%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{F}{B} \cdot \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 32.7%

      \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \color{blue}{\frac{F}{B}} \]

   if 3.2e10 < F

   1. Initial program 54.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \cdot F - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \cdot F - x}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \cdot F - x}{B} \]

      12. lower-fma.f64 37.6

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 37.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 46.0%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 5 regimes into one program.

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -8.8 \cdot 10^{+18}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B}\\ \mathbf{elif}\;F \leq -8 \cdot 10^{-185}:\\ \;\;\;\;\frac{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot F}{B}\\ \mathbf{elif}\;F \leq 2.4 \cdot 10^{-164}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 32000000000:\\ \;\;\;\;\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 42.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot \frac{F}{B}\\ \mathbf{if}\;F \leq -8.8 \cdot 10^{+18}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B}\\ \mathbf{elif}\;F \leq -8 \cdot 10^{-185}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 2.4 \cdot 10^{-164}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 32000000000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (sqrt (pow (fma F F 2.0) -1.0)) (/ F B))))
   (if (<= F -8.8e+18)
     (+ (- (/ x B)) (/ (- (* -0.16666666666666666 (* B B)) 1.0) B))
     (if (<= F -8e-185)
       t_0
       (if (<= F 2.4e-164)
         (/ (- x) B)
         (if (<= F 32000000000.0) t_0 (/ (- 1.0 x) B)))))))

C

double code(double F, double B, double x) {
	double t_0 = sqrt(pow(fma(F, F, 2.0), -1.0)) * (F / B);
	double tmp;
	if (F <= -8.8e+18) {
		tmp = -(x / B) + (((-0.16666666666666666 * (B * B)) - 1.0) / B);
	} else if (F <= -8e-185) {
		tmp = t_0;
	} else if (F <= 2.4e-164) {
		tmp = -x / B;
	} else if (F <= 32000000000.0) {
		tmp = t_0;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(sqrt((fma(F, F, 2.0) ^ -1.0)) * Float64(F / B))
	tmp = 0.0
	if (F <= -8.8e+18)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(Float64(Float64(-0.16666666666666666 * Float64(B * B)) - 1.0) / B));
	elseif (F <= -8e-185)
		tmp = t_0;
	elseif (F <= 2.4e-164)
		tmp = Float64(Float64(-x) / B);
	elseif (F <= 32000000000.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[Sqrt[N[Power[N[(F * F + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -8.8e+18], N[((-N[(x / B), $MachinePrecision]) + N[(N[(N[(-0.16666666666666666 * N[(B * B), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -8e-185], t$95$0, If[LessEqual[F, 2.4e-164], N[((-x) / B), $MachinePrecision], If[LessEqual[F, 32000000000.0], t$95$0, N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -8.8e18

   1. Initial program 56.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{\left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      2. associate-*l* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{\left({B}^{2} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)} \cdot \frac{1}{6} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      3. associate-*r* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{{B}^{2} \cdot \left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}\right)} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      4. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{{B}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      5. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}{B}} \]

   5. Applied rewrites 30.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(0.16666666666666666 \cdot F, B \cdot B, F\right)}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\frac{-1}{6} \cdot {B}^{2} - 1}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 51.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B} \]

   9. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{\frac{-1}{6} \cdot \left(B \cdot B\right) - 1}{B} \]
   10. Step-by-step derivation

       1. lower-/.f64 39.6

          \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B} \]

   11. Applied rewrites 39.6%

       \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B} \]

   if -8.8e18 < F < -7.9999999999999999e-185 or 2.39999999999999983e-164 < F < 3.2e10

   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

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \cdot F - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \cdot F - x}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \cdot F - x}{B} \]

      12. lower-fma.f64 51.4

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 51.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{F}{B} \cdot \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 35.9%

      \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \color{blue}{\frac{F}{B}} \]

   if -7.9999999999999999e-185 < F < 2.39999999999999983e-164

   1. Initial program 99.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \cdot F - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \cdot F - x}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \cdot F - x}{B} \]

      12. lower-fma.f64 47.3

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 47.3%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 39.6%

      \[\leadsto \frac{-x}{B} \]

   if 3.2e10 < F

   1. Initial program 54.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \cdot F - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \cdot F - x}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \cdot F - x}{B} \]

      12. lower-fma.f64 37.6

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 37.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 46.0%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 4 regimes into one program.

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -8.8 \cdot 10^{+18}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B}\\ \mathbf{elif}\;F \leq -8 \cdot 10^{-185}:\\ \;\;\;\;\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot \frac{F}{B}\\ \mathbf{elif}\;F \leq 2.4 \cdot 10^{-164}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 32000000000:\\ \;\;\;\;\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 51.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.02 \cdot 10^{+121}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B}\\ \mathbf{elif}\;F \leq 5000000:\\ \;\;\;\;\frac{\sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.02e+121)
   (+ (- (/ x B)) (/ (- (* -0.16666666666666666 (* B B)) 1.0) B))
   (if (<= F 5000000.0)
     (/ (- (* (sqrt (pow (fma F F (fma 2.0 x 2.0)) -1.0)) F) x) B)
     (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.02e+121) {
		tmp = -(x / B) + (((-0.16666666666666666 * (B * B)) - 1.0) / B);
	} else if (F <= 5000000.0) {
		tmp = ((sqrt(pow(fma(F, F, fma(2.0, x, 2.0)), -1.0)) * F) - x) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.02e+121)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(Float64(Float64(-0.16666666666666666 * Float64(B * B)) - 1.0) / B));
	elseif (F <= 5000000.0)
		tmp = Float64(Float64(Float64(sqrt((fma(F, F, fma(2.0, x, 2.0)) ^ -1.0)) * F) - x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.02e+121], N[((-N[(x / B), $MachinePrecision]) + N[(N[(N[(-0.16666666666666666 * N[(B * B), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5000000.0], N[(N[(N[(N[Sqrt[N[Power[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.02000000000000005e121

   1. Initial program 44.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{\left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      2. associate-*l* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{\left({B}^{2} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)} \cdot \frac{1}{6} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      3. associate-*r* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{{B}^{2} \cdot \left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}\right)} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      4. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{{B}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      5. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}{B}} \]

   5. Applied rewrites 20.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(0.16666666666666666 \cdot F, B \cdot B, F\right)}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\frac{-1}{6} \cdot {B}^{2} - 1}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 49.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B} \]

   9. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{\frac{-1}{6} \cdot \left(B \cdot B\right) - 1}{B} \]
   10. Step-by-step derivation

       1. lower-/.f64 36.8

          \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B} \]

   11. Applied rewrites 36.8%

       \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B} \]

   if -1.02000000000000005e121 < F < 5e6

   1. Initial program 97.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \cdot F - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \cdot F - x}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \cdot F - x}{B} \]

      12. lower-fma.f64 49.6

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 49.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   if 5e6 < F

   1. Initial program 54.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \cdot F - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \cdot F - x}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \cdot F - x}{B} \]

      12. lower-fma.f64 37.2

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 37.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 45.5%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.02 \cdot 10^{+121}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B}\\ \mathbf{elif}\;F \leq 5000000:\\ \;\;\;\;\frac{\sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 49.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.3 \cdot 10^{+30}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B}\\ \mathbf{elif}\;F \leq 8 \cdot 10^{+25}:\\ \;\;\;\;\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-1}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.3e+30)
   (+ (- (/ x B)) (/ (- (* -0.16666666666666666 (* B B)) 1.0) B))
   (if (<= F 8e+25)
     (/ (- (* (sqrt (pow (fma 2.0 x 2.0) -1.0)) F) x) B)
     (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.3e+30) {
		tmp = -(x / B) + (((-0.16666666666666666 * (B * B)) - 1.0) / B);
	} else if (F <= 8e+25) {
		tmp = ((sqrt(pow(fma(2.0, x, 2.0), -1.0)) * F) - x) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.3e+30)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(Float64(Float64(-0.16666666666666666 * Float64(B * B)) - 1.0) / B));
	elseif (F <= 8e+25)
		tmp = Float64(Float64(Float64(sqrt((fma(2.0, x, 2.0) ^ -1.0)) * F) - x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.3e+30], N[((-N[(x / B), $MachinePrecision]) + N[(N[(N[(-0.16666666666666666 * N[(B * B), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 8e+25], N[(N[(N[(N[Sqrt[N[Power[N[(2.0 * x + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.30000000000000026e30

   1. Initial program 56.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{\left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      2. associate-*l* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{\left({B}^{2} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)} \cdot \frac{1}{6} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      3. associate-*r* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{{B}^{2} \cdot \left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}\right)} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      4. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{{B}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      5. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}{B}} \]

   5. Applied rewrites 29.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(0.16666666666666666 \cdot F, B \cdot B, F\right)}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\frac{-1}{6} \cdot {B}^{2} - 1}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 50.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B} \]

   9. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{\frac{-1}{6} \cdot \left(B \cdot B\right) - 1}{B} \]
   10. Step-by-step derivation

       1. lower-/.f64 40.1

          \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B} \]

   11. Applied rewrites 40.1%

       \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B} \]

   if -3.30000000000000026e30 < F < 8.00000000000000072e25

   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

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \cdot F - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \cdot F - x}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \cdot F - x}{B} \]

      12. lower-fma.f64 49.3

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 49.3%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \frac{\sqrt{\frac{1}{2 + 2 \cdot x}} \cdot F - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 49.1%

      \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}} \cdot F - x}{B} \]

   if 8.00000000000000072e25 < F

   1. Initial program 53.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \cdot F - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \cdot F - x}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \cdot F - x}{B} \]

      12. lower-fma.f64 37.2

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 37.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 45.9%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.3 \cdot 10^{+30}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B}\\ \mathbf{elif}\;F \leq 8 \cdot 10^{+25}:\\ \;\;\;\;\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-1}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 42.0% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -8 \cdot 10^{-151}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B}\\ \mathbf{elif}\;F \leq 3.6 \cdot 10^{-162}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -8e-151)
   (+ (- (/ x B)) (/ (- (* -0.16666666666666666 (* B B)) 1.0) B))
   (if (<= F 3.6e-162) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -8e-151) {
		tmp = -(x / B) + (((-0.16666666666666666 * (B * B)) - 1.0) / B);
	} else if (F <= 3.6e-162) {
		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 <= (-8d-151)) then
        tmp = -(x / b) + ((((-0.16666666666666666d0) * (b * b)) - 1.0d0) / b)
    else if (f <= 3.6d-162) 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 <= -8e-151) {
		tmp = -(x / B) + (((-0.16666666666666666 * (B * B)) - 1.0) / B);
	} else if (F <= 3.6e-162) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -8e-151:
		tmp = -(x / B) + (((-0.16666666666666666 * (B * B)) - 1.0) / B)
	elif F <= 3.6e-162:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -8e-151)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(Float64(Float64(-0.16666666666666666 * Float64(B * B)) - 1.0) / B));
	elseif (F <= 3.6e-162)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -8e-151)
		tmp = -(x / B) + (((-0.16666666666666666 * (B * B)) - 1.0) / B);
	elseif (F <= 3.6e-162)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -8e-151], N[((-N[(x / B), $MachinePrecision]) + N[(N[(N[(-0.16666666666666666 * N[(B * B), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.6e-162], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -7.9999999999999995e-151

   1. Initial program 69.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{\left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      2. associate-*l* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{\left({B}^{2} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)} \cdot \frac{1}{6} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      3. associate-*r* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{{B}^{2} \cdot \left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}\right)} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      4. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{{B}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      5. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}{B}} \]

   5. Applied rewrites 39.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(0.16666666666666666 \cdot F, B \cdot B, F\right)}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\frac{-1}{6} \cdot {B}^{2} - 1}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 44.7%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B} \]

   9. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{\frac{-1}{6} \cdot \left(B \cdot B\right) - 1}{B} \]
   10. Step-by-step derivation

       1. lower-/.f64 33.6

          \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B} \]

   11. Applied rewrites 33.6%

       \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B} \]

   if -7.9999999999999995e-151 < F < 3.5999999999999998e-162

   1. Initial program 99.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \cdot F - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \cdot F - x}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \cdot F - x}{B} \]

      12. lower-fma.f64 51.5

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 51.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 36.9%

      \[\leadsto \frac{-x}{B} \]

   if 3.5999999999999998e-162 < F

   1. Initial program 65.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \cdot F - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \cdot F - x}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \cdot F - x}{B} \]

      12. lower-fma.f64 40.1

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 40.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 38.7%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 21: 42.1% accurate, 13.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -5.45 \cdot 10^{-151}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 3.6 \cdot 10^{-162}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -5.45e-151)
   (/ (- -1.0 x) B)
   (if (<= F 3.6e-162) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -5.45e-151) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 3.6e-162) {
		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 <= (-5.45d-151)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 3.6d-162) 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 <= -5.45e-151) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 3.6e-162) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -5.45e-151:
		tmp = (-1.0 - x) / B
	elif F <= 3.6e-162:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -5.45e-151)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 3.6e-162)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -5.45e-151)
		tmp = (-1.0 - x) / B;
	elseif (F <= 3.6e-162)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -5.45e-151], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 3.6e-162], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.45000000000000033e-151

   1. Initial program 69.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \cdot F - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \cdot F - x}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \cdot F - x}{B} \]

      12. lower-fma.f64 32.4

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 32.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 33.4%

      \[\leadsto \frac{-1 - x}{B} \]

   if -5.45000000000000033e-151 < F < 3.5999999999999998e-162

   1. Initial program 99.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \cdot F - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \cdot F - x}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \cdot F - x}{B} \]

      12. lower-fma.f64 51.5

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 51.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 36.9%

      \[\leadsto \frac{-x}{B} \]

   if 3.5999999999999998e-162 < F

   1. Initial program 65.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \cdot F - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \cdot F - x}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \cdot F - x}{B} \]

      12. lower-fma.f64 40.1

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 40.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 38.7%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 22: 35.8% accurate, 17.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -5.45 \cdot 10^{-151}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -5.45e-151) (/ (- -1.0 x) B) (/ (- x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -5.45e-151) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-5.45d-151)) then
        tmp = ((-1.0d0) - x) / b
    else
        tmp = -x / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -5.45e-151) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -5.45e-151:
		tmp = (-1.0 - x) / B
	else:
		tmp = -x / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -5.45e-151)
		tmp = Float64(Float64(-1.0 - x) / B);
	else
		tmp = Float64(Float64(-x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -5.45e-151)
		tmp = (-1.0 - x) / B;
	else
		tmp = -x / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -5.45e-151], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[((-x) / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -5.45000000000000033e-151

   1. Initial program 69.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \cdot F - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \cdot F - x}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \cdot F - x}{B} \]

      12. lower-fma.f64 32.4

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 32.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 33.4%

      \[\leadsto \frac{-1 - x}{B} \]

   if -5.45000000000000033e-151 < F

   1. Initial program 77.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \cdot F - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \cdot F - x}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \cdot F - x}{B} \]

      12. lower-fma.f64 44.1

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 44.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 29.9%

      \[\leadsto \frac{-x}{B} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 29.4% accurate, 26.3× speedup

Math

\[\begin{array}{l} \\ \frac{-x}{B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (/ (- x) B))

C

double code(double F, double B, double x) {
	return -x / B;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = -x / b
end function

Java

public static double code(double F, double B, double x) {
	return -x / B;
}

Python

def code(F, B, x):
	return -x / B

Julia

function code(F, B, x)
	return Float64(Float64(-x) / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = -x / B;
end

Wolfram

code[F_, B_, x_] := N[((-x) / B), $MachinePrecision]

Derivation

1. Initial program 74.3%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
2. Add Preprocessing

3. Taylor expanded in B around 0

   \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

   2. lower--.f64 N/A

      \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{B} \]

   3. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

   7. associate-+r+ N/A

      \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \cdot F - x}{B} \]

   8. +-commutative N/A

      \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

   9. unpow2 N/A

      \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \cdot F - x}{B} \]

   10. lower-fma.f64 N/A

       \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

   11. +-commutative N/A

       \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \cdot F - x}{B} \]

   12. lower-fma.f64 39.8

       \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

5. Applied rewrites 39.8%

   \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

6. Taylor expanded in F around 0

   \[\leadsto \frac{-1 \cdot x}{B} \]
7. Step-by-step derivation

8. Applied rewrites 24.1%

   \[\leadsto \frac{-x}{B} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024332 
(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))))))