VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.1% → 99.7%

Time: 17.8s

Alternatives: 28

Speedup: 1.6×

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: 76.1% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\frac{x}{\tan B}\\ \mathbf{if}\;F \leq -400000000:\\ \;\;\;\;-\frac{\mathsf{fma}\left(x, \cos B, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 20000000:\\ \;\;\;\;\mathsf{fma}\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{F}{\sin B}, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sin B}, 1, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ x (tan B)))))
   (if (<= F -400000000.0)
     (- (/ (fma x (cos B) 1.0) (sin B)))
     (if (<= F 20000000.0)
       (fma (pow (fma 2.0 x (fma F F 2.0)) -0.5) (/ F (sin B)) t_0)
       (fma (/ 1.0 (sin B)) 1.0 t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = -(x / tan(B));
	double tmp;
	if (F <= -400000000.0) {
		tmp = -(fma(x, cos(B), 1.0) / sin(B));
	} else if (F <= 20000000.0) {
		tmp = fma(pow(fma(2.0, x, fma(F, F, 2.0)), -0.5), (F / sin(B)), t_0);
	} else {
		tmp = fma((1.0 / sin(B)), 1.0, t_0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(-Float64(x / tan(B)))
	tmp = 0.0
	if (F <= -400000000.0)
		tmp = Float64(-Float64(fma(x, cos(B), 1.0) / sin(B)));
	elseif (F <= 20000000.0)
		tmp = fma((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5), Float64(F / sin(B)), t_0);
	else
		tmp = fma(Float64(1.0 / sin(B)), 1.0, t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = (-N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision])}, If[LessEqual[F, -400000000.0], (-N[(N[(x * N[Cos[B], $MachinePrecision] + 1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, 20000000.0], N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] * 1.0 + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -4e8

   1. Initial program 49.2%

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

   4. Applied rewrites 69.5%

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

   6. Applied rewrites 60.9%

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

   7. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. lft-mult-inverse N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-sin.f64 99.8

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

   9. Applied rewrites 99.8%

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

   if -4e8 < F < 2e7

   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-tan.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. neg-sub0 N/A

         \[\leadsto \color{blue}{\left(0 - x \cdot \frac{1}{\tan B}\right)} + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

      5. lift-sin.f64 N/A

         \[\leadsto \left(0 - x \cdot \frac{1}{\tan B}\right) + \frac{F}{\color{blue}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \left(0 - x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(0 - x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(\color{blue}{F \cdot F} + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

      8. lift-+.f64 N/A

         \[\leadsto \left(0 - x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\color{blue}{\left(F \cdot F + 2\right)} + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \left(0 - x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + \color{blue}{2 \cdot x}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \left(0 - x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\color{blue}{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

      11. lift-/.f64 N/A

          \[\leadsto \left(0 - x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)} \]

      12. lift-neg.f64 N/A

          \[\leadsto \left(0 - x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

      13. lift-pow.f64 N/A

          \[\leadsto \left(0 - x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

      14. lift-*.f64 N/A

          \[\leadsto \left(0 - x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

   4. Applied rewrites 99.6%

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

   if 2e7 < F

   1. Initial program 62.2%

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

   4. Applied rewrites 77.7%

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

   5. Taylor expanded in F around inf

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

   7. Applied rewrites 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -400000000:\\ \;\;\;\;-\frac{\mathsf{fma}\left(x, \cos B, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 20000000:\\ \;\;\;\;\mathsf{fma}\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{F}{\sin B}, -\frac{x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sin B}, 1, -\frac{x}{\tan B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ t_1 := t\_0 + \frac{F}{\sin B} \cdot {\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{\left(\frac{-1}{2}\right)}\\ t_2 := \sqrt{\frac{1}{2 + \mathsf{fma}\left(F, F, x \cdot 2\right)}} \cdot \frac{F}{B}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;t\_1 \leq -400:\\ \;\;\;\;t\_2 - \frac{x}{\tan B}\\ \mathbf{elif}\;t\_1 \leq 10^{-153}:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}\\ \mathbf{elif}\;t\_1 \leq 10^{-18}:\\ \;\;\;\;t\_0 + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}}\\ \mathbf{elif}\;t\_1 \leq 20:\\ \;\;\;\;\frac{1}{\frac{\sin B \cdot \sqrt{\mathsf{fma}\left(F, F, 2\right)}}{F}}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+297}:\\ \;\;\;\;t\_2 + \frac{-1}{\frac{\tan B}{x}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{\frac{1}{F} \cdot \mathsf{fma}\left(0.16666666666666666, F \cdot \left(B \cdot B\right), F\right)}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B))))
        (t_1
         (+
          t_0
          (* (/ F (sin B)) (pow (+ (* x 2.0) (+ 2.0 (* F F))) (/ -1.0 2.0)))))
        (t_2 (* (sqrt (/ 1.0 (+ 2.0 (fma F F (* x 2.0))))) (/ F B))))
   (if (<= t_1 (- INFINITY))
     (/ (- -1.0 x) B)
     (if (<= t_1 -400.0)
       (- t_2 (/ x (tan B)))
       (if (<= t_1 1e-153)
         (* F (/ (sqrt (/ 1.0 (fma F F 2.0))) (sin B)))
         (if (<= t_1 1e-18)
           (+ t_0 (* (/ F B) (sqrt (/ 1.0 (fma 2.0 x 2.0)))))
           (if (<= t_1 20.0)
             (/ 1.0 (/ (* (sin B) (sqrt (fma F F 2.0))) F))
             (if (<= t_1 5e+297)
               (+ t_2 (/ -1.0 (/ (tan B) x)))
               (+
                t_0
                (/
                 (* (/ 1.0 F) (fma 0.16666666666666666 (* F (* B B)) F))
                 B))))))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double t_1 = t_0 + ((F / sin(B)) * pow(((x * 2.0) + (2.0 + (F * F))), (-1.0 / 2.0)));
	double t_2 = sqrt((1.0 / (2.0 + fma(F, F, (x * 2.0))))) * (F / B);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (-1.0 - x) / B;
	} else if (t_1 <= -400.0) {
		tmp = t_2 - (x / tan(B));
	} else if (t_1 <= 1e-153) {
		tmp = F * (sqrt((1.0 / fma(F, F, 2.0))) / sin(B));
	} else if (t_1 <= 1e-18) {
		tmp = t_0 + ((F / B) * sqrt((1.0 / fma(2.0, x, 2.0))));
	} else if (t_1 <= 20.0) {
		tmp = 1.0 / ((sin(B) * sqrt(fma(F, F, 2.0))) / F);
	} else if (t_1 <= 5e+297) {
		tmp = t_2 + (-1.0 / (tan(B) / x));
	} else {
		tmp = t_0 + (((1.0 / F) * fma(0.16666666666666666, (F * (B * B)), F)) / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	t_1 = Float64(t_0 + Float64(Float64(F / sin(B)) * (Float64(Float64(x * 2.0) + Float64(2.0 + Float64(F * F))) ^ Float64(-1.0 / 2.0))))
	t_2 = Float64(sqrt(Float64(1.0 / Float64(2.0 + fma(F, F, Float64(x * 2.0))))) * Float64(F / B))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (t_1 <= -400.0)
		tmp = Float64(t_2 - Float64(x / tan(B)));
	elseif (t_1 <= 1e-153)
		tmp = Float64(F * Float64(sqrt(Float64(1.0 / fma(F, F, 2.0))) / sin(B)));
	elseif (t_1 <= 1e-18)
		tmp = Float64(t_0 + Float64(Float64(F / B) * sqrt(Float64(1.0 / fma(2.0, x, 2.0)))));
	elseif (t_1 <= 20.0)
		tmp = Float64(1.0 / Float64(Float64(sin(B) * sqrt(fma(F, F, 2.0))) / F));
	elseif (t_1 <= 5e+297)
		tmp = Float64(t_2 + Float64(-1.0 / Float64(tan(B) / x)));
	else
		tmp = Float64(t_0 + Float64(Float64(Float64(1.0 / F) * fma(0.16666666666666666, Float64(F * Float64(B * B)), F)) / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(x * 2.0), $MachinePrecision] + N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(F * F + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[t$95$1, -400.0], N[(t$95$2 - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-153], N[(F * N[(N[Sqrt[N[(1.0 / N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-18], N[(t$95$0 + N[(N[(F / B), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 20.0], N[(1.0 / N[(N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(F * F + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+297], N[(t$95$2 + N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(N[(N[(1.0 / F), $MachinePrecision] * N[(0.16666666666666666 * N[(F * N[(B * B), $MachinePrecision]), $MachinePrecision] + F), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < -inf.0

   1. Initial program 62.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. sub-neg N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in F around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 73.4

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

   8. Applied rewrites 73.4%

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

   if -inf.0 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < -400

   1. Initial program 99.6%

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

   3. Taylor expanded in B around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 68.0%

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

   7. Applied rewrites 67.7%

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

   8. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 99.6

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

   10. Applied rewrites 99.6%

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

   if -400 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < 1.00000000000000004e-153

   1. Initial program 59.7%

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

   3. Taylor expanded in x around 0

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-sin.f64 36.0

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

   5. Applied rewrites 36.0%

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

   if 1.00000000000000004e-153 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < 1.0000000000000001e-18

   1. Initial program 76.6%

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

   3. Taylor expanded in F around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 76.3

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

   5. Applied rewrites 76.3%

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

   6. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 45.6

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

   8. Applied rewrites 45.6%

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

   if 1.0000000000000001e-18 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < 20

   1. Initial program 94.8%

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

   4. Applied rewrites 95.0%

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

   5. Taylor expanded in x around 0

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

      1. associate-*l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 83.9

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

   7. Applied rewrites 83.9%

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

   if 20 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < 4.9999999999999998e297

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 79.5%

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

   7. Applied rewrites 79.6%

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

   8. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 99.5

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

   10. Applied rewrites 99.5%

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

       1. lift-tan.f64 N/A

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

       2. clear-num N/A

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

       3. lower-/.f64 N/A

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

       4. lower-/.f64 99.4

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

   12. Applied rewrites 99.4%

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

   if 4.9999999999999998e297 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))))))

   1. Initial program 12.7%

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

   3. Taylor expanded in B around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 71.4%

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

   6. Taylor expanded in F around inf

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

      1. lower-/.f64 79.5

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

   8. Applied rewrites 79.5%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{\frac{1}{F}} \cdot \mathsf{fma}\left(0.16666666666666666, \left(B \cdot B\right) \cdot F, F\right)}{B} \]
3. Recombined 7 regimes into one program.

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{\left(\frac{-1}{2}\right)} \leq -\infty:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{\left(\frac{-1}{2}\right)} \leq -400:\\ \;\;\;\;\sqrt{\frac{1}{2 + \mathsf{fma}\left(F, F, x \cdot 2\right)}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{\left(\frac{-1}{2}\right)} \leq 10^{-153}:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}\\ \mathbf{elif}\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{\left(\frac{-1}{2}\right)} \leq 10^{-18}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}}\\ \mathbf{elif}\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{\left(\frac{-1}{2}\right)} \leq 20:\\ \;\;\;\;\frac{1}{\frac{\sin B \cdot \sqrt{\mathsf{fma}\left(F, F, 2\right)}}{F}}\\ \mathbf{elif}\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{\left(\frac{-1}{2}\right)} \leq 5 \cdot 10^{+297}:\\ \;\;\;\;\sqrt{\frac{1}{2 + \mathsf{fma}\left(F, F, x \cdot 2\right)}} \cdot \frac{F}{B} + \frac{-1}{\frac{\tan B}{x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{\frac{1}{F} \cdot \mathsf{fma}\left(0.16666666666666666, F \cdot \left(B \cdot B\right), F\right)}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ t_1 := t\_0 + \frac{F}{\sin B} \cdot {\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{\left(\frac{-1}{2}\right)}\\ t_2 := \sqrt{\frac{1}{2 + \mathsf{fma}\left(F, F, x \cdot 2\right)}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;t\_1 \leq -400:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{-153}:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}\\ \mathbf{elif}\;t\_1 \leq 10^{-18}:\\ \;\;\;\;t\_0 + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}}\\ \mathbf{elif}\;t\_1 \leq 20:\\ \;\;\;\;\frac{1}{\frac{\sin B \cdot \sqrt{\mathsf{fma}\left(F, F, 2\right)}}{F}}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+297}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{\frac{1}{F} \cdot \mathsf{fma}\left(0.16666666666666666, F \cdot \left(B \cdot B\right), F\right)}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B))))
        (t_1
         (+
          t_0
          (* (/ F (sin B)) (pow (+ (* x 2.0) (+ 2.0 (* F F))) (/ -1.0 2.0)))))
        (t_2
         (-
          (* (sqrt (/ 1.0 (+ 2.0 (fma F F (* x 2.0))))) (/ F B))
          (/ x (tan B)))))
   (if (<= t_1 (- INFINITY))
     (/ (- -1.0 x) B)
     (if (<= t_1 -400.0)
       t_2
       (if (<= t_1 1e-153)
         (* F (/ (sqrt (/ 1.0 (fma F F 2.0))) (sin B)))
         (if (<= t_1 1e-18)
           (+ t_0 (* (/ F B) (sqrt (/ 1.0 (fma 2.0 x 2.0)))))
           (if (<= t_1 20.0)
             (/ 1.0 (/ (* (sin B) (sqrt (fma F F 2.0))) F))
             (if (<= t_1 5e+297)
               t_2
               (+
                t_0
                (/
                 (* (/ 1.0 F) (fma 0.16666666666666666 (* F (* B B)) F))
                 B))))))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double t_1 = t_0 + ((F / sin(B)) * pow(((x * 2.0) + (2.0 + (F * F))), (-1.0 / 2.0)));
	double t_2 = (sqrt((1.0 / (2.0 + fma(F, F, (x * 2.0))))) * (F / B)) - (x / tan(B));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (-1.0 - x) / B;
	} else if (t_1 <= -400.0) {
		tmp = t_2;
	} else if (t_1 <= 1e-153) {
		tmp = F * (sqrt((1.0 / fma(F, F, 2.0))) / sin(B));
	} else if (t_1 <= 1e-18) {
		tmp = t_0 + ((F / B) * sqrt((1.0 / fma(2.0, x, 2.0))));
	} else if (t_1 <= 20.0) {
		tmp = 1.0 / ((sin(B) * sqrt(fma(F, F, 2.0))) / F);
	} else if (t_1 <= 5e+297) {
		tmp = t_2;
	} else {
		tmp = t_0 + (((1.0 / F) * fma(0.16666666666666666, (F * (B * B)), F)) / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	t_1 = Float64(t_0 + Float64(Float64(F / sin(B)) * (Float64(Float64(x * 2.0) + Float64(2.0 + Float64(F * F))) ^ Float64(-1.0 / 2.0))))
	t_2 = Float64(Float64(sqrt(Float64(1.0 / Float64(2.0 + fma(F, F, Float64(x * 2.0))))) * Float64(F / B)) - Float64(x / tan(B)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (t_1 <= -400.0)
		tmp = t_2;
	elseif (t_1 <= 1e-153)
		tmp = Float64(F * Float64(sqrt(Float64(1.0 / fma(F, F, 2.0))) / sin(B)));
	elseif (t_1 <= 1e-18)
		tmp = Float64(t_0 + Float64(Float64(F / B) * sqrt(Float64(1.0 / fma(2.0, x, 2.0)))));
	elseif (t_1 <= 20.0)
		tmp = Float64(1.0 / Float64(Float64(sin(B) * sqrt(fma(F, F, 2.0))) / F));
	elseif (t_1 <= 5e+297)
		tmp = t_2;
	else
		tmp = Float64(t_0 + Float64(Float64(Float64(1.0 / F) * fma(0.16666666666666666, Float64(F * Float64(B * B)), F)) / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(x * 2.0), $MachinePrecision] + N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(F * F + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[t$95$1, -400.0], t$95$2, If[LessEqual[t$95$1, 1e-153], N[(F * N[(N[Sqrt[N[(1.0 / N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-18], N[(t$95$0 + N[(N[(F / B), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 20.0], N[(1.0 / N[(N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(F * F + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+297], t$95$2, N[(t$95$0 + N[(N[(N[(1.0 / F), $MachinePrecision] * N[(0.16666666666666666 * N[(F * N[(B * B), $MachinePrecision]), $MachinePrecision] + F), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < -inf.0

   1. Initial program 62.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. sub-neg N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in F around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 73.4

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

   8. Applied rewrites 73.4%

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

   if -inf.0 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < -400 or 20 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < 4.9999999999999998e297

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 73.7%

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

   7. Applied rewrites 73.6%

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

   8. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 99.6

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

   10. Applied rewrites 99.6%

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

   if -400 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < 1.00000000000000004e-153

   1. Initial program 59.7%

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

   3. Taylor expanded in x around 0

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-sin.f64 36.0

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

   5. Applied rewrites 36.0%

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

   if 1.00000000000000004e-153 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < 1.0000000000000001e-18

   1. Initial program 76.6%

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

   3. Taylor expanded in F around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 76.3

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

   5. Applied rewrites 76.3%

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

   6. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 45.6

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

   8. Applied rewrites 45.6%

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

   if 1.0000000000000001e-18 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < 20

   1. Initial program 94.8%

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

   4. Applied rewrites 95.0%

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

   5. Taylor expanded in x around 0

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

      1. associate-*l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 83.9

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

   7. Applied rewrites 83.9%

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

   if 4.9999999999999998e297 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))))))

   1. Initial program 12.7%

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

   3. Taylor expanded in B around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 71.4%

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

   6. Taylor expanded in F around inf

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

      1. lower-/.f64 79.5

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

   8. Applied rewrites 79.5%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{\frac{1}{F}} \cdot \mathsf{fma}\left(0.16666666666666666, \left(B \cdot B\right) \cdot F, F\right)}{B} \]
3. Recombined 6 regimes into one program.

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{\left(\frac{-1}{2}\right)} \leq -\infty:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{\left(\frac{-1}{2}\right)} \leq -400:\\ \;\;\;\;\sqrt{\frac{1}{2 + \mathsf{fma}\left(F, F, x \cdot 2\right)}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{\left(\frac{-1}{2}\right)} \leq 10^{-153}:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}\\ \mathbf{elif}\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{\left(\frac{-1}{2}\right)} \leq 10^{-18}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}}\\ \mathbf{elif}\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{\left(\frac{-1}{2}\right)} \leq 20:\\ \;\;\;\;\frac{1}{\frac{\sin B \cdot \sqrt{\mathsf{fma}\left(F, F, 2\right)}}{F}}\\ \mathbf{elif}\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{\left(\frac{-1}{2}\right)} \leq 5 \cdot 10^{+297}:\\ \;\;\;\;\sqrt{\frac{1}{2 + \mathsf{fma}\left(F, F, x \cdot 2\right)}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{\frac{1}{F} \cdot \mathsf{fma}\left(0.16666666666666666, F \cdot \left(B \cdot B\right), F\right)}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.5:\\ \;\;\;\;-\frac{\mathsf{fma}\left(x, \cos B, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 0.029:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, 2\right)}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sin B}, 1, -t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.5)
     (- (/ (fma x (cos B) 1.0) (sin B)))
     (if (<= F 0.029)
       (- (/ F (* (sin B) (sqrt (fma 2.0 x 2.0)))) t_0)
       (fma (/ 1.0 (sin B)) 1.0 (- t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.5) {
		tmp = -(fma(x, cos(B), 1.0) / sin(B));
	} else if (F <= 0.029) {
		tmp = (F / (sin(B) * sqrt(fma(2.0, x, 2.0)))) - t_0;
	} else {
		tmp = fma((1.0 / sin(B)), 1.0, -t_0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.5)
		tmp = Float64(-Float64(fma(x, cos(B), 1.0) / sin(B)));
	elseif (F <= 0.029)
		tmp = Float64(Float64(F / Float64(sin(B) * sqrt(fma(2.0, x, 2.0)))) - t_0);
	else
		tmp = fma(Float64(1.0 / sin(B)), 1.0, Float64(-t_0));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.5], (-N[(N[(x * N[Cos[B], $MachinePrecision] + 1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, 0.029], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(2.0 * x + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] * 1.0 + (-t$95$0)), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.5

   1. Initial program 50.6%

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

   4. Applied rewrites 70.3%

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

   6. Applied rewrites 61.9%

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

   7. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. lft-mult-inverse N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-sin.f64 98.9

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

   9. Applied rewrites 98.9%

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

   if -1.5 < F < 0.0290000000000000015

   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 F around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 99.4

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

   5. Applied rewrites 99.4%

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

      1. lift-tan.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-neg.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. div-inv N/A

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

      16. lift-/.f64 N/A

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

   7. Applied rewrites 99.6%

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

   if 0.0290000000000000015 < F

   1. Initial program 64.7%

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

   4. Applied rewrites 79.1%

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

   5. Taylor expanded in F around inf

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

   7. Applied rewrites 99.4%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.5:\\ \;\;\;\;-\frac{\mathsf{fma}\left(x, \cos B, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 0.029:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, 2\right)}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sin B}, 1, -\frac{x}{\tan B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -26.5:\\ \;\;\;\;-\frac{\mathsf{fma}\left(x, \cos B, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 1.66 \cdot 10^{-15}:\\ \;\;\;\;\sqrt{\frac{1}{2 + \mathsf{fma}\left(F, F, x \cdot 2\right)}} \cdot \frac{F}{B} - t\_0\\ \mathbf{elif}\;F \leq 2.6 \cdot 10^{-10}:\\ \;\;\;\;\frac{1}{\frac{\sin B \cdot \sqrt{\mathsf{fma}\left(F, F, 2\right)}}{F}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sin B}, 1, -t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -26.5)
     (- (/ (fma x (cos B) 1.0) (sin B)))
     (if (<= F 1.66e-15)
       (- (* (sqrt (/ 1.0 (+ 2.0 (fma F F (* x 2.0))))) (/ F B)) t_0)
       (if (<= F 2.6e-10)
         (/ 1.0 (/ (* (sin B) (sqrt (fma F F 2.0))) F))
         (fma (/ 1.0 (sin B)) 1.0 (- t_0)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -26.5) {
		tmp = -(fma(x, cos(B), 1.0) / sin(B));
	} else if (F <= 1.66e-15) {
		tmp = (sqrt((1.0 / (2.0 + fma(F, F, (x * 2.0))))) * (F / B)) - t_0;
	} else if (F <= 2.6e-10) {
		tmp = 1.0 / ((sin(B) * sqrt(fma(F, F, 2.0))) / F);
	} else {
		tmp = fma((1.0 / sin(B)), 1.0, -t_0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -26.5)
		tmp = Float64(-Float64(fma(x, cos(B), 1.0) / sin(B)));
	elseif (F <= 1.66e-15)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(2.0 + fma(F, F, Float64(x * 2.0))))) * Float64(F / B)) - t_0);
	elseif (F <= 2.6e-10)
		tmp = Float64(1.0 / Float64(Float64(sin(B) * sqrt(fma(F, F, 2.0))) / F));
	else
		tmp = fma(Float64(1.0 / sin(B)), 1.0, Float64(-t_0));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -26.5], (-N[(N[(x * N[Cos[B], $MachinePrecision] + 1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, 1.66e-15], N[(N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(F * F + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 2.6e-10], N[(1.0 / N[(N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(F * F + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] * 1.0 + (-t$95$0)), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -26.5

   1. Initial program 50.6%

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

   4. Applied rewrites 70.3%

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

   6. Applied rewrites 61.9%

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

   7. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. lft-mult-inverse N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-sin.f64 98.9

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

   9. Applied rewrites 98.9%

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

   if -26.5 < F < 1.65999999999999996e-15

   1. Initial program 99.5%

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

   3. Taylor expanded in B around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 65.3%

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

   7. Applied rewrites 65.4%

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

   8. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 82.9

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

   10. Applied rewrites 82.9%

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

   if 1.65999999999999996e-15 < F < 2.59999999999999981e-10

   1. Initial program 99.1%

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in x around 0

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

      1. associate-*l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if 2.59999999999999981e-10 < F

   1. Initial program 65.6%

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

   4. Applied rewrites 79.7%

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

   5. Taylor expanded in F around inf

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

   7. Applied rewrites 99.4%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -26.5:\\ \;\;\;\;-\frac{\mathsf{fma}\left(x, \cos B, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 1.66 \cdot 10^{-15}:\\ \;\;\;\;\sqrt{\frac{1}{2 + \mathsf{fma}\left(F, F, x \cdot 2\right)}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.6 \cdot 10^{-10}:\\ \;\;\;\;\frac{1}{\frac{\sin B \cdot \sqrt{\mathsf{fma}\left(F, F, 2\right)}}{F}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sin B}, 1, -\frac{x}{\tan B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -26.5:\\ \;\;\;\;-\frac{\mathsf{fma}\left(x, \cos B, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 1.66 \cdot 10^{-15}:\\ \;\;\;\;\sqrt{\frac{1}{2 + \mathsf{fma}\left(F, F, x \cdot 2\right)}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.6 \cdot 10^{-10}:\\ \;\;\;\;\frac{1}{\frac{\sin B \cdot \sqrt{\mathsf{fma}\left(F, F, 2\right)}}{F}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos B, -x, 1\right)}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -26.5)
   (- (/ (fma x (cos B) 1.0) (sin B)))
   (if (<= F 1.66e-15)
     (- (* (sqrt (/ 1.0 (+ 2.0 (fma F F (* x 2.0))))) (/ F B)) (/ x (tan B)))
     (if (<= F 2.6e-10)
       (/ 1.0 (/ (* (sin B) (sqrt (fma F F 2.0))) F))
       (/ (fma (cos B) (- x) 1.0) (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -26.5) {
		tmp = -(fma(x, cos(B), 1.0) / sin(B));
	} else if (F <= 1.66e-15) {
		tmp = (sqrt((1.0 / (2.0 + fma(F, F, (x * 2.0))))) * (F / B)) - (x / tan(B));
	} else if (F <= 2.6e-10) {
		tmp = 1.0 / ((sin(B) * sqrt(fma(F, F, 2.0))) / F);
	} else {
		tmp = fma(cos(B), -x, 1.0) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -26.5)
		tmp = Float64(-Float64(fma(x, cos(B), 1.0) / sin(B)));
	elseif (F <= 1.66e-15)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(2.0 + fma(F, F, Float64(x * 2.0))))) * Float64(F / B)) - Float64(x / tan(B)));
	elseif (F <= 2.6e-10)
		tmp = Float64(1.0 / Float64(Float64(sin(B) * sqrt(fma(F, F, 2.0))) / F));
	else
		tmp = Float64(fma(cos(B), Float64(-x), 1.0) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -26.5], (-N[(N[(x * N[Cos[B], $MachinePrecision] + 1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, 1.66e-15], N[(N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(F * F + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.6e-10], N[(1.0 / N[(N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(F * F + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[B], $MachinePrecision] * (-x) + 1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -26.5

   1. Initial program 50.6%

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

   4. Applied rewrites 70.3%

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

   6. Applied rewrites 61.9%

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

   7. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. lft-mult-inverse N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-sin.f64 98.9

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

   9. Applied rewrites 98.9%

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

   if -26.5 < F < 1.65999999999999996e-15

   1. Initial program 99.5%

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

   3. Taylor expanded in B around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 65.3%

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

   7. Applied rewrites 65.4%

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

   8. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 82.9

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

   10. Applied rewrites 82.9%

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

   if 1.65999999999999996e-15 < F < 2.59999999999999981e-10

   1. Initial program 99.1%

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in x around 0

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

      1. associate-*l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if 2.59999999999999981e-10 < F

   1. Initial program 65.6%

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

   4. Applied rewrites 79.7%

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

   6. Applied rewrites 71.1%

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

   7. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. distribute-lft-in N/A

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

      3. rgt-mult-inverse N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sin.f64 99.3

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

   9. Applied rewrites 99.3%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -26.5:\\ \;\;\;\;-\frac{\mathsf{fma}\left(x, \cos B, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 1.66 \cdot 10^{-15}:\\ \;\;\;\;\sqrt{\frac{1}{2 + \mathsf{fma}\left(F, F, x \cdot 2\right)}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.6 \cdot 10^{-10}:\\ \;\;\;\;\frac{1}{\frac{\sin B \cdot \sqrt{\mathsf{fma}\left(F, F, 2\right)}}{F}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos B, -x, 1\right)}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\frac{x \cdot \cos B}{\sin B}\\ \mathbf{if}\;x \leq -880000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-197}:\\ \;\;\;\;\sqrt{\frac{1}{2 + \mathsf{fma}\left(F, F, x \cdot 2\right)}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{-81}:\\ \;\;\;\;\frac{1}{\frac{\sin B \cdot \sqrt{\mathsf{fma}\left(F, F, 2\right)}}{F}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ (* x (cos B)) (sin B)))))
   (if (<= x -880000.0)
     t_0
     (if (<= x -8e-197)
       (- (* (sqrt (/ 1.0 (+ 2.0 (fma F F (* x 2.0))))) (/ F B)) (/ x (tan B)))
       (if (<= x 7.6e-81)
         (/ 1.0 (/ (* (sin B) (sqrt (fma F F 2.0))) F))
         t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = -((x * cos(B)) / sin(B));
	double tmp;
	if (x <= -880000.0) {
		tmp = t_0;
	} else if (x <= -8e-197) {
		tmp = (sqrt((1.0 / (2.0 + fma(F, F, (x * 2.0))))) * (F / B)) - (x / tan(B));
	} else if (x <= 7.6e-81) {
		tmp = 1.0 / ((sin(B) * sqrt(fma(F, F, 2.0))) / F);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(-Float64(Float64(x * cos(B)) / sin(B)))
	tmp = 0.0
	if (x <= -880000.0)
		tmp = t_0;
	elseif (x <= -8e-197)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(2.0 + fma(F, F, Float64(x * 2.0))))) * Float64(F / B)) - Float64(x / tan(B)));
	elseif (x <= 7.6e-81)
		tmp = Float64(1.0 / Float64(Float64(sin(B) * sqrt(fma(F, F, 2.0))) / F));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = (-N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision])}, If[LessEqual[x, -880000.0], t$95$0, If[LessEqual[x, -8e-197], N[(N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(F * F + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.6e-81], N[(1.0 / N[(N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(F * F + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.8e5 or 7.5999999999999997e-81 < x

   1. Initial program 82.2%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-sin.f64 93.4

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

   5. Applied rewrites 93.4%

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

   if -8.8e5 < x < -7.9999999999999999e-197

   1. Initial program 69.9%

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

   3. Taylor expanded in B around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 54.6%

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

   7. Applied rewrites 54.4%

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

   8. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 56.7

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

   10. Applied rewrites 56.7%

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

   if -7.9999999999999999e-197 < x < 7.5999999999999997e-81

   1. Initial program 67.8%

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

   4. Applied rewrites 67.8%

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

   5. Taylor expanded in x around 0

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

      1. associate-*l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 65.5

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

   7. Applied rewrites 65.5%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -880000:\\ \;\;\;\;-\frac{x \cdot \cos B}{\sin B}\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-197}:\\ \;\;\;\;\sqrt{\frac{1}{2 + \mathsf{fma}\left(F, F, x \cdot 2\right)}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{-81}:\\ \;\;\;\;\frac{1}{\frac{\sin B \cdot \sqrt{\mathsf{fma}\left(F, F, 2\right)}}{F}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x \cdot \cos B}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -26.5:\\ \;\;\;\;-\frac{\mathsf{fma}\left(x, \cos B, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{\frac{1}{2 + \mathsf{fma}\left(F, F, x \cdot 2\right)}} \cdot \frac{F}{B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -26.5)
     (- (/ (fma x (cos B) 1.0) (sin B)))
     (if (<= F 5e+79)
       (- (* (sqrt (/ 1.0 (+ 2.0 (fma F F (* x 2.0))))) (/ F B)) t_0)
       (- (/ (fma (* B B) 0.16666666666666666 1.0) B) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -26.5) {
		tmp = -(fma(x, cos(B), 1.0) / sin(B));
	} else if (F <= 5e+79) {
		tmp = (sqrt((1.0 / (2.0 + fma(F, F, (x * 2.0))))) * (F / B)) - t_0;
	} else {
		tmp = (fma((B * B), 0.16666666666666666, 1.0) / B) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -26.5)
		tmp = Float64(-Float64(fma(x, cos(B), 1.0) / sin(B)));
	elseif (F <= 5e+79)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(2.0 + fma(F, F, Float64(x * 2.0))))) * Float64(F / B)) - t_0);
	else
		tmp = Float64(Float64(fma(Float64(B * B), 0.16666666666666666, 1.0) / B) - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -26.5], (-N[(N[(x * N[Cos[B], $MachinePrecision] + 1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, 5e+79], N[(N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(F * F + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[(N[(B * B), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] / B), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -26.5

   1. Initial program 50.6%

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

   4. Applied rewrites 70.3%

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

   6. Applied rewrites 61.9%

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

   7. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. lft-mult-inverse N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-sin.f64 98.9

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

   9. Applied rewrites 98.9%

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

   if -26.5 < F < 5e79

   1. Initial program 98.7%

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

   3. Taylor expanded in B around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 62.9%

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

   7. Applied rewrites 62.8%

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

   8. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 78.7

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

   10. Applied rewrites 78.7%

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

   if 5e79 < F

   1. Initial program 51.1%

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

   3. Taylor expanded in B around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 44.8%

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

   7. Applied rewrites 44.7%

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

   8. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 + \frac{1}{6} \cdot {B}^{2}}{B}} - \frac{x}{\tan B} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{6} \cdot {B}^{2} + 1}}{B} - \frac{x}{\tan B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{B}^{2} \cdot \frac{1}{6}} + 1}{B} - \frac{x}{\tan B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({B}^{2}, \frac{1}{6}, 1\right)}}{B} - \frac{x}{\tan B} \]

      5. unpow2 N/A

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

      6. lower-*.f64 62.7

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

   10. Applied rewrites 62.7%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B}} - \frac{x}{\tan B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -26.5:\\ \;\;\;\;-\frac{\mathsf{fma}\left(x, \cos B, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{\frac{1}{2 + \mathsf{fma}\left(F, F, x \cdot 2\right)}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;x \leq -4.2 \cdot 10^{-13}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-81}:\\ \;\;\;\;\frac{1}{\frac{\sin B \cdot \sqrt{\mathsf{fma}\left(F, F, 2\right)}}{F}}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+15}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ -1.0 B) (/ x (tan B)))))
   (if (<= x -4.2e-13)
     t_0
     (if (<= x 7e-81)
       (/ 1.0 (/ (* (sin B) (sqrt (fma F F 2.0))) F))
       (if (<= x 1.05e+15)
         (+ (* x (/ -1.0 (tan B))) (* (/ F B) (sqrt (/ 1.0 (fma 2.0 x 2.0)))))
         t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 / B) - (x / tan(B));
	double tmp;
	if (x <= -4.2e-13) {
		tmp = t_0;
	} else if (x <= 7e-81) {
		tmp = 1.0 / ((sin(B) * sqrt(fma(F, F, 2.0))) / F);
	} else if (x <= 1.05e+15) {
		tmp = (x * (-1.0 / tan(B))) + ((F / B) * sqrt((1.0 / fma(2.0, x, 2.0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 / B) - Float64(x / tan(B)))
	tmp = 0.0
	if (x <= -4.2e-13)
		tmp = t_0;
	elseif (x <= 7e-81)
		tmp = Float64(1.0 / Float64(Float64(sin(B) * sqrt(fma(F, F, 2.0))) / F));
	elseif (x <= 1.05e+15)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(Float64(F / B) * sqrt(Float64(1.0 / fma(2.0, x, 2.0)))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.2e-13], t$95$0, If[LessEqual[x, 7e-81], N[(1.0 / N[(N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(F * F + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.05e+15], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(F / B), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.19999999999999977e-13 or 1.05e15 < x

   1. Initial program 82.6%

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

   3. Taylor expanded in B around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 70.1%

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

   7. Applied rewrites 70.0%

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

   8. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 82.7

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

   10. Applied rewrites 82.7%

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

   11. Taylor expanded in F around -inf

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

       1. lower-/.f64 98.9

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

   13. Applied rewrites 98.9%

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

   if -4.19999999999999977e-13 < x < 6.99999999999999973e-81

   1. Initial program 68.5%

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

   4. Applied rewrites 68.5%

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

   5. Taylor expanded in x around 0

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

      1. associate-*l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 58.6

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

   7. Applied rewrites 58.6%

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

   if 6.99999999999999973e-81 < x < 1.05e15

   1. Initial program 79.2%

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

   3. Taylor expanded in F around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 68.3

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

   5. Applied rewrites 68.3%

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

   6. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 68.6

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

   8. Applied rewrites 68.6%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-13}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-81}:\\ \;\;\;\;\frac{1}{\frac{\sin B \cdot \sqrt{\mathsf{fma}\left(F, F, 2\right)}}{F}}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+15}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ t_1 := \frac{-1}{B} - t\_0\\ \mathbf{if}\;x \leq -4.2 \cdot 10^{-13}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-55}:\\ \;\;\;\;\frac{1}{\frac{\sin B \cdot \sqrt{\mathsf{fma}\left(F, F, 2\right)}}{F}}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+15}:\\ \;\;\;\;\frac{F}{B} \cdot \sqrt{\frac{0.5}{x}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))) (t_1 (- (/ -1.0 B) t_0)))
   (if (<= x -4.2e-13)
     t_1
     (if (<= x 1.45e-55)
       (/ 1.0 (/ (* (sin B) (sqrt (fma F F 2.0))) F))
       (if (<= x 1.05e+15) (- (* (/ F B) (sqrt (/ 0.5 x))) t_0) t_1)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double t_1 = (-1.0 / B) - t_0;
	double tmp;
	if (x <= -4.2e-13) {
		tmp = t_1;
	} else if (x <= 1.45e-55) {
		tmp = 1.0 / ((sin(B) * sqrt(fma(F, F, 2.0))) / F);
	} else if (x <= 1.05e+15) {
		tmp = ((F / B) * sqrt((0.5 / x))) - t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	t_1 = Float64(Float64(-1.0 / B) - t_0)
	tmp = 0.0
	if (x <= -4.2e-13)
		tmp = t_1;
	elseif (x <= 1.45e-55)
		tmp = Float64(1.0 / Float64(Float64(sin(B) * sqrt(fma(F, F, 2.0))) / F));
	elseif (x <= 1.05e+15)
		tmp = Float64(Float64(Float64(F / B) * sqrt(Float64(0.5 / x))) - t_0);
	else
		tmp = t_1;
	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[(N[(-1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[x, -4.2e-13], t$95$1, If[LessEqual[x, 1.45e-55], N[(1.0 / N[(N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(F * F + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.05e+15], N[(N[(N[(F / B), $MachinePrecision] * N[Sqrt[N[(0.5 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.19999999999999977e-13 or 1.05e15 < x

   1. Initial program 82.6%

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

   3. Taylor expanded in B around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 70.1%

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

   7. Applied rewrites 70.0%

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

   8. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 82.7

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

   10. Applied rewrites 82.7%

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

   11. Taylor expanded in F around -inf

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

       1. lower-/.f64 98.9

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

   13. Applied rewrites 98.9%

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

   if -4.19999999999999977e-13 < x < 1.45e-55

   1. Initial program 67.9%

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

   4. Applied rewrites 68.0%

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

   5. Taylor expanded in x around 0

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

      1. associate-*l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 57.3

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

   7. Applied rewrites 57.3%

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

   if 1.45e-55 < x < 1.05e15

   1. Initial program 88.1%

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

   3. Taylor expanded in B around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 82.6%

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

   7. Applied rewrites 82.7%

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

   8. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 88.3

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

   10. Applied rewrites 88.3%

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

   11. Taylor expanded in x around inf

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

       1. lower-/.f64 83.1

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

   13. Applied rewrites 83.1%

       \[\leadsto \sqrt{\color{blue}{\frac{0.5}{x}}} \cdot \frac{F}{B} - \frac{x}{\tan B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-13}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-55}:\\ \;\;\;\;\frac{1}{\frac{\sin B \cdot \sqrt{\mathsf{fma}\left(F, F, 2\right)}}{F}}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+15}:\\ \;\;\;\;\frac{F}{B} \cdot \sqrt{\frac{0.5}{x}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 71.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;x \leq -4.2 \cdot 10^{-13}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{-81}:\\ \;\;\;\;\frac{1}{\frac{\sin B \cdot \sqrt{\mathsf{fma}\left(F, F, 2\right)}}{F}}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{x}{F}}{\tan B \cdot \frac{-1}{F}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ -1.0 B) (/ x (tan B)))))
   (if (<= x -4.2e-13)
     t_0
     (if (<= x 7.6e-81)
       (/ 1.0 (/ (* (sin B) (sqrt (fma F F 2.0))) F))
       (if (<= x 1.05e+15) (/ (/ x F) (* (tan B) (/ -1.0 F))) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 / B) - (x / tan(B));
	double tmp;
	if (x <= -4.2e-13) {
		tmp = t_0;
	} else if (x <= 7.6e-81) {
		tmp = 1.0 / ((sin(B) * sqrt(fma(F, F, 2.0))) / F);
	} else if (x <= 1.05e+15) {
		tmp = (x / F) / (tan(B) * (-1.0 / F));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 / B) - Float64(x / tan(B)))
	tmp = 0.0
	if (x <= -4.2e-13)
		tmp = t_0;
	elseif (x <= 7.6e-81)
		tmp = Float64(1.0 / Float64(Float64(sin(B) * sqrt(fma(F, F, 2.0))) / F));
	elseif (x <= 1.05e+15)
		tmp = Float64(Float64(x / F) / Float64(tan(B) * Float64(-1.0 / F)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.2e-13], t$95$0, If[LessEqual[x, 7.6e-81], N[(1.0 / N[(N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(F * F + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.05e+15], N[(N[(x / F), $MachinePrecision] / N[(N[Tan[B], $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.19999999999999977e-13 or 1.05e15 < x

   1. Initial program 82.6%

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

   3. Taylor expanded in B around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 70.1%

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

   7. Applied rewrites 70.0%

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

   8. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 82.7

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

   10. Applied rewrites 82.7%

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

   11. Taylor expanded in F around -inf

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

       1. lower-/.f64 98.9

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

   13. Applied rewrites 98.9%

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

   if -4.19999999999999977e-13 < x < 7.5999999999999997e-81

   1. Initial program 68.5%

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

   4. Applied rewrites 68.5%

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

   5. Taylor expanded in x around 0

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

      1. associate-*l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 58.6

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

   7. Applied rewrites 58.6%

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

   if 7.5999999999999997e-81 < x < 1.05e15

   1. Initial program 79.2%

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

   4. Applied rewrites 83.8%

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

   6. Applied rewrites 79.9%

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

   7. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 68.2

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

   9. Applied rewrites 68.2%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-13}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{-81}:\\ \;\;\;\;\frac{1}{\frac{\sin B \cdot \sqrt{\mathsf{fma}\left(F, F, 2\right)}}{F}}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{x}{F}}{\tan B \cdot \frac{-1}{F}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 70.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;x \leq -4.2 \cdot 10^{-13}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{-81}:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{x}{F}}{\tan B \cdot \frac{-1}{F}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ -1.0 B) (/ x (tan B)))))
   (if (<= x -4.2e-13)
     t_0
     (if (<= x 7.6e-81)
       (* F (/ (sqrt (/ 1.0 (fma F F 2.0))) (sin B)))
       (if (<= x 1.05e+15) (/ (/ x F) (* (tan B) (/ -1.0 F))) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 / B) - (x / tan(B));
	double tmp;
	if (x <= -4.2e-13) {
		tmp = t_0;
	} else if (x <= 7.6e-81) {
		tmp = F * (sqrt((1.0 / fma(F, F, 2.0))) / sin(B));
	} else if (x <= 1.05e+15) {
		tmp = (x / F) / (tan(B) * (-1.0 / F));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 / B) - Float64(x / tan(B)))
	tmp = 0.0
	if (x <= -4.2e-13)
		tmp = t_0;
	elseif (x <= 7.6e-81)
		tmp = Float64(F * Float64(sqrt(Float64(1.0 / fma(F, F, 2.0))) / sin(B)));
	elseif (x <= 1.05e+15)
		tmp = Float64(Float64(x / F) / Float64(tan(B) * Float64(-1.0 / F)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.2e-13], t$95$0, If[LessEqual[x, 7.6e-81], N[(F * N[(N[Sqrt[N[(1.0 / N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.05e+15], N[(N[(x / F), $MachinePrecision] / N[(N[Tan[B], $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.19999999999999977e-13 or 1.05e15 < x

   1. Initial program 82.6%

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

   3. Taylor expanded in B around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 70.1%

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

   7. Applied rewrites 70.0%

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

   8. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 82.7

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

   10. Applied rewrites 82.7%

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

   11. Taylor expanded in F around -inf

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

       1. lower-/.f64 98.9

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

   13. Applied rewrites 98.9%

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

   if -4.19999999999999977e-13 < x < 7.5999999999999997e-81

   1. Initial program 68.5%

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

   3. Taylor expanded in x around 0

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-sin.f64 58.4

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

   5. Applied rewrites 58.4%

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

   if 7.5999999999999997e-81 < x < 1.05e15

   1. Initial program 79.2%

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

   4. Applied rewrites 83.8%

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

   6. Applied rewrites 79.9%

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

   7. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 68.2

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

   9. Applied rewrites 68.2%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-13}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{-81}:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{x}{F}}{\tan B \cdot \frac{-1}{F}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 57.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.13:\\ \;\;\;\;\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}, 0.3333333333333333 \cdot \left(x \cdot \left(B \cdot B\right)\right) - x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{B} \cdot \frac{1}{F} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 0.13)
   (/
    (fma
     F
     (sqrt (/ 1.0 (fma 2.0 x (fma F F 2.0))))
     (- (* 0.3333333333333333 (* x (* B B))) x))
    B)
   (- (* (/ F B) (/ 1.0 F)) (/ x (tan B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 0.13) {
		tmp = fma(F, sqrt((1.0 / fma(2.0, x, fma(F, F, 2.0)))), ((0.3333333333333333 * (x * (B * B))) - x)) / B;
	} else {
		tmp = ((F / B) * (1.0 / F)) - (x / tan(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 0.13)
		tmp = Float64(fma(F, sqrt(Float64(1.0 / fma(2.0, x, fma(F, F, 2.0)))), Float64(Float64(0.3333333333333333 * Float64(x * Float64(B * B))) - x)) / B);
	else
		tmp = Float64(Float64(Float64(F / B) * Float64(1.0 / F)) - Float64(x / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 0.13], N[(N[(F * N[Sqrt[N[(1.0 / N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[(0.3333333333333333 * N[(x * N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(F / B), $MachinePrecision] * N[(1.0 / F), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 0.13

   1. Initial program 73.1%

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

   3. Taylor expanded in B around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 67.0%

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

   7. Applied rewrites 66.9%

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

   8. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 62.8

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

   10. Applied rewrites 62.8%

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

   11. Taylor expanded in B around 0

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

       1. lower-/.f64 N/A

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

   13. Applied rewrites 63.1%

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

   if 0.13 < B

   1. Initial program 79.9%

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

   3. Taylor expanded in B around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 20.0%

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

   7. Applied rewrites 19.9%

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

   8. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 52.5

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

   10. Applied rewrites 52.5%

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

   11. Taylor expanded in F around inf

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

       1. lower-/.f64 50.3

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

   13. Applied rewrites 50.3%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 0.13:\\ \;\;\;\;\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}, 0.3333333333333333 \cdot \left(x \cdot \left(B \cdot B\right)\right) - x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{B} \cdot \frac{1}{F} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 56.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{F}}{\tan B \cdot \frac{-1}{F}}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 4e-6)
   (/ (- (/ F (sqrt (fma 2.0 x (fma F F 2.0)))) x) B)
   (/ (/ x F) (* (tan B) (/ -1.0 F)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 4e-6) {
		tmp = ((F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) - x) / B;
	} else {
		tmp = (x / F) / (tan(B) * (-1.0 / F));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 4e-6)
		tmp = Float64(Float64(Float64(F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) - x) / B);
	else
		tmp = Float64(Float64(x / F) / Float64(tan(B) * Float64(-1.0 / F)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 4e-6], N[(N[(N[(F / N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(x / F), $MachinePrecision] / N[(N[Tan[B], $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 3.99999999999999982e-6

   1. Initial program 73.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. sub-neg N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-neg.f64 63.5

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

   5. Applied rewrites 63.5%

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

   7. Applied rewrites 63.5%

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

   if 3.99999999999999982e-6 < B

   1. Initial program 79.1%

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

   4. Applied rewrites 79.1%

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

   6. Applied rewrites 75.3%

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

   7. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 48.5

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

   9. Applied rewrites 48.5%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{F}}{\tan B \cdot \frac{-1}{F}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 55.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 1.18 \cdot 10^{-45}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 1.18e-45)
   (/ (- (/ F (sqrt (fma 2.0 x (fma F F 2.0)))) x) B)
   (- (/ -1.0 B) (/ x (tan B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 1.18e-45) {
		tmp = ((F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) - x) / B;
	} else {
		tmp = (-1.0 / B) - (x / tan(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 1.18e-45)
		tmp = Float64(Float64(Float64(F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) - x) / B);
	else
		tmp = Float64(Float64(-1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 1.18e-45], N[(N[(N[(F / N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.18e-45

   1. Initial program 73.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. sub-neg N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-neg.f64 62.5

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

   5. Applied rewrites 62.5%

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

   7. Applied rewrites 62.5%

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

   if 1.18e-45 < B

   1. Initial program 79.2%

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

   3. Taylor expanded in B around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}{B}} \]

   5. Applied rewrites 28.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \mathsf{fma}\left(0.16666666666666666, \left(B \cdot B\right) \cdot F, F\right)}{B}} \]

   7. Applied rewrites 28.2%

      \[\leadsto \color{blue}{\frac{1 \cdot \mathsf{fma}\left(0.16666666666666666, F \cdot \left(B \cdot B\right), F\right)}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)} \cdot B} - \frac{x}{\tan B}} \]

   8. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} - \frac{x}{\tan B} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \frac{F}{B}} - \frac{x}{\tan B} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \frac{F}{B}} - \frac{x}{\tan B} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot \frac{F}{B} - \frac{x}{\tan B} \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot \frac{F}{B} - \frac{x}{\tan B} \]

      5. lower-+.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot \frac{F}{B} - \frac{x}{\tan B} \]

      6. +-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2 + \color{blue}{\left({F}^{2} + 2 \cdot x\right)}}} \cdot \frac{F}{B} - \frac{x}{\tan B} \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{2 + \left(\color{blue}{F \cdot F} + 2 \cdot x\right)}} \cdot \frac{F}{B} - \frac{x}{\tan B} \]

      8. lower-fma.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + \color{blue}{\mathsf{fma}\left(F, F, 2 \cdot x\right)}}} \cdot \frac{F}{B} - \frac{x}{\tan B} \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + \mathsf{fma}\left(F, F, \color{blue}{2 \cdot x}\right)}} \cdot \frac{F}{B} - \frac{x}{\tan B} \]

      10. lower-/.f64 55.9

          \[\leadsto \sqrt{\frac{1}{2 + \mathsf{fma}\left(F, F, 2 \cdot x\right)}} \cdot \color{blue}{\frac{F}{B}} - \frac{x}{\tan B} \]

   10. Applied rewrites 55.9%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{2 + \mathsf{fma}\left(F, F, 2 \cdot x\right)}} \cdot \frac{F}{B}} - \frac{x}{\tan B} \]

   11. Taylor expanded in F around -inf

       \[\leadsto \color{blue}{\frac{-1}{B}} - \frac{x}{\tan B} \]
   12. Step-by-step derivation

       1. lower-/.f64 48.7

          \[\leadsto \color{blue}{\frac{-1}{B}} - \frac{x}{\tan B} \]

   13. Applied rewrites 48.7%

       \[\leadsto \color{blue}{\frac{-1}{B}} - \frac{x}{\tan B} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 49.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.135:\\ \;\;\;\;\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}, 0.3333333333333333 \cdot \left(x \cdot \left(B \cdot B\right)\right) - x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot 0.16666666666666666 - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 0.135)
   (/
    (fma
     F
     (sqrt (/ 1.0 (fma 2.0 x (fma F F 2.0))))
     (- (* 0.3333333333333333 (* x (* B B))) x))
    B)
   (- (* B 0.16666666666666666) (/ x (tan B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 0.135) {
		tmp = fma(F, sqrt((1.0 / fma(2.0, x, fma(F, F, 2.0)))), ((0.3333333333333333 * (x * (B * B))) - x)) / B;
	} else {
		tmp = (B * 0.16666666666666666) - (x / tan(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 0.135)
		tmp = Float64(fma(F, sqrt(Float64(1.0 / fma(2.0, x, fma(F, F, 2.0)))), Float64(Float64(0.3333333333333333 * Float64(x * Float64(B * B))) - x)) / B);
	else
		tmp = Float64(Float64(B * 0.16666666666666666) - Float64(x / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 0.135], N[(N[(F * N[Sqrt[N[(1.0 / N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[(0.3333333333333333 * N[(x * N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], N[(N[(B * 0.16666666666666666), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 0.13500000000000001

   1. Initial program 73.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{\frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)}}{B} \]

      2. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{\left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}}}{B} \]

      3. associate-*l* N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{\left({B}^{2} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)} \cdot \frac{1}{6}}{B} \]

      4. associate-*r* N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{{B}^{2} \cdot \left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}\right)}}{B} \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}{B}} \]

   5. Applied rewrites 67.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \mathsf{fma}\left(0.16666666666666666, \left(B \cdot B\right) \cdot F, F\right)}{B}} \]

   7. Applied rewrites 66.9%

      \[\leadsto \color{blue}{\frac{1 \cdot \mathsf{fma}\left(0.16666666666666666, F \cdot \left(B \cdot B\right), F\right)}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)} \cdot B} - \frac{x}{\tan B}} \]

   8. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} - \frac{x}{\tan B} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \frac{F}{B}} - \frac{x}{\tan B} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \frac{F}{B}} - \frac{x}{\tan B} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot \frac{F}{B} - \frac{x}{\tan B} \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot \frac{F}{B} - \frac{x}{\tan B} \]

      5. lower-+.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot \frac{F}{B} - \frac{x}{\tan B} \]

      6. +-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2 + \color{blue}{\left({F}^{2} + 2 \cdot x\right)}}} \cdot \frac{F}{B} - \frac{x}{\tan B} \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{2 + \left(\color{blue}{F \cdot F} + 2 \cdot x\right)}} \cdot \frac{F}{B} - \frac{x}{\tan B} \]

      8. lower-fma.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + \color{blue}{\mathsf{fma}\left(F, F, 2 \cdot x\right)}}} \cdot \frac{F}{B} - \frac{x}{\tan B} \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + \mathsf{fma}\left(F, F, \color{blue}{2 \cdot x}\right)}} \cdot \frac{F}{B} - \frac{x}{\tan B} \]

      10. lower-/.f64 62.8

          \[\leadsto \sqrt{\frac{1}{2 + \mathsf{fma}\left(F, F, 2 \cdot x\right)}} \cdot \color{blue}{\frac{F}{B}} - \frac{x}{\tan B} \]

   10. Applied rewrites 62.8%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{2 + \mathsf{fma}\left(F, F, 2 \cdot x\right)}} \cdot \frac{F}{B}} - \frac{x}{\tan B} \]

   11. Taylor expanded in B around 0

       \[\leadsto \color{blue}{\frac{\left(\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) - x}{B}} \]
   12. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left(\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) - x}{B}} \]

   13. Applied rewrites 63.1%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}, 0.3333333333333333 \cdot \left(x \cdot \left(B \cdot B\right)\right) - x\right)}{B}} \]

   if 0.13500000000000001 < B

   1. Initial program 79.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{\frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)}}{B} \]

      2. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{\left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}}}{B} \]

      3. associate-*l* N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{\left({B}^{2} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)} \cdot \frac{1}{6}}{B} \]

      4. associate-*r* N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{{B}^{2} \cdot \left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}\right)}}{B} \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}{B}} \]

   5. Applied rewrites 20.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \mathsf{fma}\left(0.16666666666666666, \left(B \cdot B\right) \cdot F, F\right)}{B}} \]

   7. Applied rewrites 19.9%

      \[\leadsto \color{blue}{\frac{1 \cdot \mathsf{fma}\left(0.16666666666666666, F \cdot \left(B \cdot B\right), F\right)}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)} \cdot B} - \frac{x}{\tan B}} \]

   8. Taylor expanded in F around inf

      \[\leadsto \frac{1 \cdot \mathsf{fma}\left(\frac{1}{6}, F \cdot \left(B \cdot B\right), F\right)}{\color{blue}{B \cdot F}} - \frac{x}{\tan B} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1 \cdot \mathsf{fma}\left(\frac{1}{6}, F \cdot \left(B \cdot B\right), F\right)}{\color{blue}{F \cdot B}} - \frac{x}{\tan B} \]

      2. lower-*.f64 11.1

         \[\leadsto \frac{1 \cdot \mathsf{fma}\left(0.16666666666666666, F \cdot \left(B \cdot B\right), F\right)}{\color{blue}{F \cdot B}} - \frac{x}{\tan B} \]

   10. Applied rewrites 11.1%

       \[\leadsto \frac{1 \cdot \mathsf{fma}\left(0.16666666666666666, F \cdot \left(B \cdot B\right), F\right)}{\color{blue}{F \cdot B}} - \frac{x}{\tan B} \]

   11. Taylor expanded in B around inf

       \[\leadsto \color{blue}{\frac{1}{6} \cdot B} - \frac{x}{\tan B} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{B \cdot \frac{1}{6}} - \frac{x}{\tan B} \]

       2. lower-*.f64 18.0

          \[\leadsto \color{blue}{B \cdot 0.16666666666666666} - \frac{x}{\tan B} \]

   13. Applied rewrites 18.0%

       \[\leadsto \color{blue}{B \cdot 0.16666666666666666} - \frac{x}{\tan B} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 52.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}\\ \mathbf{if}\;F \leq -110:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, t\_0, -1 - x\right)}{B}\\ \mathbf{elif}\;F \leq 3000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(F, \mathsf{fma}\left(-0.5 \cdot \left(F \cdot F\right), \sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right) \cdot \left(\mathsf{fma}\left(2, x, 2\right) \cdot \mathsf{fma}\left(2, x, 2\right)\right)}}, \sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}}\right), -x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, t\_0, 1\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (fma 2.0 x 2.0) (* F F))))
   (if (<= F -110.0)
     (/ (fma 0.5 t_0 (- -1.0 x)) B)
     (if (<= F 3000000.0)
       (/
        (fma
         F
         (fma
          (* -0.5 (* F F))
          (sqrt
           (/ 1.0 (* (fma 2.0 x 2.0) (* (fma 2.0 x 2.0) (fma 2.0 x 2.0)))))
          (sqrt (/ 1.0 (fma 2.0 x 2.0))))
         (- x))
        B)
       (/ (- (fma -0.5 t_0 1.0) x) B)))))

C

double code(double F, double B, double x) {
	double t_0 = fma(2.0, x, 2.0) / (F * F);
	double tmp;
	if (F <= -110.0) {
		tmp = fma(0.5, t_0, (-1.0 - x)) / B;
	} else if (F <= 3000000.0) {
		tmp = fma(F, fma((-0.5 * (F * F)), sqrt((1.0 / (fma(2.0, x, 2.0) * (fma(2.0, x, 2.0) * fma(2.0, x, 2.0))))), sqrt((1.0 / fma(2.0, x, 2.0)))), -x) / B;
	} else {
		tmp = (fma(-0.5, t_0, 1.0) - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(fma(2.0, x, 2.0) / Float64(F * F))
	tmp = 0.0
	if (F <= -110.0)
		tmp = Float64(fma(0.5, t_0, Float64(-1.0 - x)) / B);
	elseif (F <= 3000000.0)
		tmp = Float64(fma(F, fma(Float64(-0.5 * Float64(F * F)), sqrt(Float64(1.0 / Float64(fma(2.0, x, 2.0) * Float64(fma(2.0, x, 2.0) * fma(2.0, x, 2.0))))), sqrt(Float64(1.0 / fma(2.0, x, 2.0)))), Float64(-x)) / B);
	else
		tmp = Float64(Float64(fma(-0.5, t_0, 1.0) - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(2.0 * x + 2.0), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -110.0], N[(N[(0.5 * t$95$0 + N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 3000000.0], N[(N[(F * N[(N[(-0.5 * N[(F * F), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(N[(2.0 * x + 2.0), $MachinePrecision] * N[(N[(2.0 * x + 2.0), $MachinePrecision] * N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Sqrt[N[(1.0 / N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + (-x)), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(-0.5 * t$95$0 + 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -110

   1. Initial program 49.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   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. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, \mathsf{neg}\left(x\right)\right)}}{B} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      14. lower-neg.f64 46.5

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 46.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}} - \left(1 + x\right)}}{B} \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}} + \left(\mathsf{neg}\left(\left(1 + x\right)\right)\right)}}{B} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \frac{2 + 2 \cdot x}{{F}^{2}}, \mathsf{neg}\left(\left(1 + x\right)\right)\right)}}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\frac{2 + 2 \cdot x}{{F}^{2}}}, \mathsf{neg}\left(\left(1 + x\right)\right)\right)}{B} \]

      4. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2}, \frac{\color{blue}{2 \cdot x + 2}}{{F}^{2}}, \mathsf{neg}\left(\left(1 + x\right)\right)\right)}{B} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2}, \frac{\color{blue}{\mathsf{fma}\left(2, x, 2\right)}}{{F}^{2}}, \mathsf{neg}\left(\left(1 + x\right)\right)\right)}{B} \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{\color{blue}{F \cdot F}}, \mathsf{neg}\left(\left(1 + x\right)\right)\right)}{B} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{\color{blue}{F \cdot F}}, \mathsf{neg}\left(\left(1 + x\right)\right)\right)}{B} \]

      8. distribute-neg-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, \color{blue}{\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(x\right)\right)}\right)}{B} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, \color{blue}{-1} + \left(\mathsf{neg}\left(x\right)\right)\right)}{B} \]

      10. unsub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, \color{blue}{-1 - x}\right)}{B} \]

      11. lower--.f64 55.6

          \[\leadsto \frac{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, \color{blue}{-1 - x}\right)}{B} \]

   8. Applied rewrites 55.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, -1 - x\right)}}{B} \]

   if -110 < F < 3e6

   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. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, \mathsf{neg}\left(x\right)\right)}}{B} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      14. lower-neg.f64 47.3

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 47.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \frac{\color{blue}{F \cdot \left(\sqrt{\frac{1}{2 + 2 \cdot x}} + \frac{-1}{2} \cdot \left({F}^{2} \cdot \sqrt{\frac{1}{{\left(2 + 2 \cdot x\right)}^{3}}}\right)\right) - x}}{B} \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \left(\sqrt{\frac{1}{2 + 2 \cdot x}} + \frac{-1}{2} \cdot \left({F}^{2} \cdot \sqrt{\frac{1}{{\left(2 + 2 \cdot x\right)}^{3}}}\right)\right) + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + 2 \cdot x}} + \frac{-1}{2} \cdot \left({F}^{2} \cdot \sqrt{\frac{1}{{\left(2 + 2 \cdot x\right)}^{3}}}\right), \mathsf{neg}\left(x\right)\right)}}{B} \]

   8. Applied rewrites 47.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(F, \mathsf{fma}\left(-0.5 \cdot \left(F \cdot F\right), \sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right) \cdot \left(\mathsf{fma}\left(2, x, 2\right) \cdot \mathsf{fma}\left(2, x, 2\right)\right)}}, \sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}}\right), -x\right)}}{B} \]

   if 3e6 < F

   1. Initial program 63.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, \mathsf{neg}\left(x\right)\right)}}{B} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      14. lower-neg.f64 47.5

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 47.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right) - x}}{B} \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right) - x}}{B} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}} + 1\right)} - x}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \frac{2 + 2 \cdot x}{{F}^{2}}, 1\right)} - x}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{2 + 2 \cdot x}{{F}^{2}}}, 1\right) - x}{B} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{2 \cdot x + 2}}{{F}^{2}}, 1\right) - x}{B} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{\mathsf{fma}\left(2, x, 2\right)}}{{F}^{2}}, 1\right) - x}{B} \]

      7. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{\color{blue}{F \cdot F}}, 1\right) - x}{B} \]

      8. lower-*.f64 55.5

         \[\leadsto \frac{\mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{\color{blue}{F \cdot F}}, 1\right) - x}{B} \]

   8. Applied rewrites 55.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, 1\right) - x}}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 52.5% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}\\ \mathbf{if}\;F \leq -106:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, t\_0, -1 - x\right)}{B}\\ \mathbf{elif}\;F \leq 58000000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}, 0.3333333333333333 \cdot \left(x \cdot \left(B \cdot B\right)\right) - x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, t\_0, 1\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (fma 2.0 x 2.0) (* F F))))
   (if (<= F -106.0)
     (/ (fma 0.5 t_0 (- -1.0 x)) B)
     (if (<= F 58000000000000.0)
       (/
        (fma
         F
         (sqrt (/ 1.0 (fma 2.0 x (fma F F 2.0))))
         (- (* 0.3333333333333333 (* x (* B B))) x))
        B)
       (/ (- (fma -0.5 t_0 1.0) x) B)))))

C

double code(double F, double B, double x) {
	double t_0 = fma(2.0, x, 2.0) / (F * F);
	double tmp;
	if (F <= -106.0) {
		tmp = fma(0.5, t_0, (-1.0 - x)) / B;
	} else if (F <= 58000000000000.0) {
		tmp = fma(F, sqrt((1.0 / fma(2.0, x, fma(F, F, 2.0)))), ((0.3333333333333333 * (x * (B * B))) - x)) / B;
	} else {
		tmp = (fma(-0.5, t_0, 1.0) - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(fma(2.0, x, 2.0) / Float64(F * F))
	tmp = 0.0
	if (F <= -106.0)
		tmp = Float64(fma(0.5, t_0, Float64(-1.0 - x)) / B);
	elseif (F <= 58000000000000.0)
		tmp = Float64(fma(F, sqrt(Float64(1.0 / fma(2.0, x, fma(F, F, 2.0)))), Float64(Float64(0.3333333333333333 * Float64(x * Float64(B * B))) - x)) / B);
	else
		tmp = Float64(Float64(fma(-0.5, t_0, 1.0) - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(2.0 * x + 2.0), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -106.0], N[(N[(0.5 * t$95$0 + N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 58000000000000.0], N[(N[(F * N[Sqrt[N[(1.0 / N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[(0.3333333333333333 * N[(x * N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(-0.5 * t$95$0 + 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -106

   1. Initial program 50.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, \mathsf{neg}\left(x\right)\right)}}{B} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      14. lower-neg.f64 45.9

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 45.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}} - \left(1 + x\right)}}{B} \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}} + \left(\mathsf{neg}\left(\left(1 + x\right)\right)\right)}}{B} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \frac{2 + 2 \cdot x}{{F}^{2}}, \mathsf{neg}\left(\left(1 + x\right)\right)\right)}}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\frac{2 + 2 \cdot x}{{F}^{2}}}, \mathsf{neg}\left(\left(1 + x\right)\right)\right)}{B} \]

      4. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2}, \frac{\color{blue}{2 \cdot x + 2}}{{F}^{2}}, \mathsf{neg}\left(\left(1 + x\right)\right)\right)}{B} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2}, \frac{\color{blue}{\mathsf{fma}\left(2, x, 2\right)}}{{F}^{2}}, \mathsf{neg}\left(\left(1 + x\right)\right)\right)}{B} \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{\color{blue}{F \cdot F}}, \mathsf{neg}\left(\left(1 + x\right)\right)\right)}{B} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{\color{blue}{F \cdot F}}, \mathsf{neg}\left(\left(1 + x\right)\right)\right)}{B} \]

      8. distribute-neg-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, \color{blue}{\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(x\right)\right)}\right)}{B} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, \color{blue}{-1} + \left(\mathsf{neg}\left(x\right)\right)\right)}{B} \]

      10. unsub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, \color{blue}{-1 - x}\right)}{B} \]

      11. lower--.f64 54.9

          \[\leadsto \frac{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, \color{blue}{-1 - x}\right)}{B} \]

   8. Applied rewrites 54.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, -1 - x\right)}}{B} \]

   if -106 < F < 5.8e13

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{\frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)}}{B} \]

      2. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{\left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}}}{B} \]

      3. associate-*l* N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{\left({B}^{2} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)} \cdot \frac{1}{6}}{B} \]

      4. associate-*r* N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{{B}^{2} \cdot \left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}\right)}}{B} \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}{B}} \]

   5. Applied rewrites 62.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \mathsf{fma}\left(0.16666666666666666, \left(B \cdot B\right) \cdot F, F\right)}{B}} \]

   7. Applied rewrites 62.5%

      \[\leadsto \color{blue}{\frac{1 \cdot \mathsf{fma}\left(0.16666666666666666, F \cdot \left(B \cdot B\right), F\right)}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)} \cdot B} - \frac{x}{\tan B}} \]

   8. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} - \frac{x}{\tan B} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \frac{F}{B}} - \frac{x}{\tan B} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \frac{F}{B}} - \frac{x}{\tan B} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot \frac{F}{B} - \frac{x}{\tan B} \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot \frac{F}{B} - \frac{x}{\tan B} \]

      5. lower-+.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot \frac{F}{B} - \frac{x}{\tan B} \]

      6. +-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2 + \color{blue}{\left({F}^{2} + 2 \cdot x\right)}}} \cdot \frac{F}{B} - \frac{x}{\tan B} \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{2 + \left(\color{blue}{F \cdot F} + 2 \cdot x\right)}} \cdot \frac{F}{B} - \frac{x}{\tan B} \]

      8. lower-fma.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + \color{blue}{\mathsf{fma}\left(F, F, 2 \cdot x\right)}}} \cdot \frac{F}{B} - \frac{x}{\tan B} \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + \mathsf{fma}\left(F, F, \color{blue}{2 \cdot x}\right)}} \cdot \frac{F}{B} - \frac{x}{\tan B} \]

      10. lower-/.f64 78.9

          \[\leadsto \sqrt{\frac{1}{2 + \mathsf{fma}\left(F, F, 2 \cdot x\right)}} \cdot \color{blue}{\frac{F}{B}} - \frac{x}{\tan B} \]

   10. Applied rewrites 78.9%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{2 + \mathsf{fma}\left(F, F, 2 \cdot x\right)}} \cdot \frac{F}{B}} - \frac{x}{\tan B} \]

   11. Taylor expanded in B around 0

       \[\leadsto \color{blue}{\frac{\left(\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) - x}{B}} \]
   12. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left(\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) - x}{B}} \]

   13. Applied rewrites 47.4%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}, 0.3333333333333333 \cdot \left(x \cdot \left(B \cdot B\right)\right) - x\right)}{B}} \]

   if 5.8e13 < F

   1. Initial program 61.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   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. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, \mathsf{neg}\left(x\right)\right)}}{B} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      14. lower-neg.f64 48.0

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 48.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right) - x}}{B} \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right) - x}}{B} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}} + 1\right)} - x}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \frac{2 + 2 \cdot x}{{F}^{2}}, 1\right)} - x}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{2 + 2 \cdot x}{{F}^{2}}}, 1\right) - x}{B} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{2 \cdot x + 2}}{{F}^{2}}, 1\right) - x}{B} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{\mathsf{fma}\left(2, x, 2\right)}}{{F}^{2}}, 1\right) - x}{B} \]

      7. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{\color{blue}{F \cdot F}}, 1\right) - x}{B} \]

      8. lower-*.f64 56.4

         \[\leadsto \frac{\mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{\color{blue}{F \cdot F}}, 1\right) - x}{B} \]

   8. Applied rewrites 56.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, 1\right) - x}}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 52.5% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}\\ \mathbf{if}\;F \leq -106:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, t\_0, -1 - x\right)}{B}\\ \mathbf{elif}\;F \leq 0.33:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, t\_0, 1\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (fma 2.0 x 2.0) (* F F))))
   (if (<= F -106.0)
     (/ (fma 0.5 t_0 (- -1.0 x)) B)
     (if (<= F 0.33)
       (/ (- (/ F (sqrt (fma 2.0 x (fma F F 2.0)))) x) B)
       (/ (- (fma -0.5 t_0 1.0) x) B)))))

C

double code(double F, double B, double x) {
	double t_0 = fma(2.0, x, 2.0) / (F * F);
	double tmp;
	if (F <= -106.0) {
		tmp = fma(0.5, t_0, (-1.0 - x)) / B;
	} else if (F <= 0.33) {
		tmp = ((F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) - x) / B;
	} else {
		tmp = (fma(-0.5, t_0, 1.0) - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(fma(2.0, x, 2.0) / Float64(F * F))
	tmp = 0.0
	if (F <= -106.0)
		tmp = Float64(fma(0.5, t_0, Float64(-1.0 - x)) / B);
	elseif (F <= 0.33)
		tmp = Float64(Float64(Float64(F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) - x) / B);
	else
		tmp = Float64(Float64(fma(-0.5, t_0, 1.0) - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(2.0 * x + 2.0), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -106.0], N[(N[(0.5 * t$95$0 + N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 0.33], N[(N[(N[(F / N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(-0.5 * t$95$0 + 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -106

   1. Initial program 50.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, \mathsf{neg}\left(x\right)\right)}}{B} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      14. lower-neg.f64 45.9

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 45.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}} - \left(1 + x\right)}}{B} \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}} + \left(\mathsf{neg}\left(\left(1 + x\right)\right)\right)}}{B} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \frac{2 + 2 \cdot x}{{F}^{2}}, \mathsf{neg}\left(\left(1 + x\right)\right)\right)}}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\frac{2 + 2 \cdot x}{{F}^{2}}}, \mathsf{neg}\left(\left(1 + x\right)\right)\right)}{B} \]

      4. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2}, \frac{\color{blue}{2 \cdot x + 2}}{{F}^{2}}, \mathsf{neg}\left(\left(1 + x\right)\right)\right)}{B} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2}, \frac{\color{blue}{\mathsf{fma}\left(2, x, 2\right)}}{{F}^{2}}, \mathsf{neg}\left(\left(1 + x\right)\right)\right)}{B} \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{\color{blue}{F \cdot F}}, \mathsf{neg}\left(\left(1 + x\right)\right)\right)}{B} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{\color{blue}{F \cdot F}}, \mathsf{neg}\left(\left(1 + x\right)\right)\right)}{B} \]

      8. distribute-neg-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, \color{blue}{\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(x\right)\right)}\right)}{B} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, \color{blue}{-1} + \left(\mathsf{neg}\left(x\right)\right)\right)}{B} \]

      10. unsub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, \color{blue}{-1 - x}\right)}{B} \]

      11. lower--.f64 54.9

          \[\leadsto \frac{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, \color{blue}{-1 - x}\right)}{B} \]

   8. Applied rewrites 54.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, -1 - x\right)}}{B} \]

   if -106 < F < 0.330000000000000016

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   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. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, \mathsf{neg}\left(x\right)\right)}}{B} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      14. lower-neg.f64 47.7

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 47.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{B}} \]

   7. Applied rewrites 47.7%

      \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}} \]

   if 0.330000000000000016 < F

   1. Initial program 64.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, \mathsf{neg}\left(x\right)\right)}}{B} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      14. lower-neg.f64 47.6

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 47.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right) - x}}{B} \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right) - x}}{B} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}} + 1\right)} - x}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \frac{2 + 2 \cdot x}{{F}^{2}}, 1\right)} - x}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{2 + 2 \cdot x}{{F}^{2}}}, 1\right) - x}{B} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{2 \cdot x + 2}}{{F}^{2}}, 1\right) - x}{B} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{\mathsf{fma}\left(2, x, 2\right)}}{{F}^{2}}, 1\right) - x}{B} \]

      7. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{\color{blue}{F \cdot F}}, 1\right) - x}{B} \]

      8. lower-*.f64 55.4

         \[\leadsto \frac{\mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{\color{blue}{F \cdot F}}, 1\right) - x}{B} \]

   8. Applied rewrites 55.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, 1\right) - x}}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 52.4% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}\\ \mathbf{if}\;F \leq -37:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, t\_0, -1 - x\right)}{B}\\ \mathbf{elif}\;F \leq 0.33:\\ \;\;\;\;\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}}, -x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, t\_0, 1\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (fma 2.0 x 2.0) (* F F))))
   (if (<= F -37.0)
     (/ (fma 0.5 t_0 (- -1.0 x)) B)
     (if (<= F 0.33)
       (/ (fma F (sqrt (/ 1.0 (fma 2.0 x 2.0))) (- x)) B)
       (/ (- (fma -0.5 t_0 1.0) x) B)))))

C

double code(double F, double B, double x) {
	double t_0 = fma(2.0, x, 2.0) / (F * F);
	double tmp;
	if (F <= -37.0) {
		tmp = fma(0.5, t_0, (-1.0 - x)) / B;
	} else if (F <= 0.33) {
		tmp = fma(F, sqrt((1.0 / fma(2.0, x, 2.0))), -x) / B;
	} else {
		tmp = (fma(-0.5, t_0, 1.0) - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(fma(2.0, x, 2.0) / Float64(F * F))
	tmp = 0.0
	if (F <= -37.0)
		tmp = Float64(fma(0.5, t_0, Float64(-1.0 - x)) / B);
	elseif (F <= 0.33)
		tmp = Float64(fma(F, sqrt(Float64(1.0 / fma(2.0, x, 2.0))), Float64(-x)) / B);
	else
		tmp = Float64(Float64(fma(-0.5, t_0, 1.0) - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(2.0 * x + 2.0), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -37.0], N[(N[(0.5 * t$95$0 + N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 0.33], N[(N[(F * N[Sqrt[N[(1.0 / N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + (-x)), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(-0.5 * t$95$0 + 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -37

   1. Initial program 50.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, \mathsf{neg}\left(x\right)\right)}}{B} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      14. lower-neg.f64 45.9

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 45.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}} - \left(1 + x\right)}}{B} \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}} + \left(\mathsf{neg}\left(\left(1 + x\right)\right)\right)}}{B} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{2}, \frac{2 + 2 \cdot x}{{F}^{2}}, \mathsf{neg}\left(\left(1 + x\right)\right)\right)}}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\frac{2 + 2 \cdot x}{{F}^{2}}}, \mathsf{neg}\left(\left(1 + x\right)\right)\right)}{B} \]

      4. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2}, \frac{\color{blue}{2 \cdot x + 2}}{{F}^{2}}, \mathsf{neg}\left(\left(1 + x\right)\right)\right)}{B} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2}, \frac{\color{blue}{\mathsf{fma}\left(2, x, 2\right)}}{{F}^{2}}, \mathsf{neg}\left(\left(1 + x\right)\right)\right)}{B} \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{\color{blue}{F \cdot F}}, \mathsf{neg}\left(\left(1 + x\right)\right)\right)}{B} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{\color{blue}{F \cdot F}}, \mathsf{neg}\left(\left(1 + x\right)\right)\right)}{B} \]

      8. distribute-neg-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, \color{blue}{\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(x\right)\right)}\right)}{B} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, \color{blue}{-1} + \left(\mathsf{neg}\left(x\right)\right)\right)}{B} \]

      10. unsub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, \color{blue}{-1 - x}\right)}{B} \]

      11. lower--.f64 54.9

          \[\leadsto \frac{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, \color{blue}{-1 - x}\right)}{B} \]

   8. Applied rewrites 54.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, -1 - x\right)}}{B} \]

   if -37 < F < 0.330000000000000016

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   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. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, \mathsf{neg}\left(x\right)\right)}}{B} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      14. lower-neg.f64 47.7

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 47.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B} + \frac{F}{B} \cdot \sqrt{\frac{1}{2 + 2 \cdot x}}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + 2 \cdot x}} + -1 \cdot \frac{x}{B}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{F}{B} \cdot \sqrt{\frac{1}{2 + 2 \cdot x}} + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{B}\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + 2 \cdot x}} - \frac{x}{B}} \]

      4. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + 2 \cdot x}}}{B}} - \frac{x}{B} \]

      5. div-sub N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + 2 \cdot x}} - x}{B}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + 2 \cdot x}} - x}{B}} \]

      7. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + 2 \cdot x}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + 2 \cdot x}}, \mathsf{neg}\left(x\right)\right)}}{B} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + 2 \cdot x}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + 2 \cdot x}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{2 \cdot x + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, x, 2\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-neg.f64 47.7

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}}, \color{blue}{-x}\right)}{B} \]

   8. Applied rewrites 47.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}}, -x\right)}{B}} \]

   if 0.330000000000000016 < F

   1. Initial program 64.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, \mathsf{neg}\left(x\right)\right)}}{B} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      14. lower-neg.f64 47.6

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 47.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right) - x}}{B} \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right) - x}}{B} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}} + 1\right)} - x}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \frac{2 + 2 \cdot x}{{F}^{2}}, 1\right)} - x}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{2 + 2 \cdot x}{{F}^{2}}}, 1\right) - x}{B} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{2 \cdot x + 2}}{{F}^{2}}, 1\right) - x}{B} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{\mathsf{fma}\left(2, x, 2\right)}}{{F}^{2}}, 1\right) - x}{B} \]

      7. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{\color{blue}{F \cdot F}}, 1\right) - x}{B} \]

      8. lower-*.f64 55.4

         \[\leadsto \frac{\mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{\color{blue}{F \cdot F}}, 1\right) - x}{B} \]

   8. Applied rewrites 55.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, 1\right) - x}}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 21: 44.7% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -6.2 \cdot 10^{-54}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 3000000:\\ \;\;\;\;\left(-x\right) \cdot \mathsf{fma}\left(B, -0.3333333333333333, \frac{1}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, 1\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -6.2e-54)
   (/ (- -1.0 x) B)
   (if (<= F 3000000.0)
     (* (- x) (fma B -0.3333333333333333 (/ 1.0 B)))
     (/ (- (fma -0.5 (/ (fma 2.0 x 2.0) (* F F)) 1.0) x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -6.2e-54) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 3000000.0) {
		tmp = -x * fma(B, -0.3333333333333333, (1.0 / B));
	} else {
		tmp = (fma(-0.5, (fma(2.0, x, 2.0) / (F * F)), 1.0) - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -6.2e-54)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 3000000.0)
		tmp = Float64(Float64(-x) * fma(B, -0.3333333333333333, Float64(1.0 / B)));
	else
		tmp = Float64(Float64(fma(-0.5, Float64(fma(2.0, x, 2.0) / Float64(F * F)), 1.0) - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -6.2e-54], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 3000000.0], N[((-x) * N[(B * -0.3333333333333333 + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.5 * N[(N[(2.0 * x + 2.0), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -6.20000000000000008e-54

   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. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, \mathsf{neg}\left(x\right)\right)}}{B} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      14. lower-neg.f64 46.2

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 46.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\left(1 + x\right)\right)}}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(1 + x\right)\right)}{B}} \]

      4. distribute-neg-in N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-1} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      6. unsub-neg N/A

         \[\leadsto \frac{\color{blue}{-1 - x}}{B} \]

      7. lower--.f64 52.4

         \[\leadsto \frac{\color{blue}{-1 - x}}{B} \]

   8. Applied rewrites 52.4%

      \[\leadsto \color{blue}{\frac{-1 - x}{B}} \]

   if -6.20000000000000008e-54 < F < 3e6

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{\frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)}}{B} \]

      2. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{\left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}}}{B} \]

      3. associate-*l* N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{\left({B}^{2} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)} \cdot \frac{1}{6}}{B} \]

      4. associate-*r* N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{{B}^{2} \cdot \left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}\right)}}{B} \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}{B}} \]

   5. Applied rewrites 63.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \mathsf{fma}\left(0.16666666666666666, \left(B \cdot B\right) \cdot F, F\right)}{B}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \left(\mathsf{neg}\left(x \cdot \color{blue}{\frac{1 + \frac{-1}{3} \cdot {B}^{2}}{B}}\right)\right) + \frac{\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \mathsf{fma}\left(\frac{1}{6}, \left(B \cdot B\right) \cdot F, F\right)}{B} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \color{blue}{\frac{1 + \frac{-1}{3} \cdot {B}^{2}}{B}}\right)\right) + \frac{\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \mathsf{fma}\left(\frac{1}{6}, \left(B \cdot B\right) \cdot F, F\right)}{B} \]

      2. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{\color{blue}{\frac{-1}{3} \cdot {B}^{2} + 1}}{B}\right)\right) + \frac{\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \mathsf{fma}\left(\frac{1}{6}, \left(B \cdot B\right) \cdot F, F\right)}{B} \]

      3. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{\color{blue}{{B}^{2} \cdot \frac{-1}{3}} + 1}{B}\right)\right) + \frac{\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \mathsf{fma}\left(\frac{1}{6}, \left(B \cdot B\right) \cdot F, F\right)}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{\color{blue}{\mathsf{fma}\left({B}^{2}, \frac{-1}{3}, 1\right)}}{B}\right)\right) + \frac{\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \mathsf{fma}\left(\frac{1}{6}, \left(B \cdot B\right) \cdot F, F\right)}{B} \]

      5. unpow2 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{\mathsf{fma}\left(\color{blue}{B \cdot B}, \frac{-1}{3}, 1\right)}{B}\right)\right) + \frac{\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \mathsf{fma}\left(\frac{1}{6}, \left(B \cdot B\right) \cdot F, F\right)}{B} \]

      6. lower-*.f64 47.0

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\color{blue}{B \cdot B}, -0.3333333333333333, 1\right)}{B}\right) + \frac{\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \mathsf{fma}\left(0.16666666666666666, \left(B \cdot B\right) \cdot F, F\right)}{B} \]

   8. Applied rewrites 47.0%

      \[\leadsto \left(-x \cdot \color{blue}{\frac{\mathsf{fma}\left(B \cdot B, -0.3333333333333333, 1\right)}{B}}\right) + \frac{\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \mathsf{fma}\left(0.16666666666666666, \left(B \cdot B\right) \cdot F, F\right)}{B} \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(\frac{-1}{3} \cdot B + \frac{1}{B}\right)\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(\frac{-1}{3} \cdot B + \frac{1}{B}\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{-1}{3} \cdot B + \frac{1}{B}\right) \cdot x}\right) \]

       3. distribute-rgt-neg-in N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{3} \cdot B + \frac{1}{B}\right) \cdot \left(\mathsf{neg}\left(x\right)\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{3} \cdot B + \frac{1}{B}\right) \cdot \left(\mathsf{neg}\left(x\right)\right)} \]

       5. *-commutative N/A

          \[\leadsto \left(\color{blue}{B \cdot \frac{-1}{3}} + \frac{1}{B}\right) \cdot \left(\mathsf{neg}\left(x\right)\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(B, \frac{-1}{3}, \frac{1}{B}\right)} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

       7. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(B, \frac{-1}{3}, \color{blue}{\frac{1}{B}}\right) \cdot \left(\mathsf{neg}\left(x\right)\right) \]

       8. lower-neg.f64 38.9

          \[\leadsto \mathsf{fma}\left(B, -0.3333333333333333, \frac{1}{B}\right) \cdot \color{blue}{\left(-x\right)} \]

   11. Applied rewrites 38.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(B, -0.3333333333333333, \frac{1}{B}\right) \cdot \left(-x\right)} \]

   if 3e6 < F

   1. Initial program 63.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, \mathsf{neg}\left(x\right)\right)}}{B} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      14. lower-neg.f64 47.5

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 47.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right) - x}}{B} \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{-1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right) - x}}{B} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}} + 1\right)} - x}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{2}, \frac{2 + 2 \cdot x}{{F}^{2}}, 1\right)} - x}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{2 + 2 \cdot x}{{F}^{2}}}, 1\right) - x}{B} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{2 \cdot x + 2}}{{F}^{2}}, 1\right) - x}{B} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{\mathsf{fma}\left(2, x, 2\right)}}{{F}^{2}}, 1\right) - x}{B} \]

      7. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{\color{blue}{F \cdot F}}, 1\right) - x}{B} \]

      8. lower-*.f64 55.5

         \[\leadsto \frac{\mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{\color{blue}{F \cdot F}}, 1\right) - x}{B} \]

   8. Applied rewrites 55.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, 1\right) - x}}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6.2 \cdot 10^{-54}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 3000000:\\ \;\;\;\;\left(-x\right) \cdot \mathsf{fma}\left(B, -0.3333333333333333, \frac{1}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, 1\right) - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 44.7% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -6.2 \cdot 10^{-54}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 3000000:\\ \;\;\;\;\left(-x\right) \cdot \mathsf{fma}\left(B, -0.3333333333333333, \frac{1}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -6.2e-54)
   (/ (- -1.0 x) B)
   (if (<= F 3000000.0)
     (* (- x) (fma B -0.3333333333333333 (/ 1.0 B)))
     (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -6.2e-54) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 3000000.0) {
		tmp = -x * fma(B, -0.3333333333333333, (1.0 / B));
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -6.2e-54)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 3000000.0)
		tmp = Float64(Float64(-x) * fma(B, -0.3333333333333333, Float64(1.0 / B)));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -6.2e-54], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 3000000.0], N[((-x) * N[(B * -0.3333333333333333 + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -6.20000000000000008e-54

   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. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, \mathsf{neg}\left(x\right)\right)}}{B} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      14. lower-neg.f64 46.2

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 46.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\left(1 + x\right)\right)}}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(1 + x\right)\right)}{B}} \]

      4. distribute-neg-in N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-1} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      6. unsub-neg N/A

         \[\leadsto \frac{\color{blue}{-1 - x}}{B} \]

      7. lower--.f64 52.4

         \[\leadsto \frac{\color{blue}{-1 - x}}{B} \]

   8. Applied rewrites 52.4%

      \[\leadsto \color{blue}{\frac{-1 - x}{B}} \]

   if -6.20000000000000008e-54 < F < 3e6

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{\frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)}}{B} \]

      2. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{\left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}}}{B} \]

      3. associate-*l* N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{\left({B}^{2} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)} \cdot \frac{1}{6}}{B} \]

      4. associate-*r* N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{{B}^{2} \cdot \left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}\right)}}{B} \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}{B}} \]

   5. Applied rewrites 63.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \mathsf{fma}\left(0.16666666666666666, \left(B \cdot B\right) \cdot F, F\right)}{B}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \left(\mathsf{neg}\left(x \cdot \color{blue}{\frac{1 + \frac{-1}{3} \cdot {B}^{2}}{B}}\right)\right) + \frac{\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \mathsf{fma}\left(\frac{1}{6}, \left(B \cdot B\right) \cdot F, F\right)}{B} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \color{blue}{\frac{1 + \frac{-1}{3} \cdot {B}^{2}}{B}}\right)\right) + \frac{\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \mathsf{fma}\left(\frac{1}{6}, \left(B \cdot B\right) \cdot F, F\right)}{B} \]

      2. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{\color{blue}{\frac{-1}{3} \cdot {B}^{2} + 1}}{B}\right)\right) + \frac{\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \mathsf{fma}\left(\frac{1}{6}, \left(B \cdot B\right) \cdot F, F\right)}{B} \]

      3. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{\color{blue}{{B}^{2} \cdot \frac{-1}{3}} + 1}{B}\right)\right) + \frac{\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \mathsf{fma}\left(\frac{1}{6}, \left(B \cdot B\right) \cdot F, F\right)}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{\color{blue}{\mathsf{fma}\left({B}^{2}, \frac{-1}{3}, 1\right)}}{B}\right)\right) + \frac{\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \mathsf{fma}\left(\frac{1}{6}, \left(B \cdot B\right) \cdot F, F\right)}{B} \]

      5. unpow2 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{\mathsf{fma}\left(\color{blue}{B \cdot B}, \frac{-1}{3}, 1\right)}{B}\right)\right) + \frac{\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \mathsf{fma}\left(\frac{1}{6}, \left(B \cdot B\right) \cdot F, F\right)}{B} \]

      6. lower-*.f64 47.0

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\color{blue}{B \cdot B}, -0.3333333333333333, 1\right)}{B}\right) + \frac{\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \mathsf{fma}\left(0.16666666666666666, \left(B \cdot B\right) \cdot F, F\right)}{B} \]

   8. Applied rewrites 47.0%

      \[\leadsto \left(-x \cdot \color{blue}{\frac{\mathsf{fma}\left(B \cdot B, -0.3333333333333333, 1\right)}{B}}\right) + \frac{\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \mathsf{fma}\left(0.16666666666666666, \left(B \cdot B\right) \cdot F, F\right)}{B} \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(\frac{-1}{3} \cdot B + \frac{1}{B}\right)\right)} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(\frac{-1}{3} \cdot B + \frac{1}{B}\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{-1}{3} \cdot B + \frac{1}{B}\right) \cdot x}\right) \]

       3. distribute-rgt-neg-in N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{3} \cdot B + \frac{1}{B}\right) \cdot \left(\mathsf{neg}\left(x\right)\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{3} \cdot B + \frac{1}{B}\right) \cdot \left(\mathsf{neg}\left(x\right)\right)} \]

       5. *-commutative N/A

          \[\leadsto \left(\color{blue}{B \cdot \frac{-1}{3}} + \frac{1}{B}\right) \cdot \left(\mathsf{neg}\left(x\right)\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(B, \frac{-1}{3}, \frac{1}{B}\right)} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

       7. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(B, \frac{-1}{3}, \color{blue}{\frac{1}{B}}\right) \cdot \left(\mathsf{neg}\left(x\right)\right) \]

       8. lower-neg.f64 38.9

          \[\leadsto \mathsf{fma}\left(B, -0.3333333333333333, \frac{1}{B}\right) \cdot \color{blue}{\left(-x\right)} \]

   11. Applied rewrites 38.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(B, -0.3333333333333333, \frac{1}{B}\right) \cdot \left(-x\right)} \]

   if 3e6 < F

   1. Initial program 63.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, \mathsf{neg}\left(x\right)\right)}}{B} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      14. lower-neg.f64 47.5

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 47.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

      2. lower--.f64 55.3

         \[\leadsto \frac{\color{blue}{1 - x}}{B} \]

   8. Applied rewrites 55.3%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6.2 \cdot 10^{-54}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 3000000:\\ \;\;\;\;\left(-x\right) \cdot \mathsf{fma}\left(B, -0.3333333333333333, \frac{1}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 37.7% accurate, 11.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{B}\\ \mathbf{if}\;F \leq -6.2 \cdot 10^{-54}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.2 \cdot 10^{+17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 8.2 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) B)))
   (if (<= F -6.2e-54)
     (/ (- -1.0 x) B)
     (if (<= F 2.2e+17) t_0 (if (<= F 8.2e+154) (/ 1.0 B) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = -x / B;
	double tmp;
	if (F <= -6.2e-54) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 2.2e+17) {
		tmp = t_0;
	} else if (F <= 8.2e+154) {
		tmp = 1.0 / B;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -x / b
    if (f <= (-6.2d-54)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 2.2d+17) then
        tmp = t_0
    else if (f <= 8.2d+154) then
        tmp = 1.0d0 / b
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = -x / B;
	double tmp;
	if (F <= -6.2e-54) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 2.2e+17) {
		tmp = t_0;
	} else if (F <= 8.2e+154) {
		tmp = 1.0 / B;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -x / B
	tmp = 0
	if F <= -6.2e-54:
		tmp = (-1.0 - x) / B
	elif F <= 2.2e+17:
		tmp = t_0
	elif F <= 8.2e+154:
		tmp = 1.0 / B
	else:
		tmp = t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / B)
	tmp = 0.0
	if (F <= -6.2e-54)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 2.2e+17)
		tmp = t_0;
	elseif (F <= 8.2e+154)
		tmp = Float64(1.0 / B);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = -x / B;
	tmp = 0.0;
	if (F <= -6.2e-54)
		tmp = (-1.0 - x) / B;
	elseif (F <= 2.2e+17)
		tmp = t_0;
	elseif (F <= 8.2e+154)
		tmp = 1.0 / B;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-x) / B), $MachinePrecision]}, If[LessEqual[F, -6.2e-54], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 2.2e+17], t$95$0, If[LessEqual[F, 8.2e+154], N[(1.0 / B), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if F < -6.20000000000000008e-54

   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. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, \mathsf{neg}\left(x\right)\right)}}{B} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      14. lower-neg.f64 46.2

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 46.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\left(1 + x\right)\right)}}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(1 + x\right)\right)}{B}} \]

      4. distribute-neg-in N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-1} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      6. unsub-neg N/A

         \[\leadsto \frac{\color{blue}{-1 - x}}{B} \]

      7. lower--.f64 52.4

         \[\leadsto \frac{\color{blue}{-1 - x}}{B} \]

   8. Applied rewrites 52.4%

      \[\leadsto \color{blue}{\frac{-1 - x}{B}} \]

   if -6.20000000000000008e-54 < F < 2.2e17 or 8.2e154 < F

   1. Initial program 85.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, \mathsf{neg}\left(x\right)\right)}}{B} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      14. lower-neg.f64 44.4

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 44.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot x}{B}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(x\right)}}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(x\right)}{B}} \]

      4. lower-neg.f64 37.1

         \[\leadsto \frac{\color{blue}{-x}}{B} \]

   8. Applied rewrites 37.1%

      \[\leadsto \color{blue}{\frac{-x}{B}} \]

   if 2.2e17 < F < 8.2e154

   1. Initial program 83.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   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. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, \mathsf{neg}\left(x\right)\right)}}{B} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      14. lower-neg.f64 59.7

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 59.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{B}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{F}{B}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{F}{B}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}} \cdot \frac{F}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{2 + {F}^{2}}}} \cdot \frac{F}{B} \]

      5. +-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{{F}^{2} + 2}}} \cdot \frac{F}{B} \]

      6. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}} \cdot \frac{F}{B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2\right)}}} \cdot \frac{F}{B} \]

      8. lower-/.f64 32.8

         \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \color{blue}{\frac{F}{B}} \]

   8. Applied rewrites 32.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{B}} \]

   9. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{B}} \]
   10. Step-by-step derivation

       1. lower-/.f64 46.1

          \[\leadsto \color{blue}{\frac{1}{B}} \]

   11. Applied rewrites 46.1%

       \[\leadsto \color{blue}{\frac{1}{B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 24: 31.9% accurate, 11.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{B}\\ \mathbf{if}\;x \leq -8.2 \cdot 10^{-89}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-280}:\\ \;\;\;\;\frac{1}{B}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-129}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) B)))
   (if (<= x -8.2e-89)
     t_0
     (if (<= x 1.05e-280) (/ 1.0 B) (if (<= x 6.2e-129) (/ -1.0 B) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = -x / B;
	double tmp;
	if (x <= -8.2e-89) {
		tmp = t_0;
	} else if (x <= 1.05e-280) {
		tmp = 1.0 / B;
	} else if (x <= 6.2e-129) {
		tmp = -1.0 / B;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -x / b
    if (x <= (-8.2d-89)) then
        tmp = t_0
    else if (x <= 1.05d-280) then
        tmp = 1.0d0 / b
    else if (x <= 6.2d-129) then
        tmp = (-1.0d0) / b
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = -x / B;
	double tmp;
	if (x <= -8.2e-89) {
		tmp = t_0;
	} else if (x <= 1.05e-280) {
		tmp = 1.0 / B;
	} else if (x <= 6.2e-129) {
		tmp = -1.0 / B;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -x / B
	tmp = 0
	if x <= -8.2e-89:
		tmp = t_0
	elif x <= 1.05e-280:
		tmp = 1.0 / B
	elif x <= 6.2e-129:
		tmp = -1.0 / B
	else:
		tmp = t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / B)
	tmp = 0.0
	if (x <= -8.2e-89)
		tmp = t_0;
	elseif (x <= 1.05e-280)
		tmp = Float64(1.0 / B);
	elseif (x <= 6.2e-129)
		tmp = Float64(-1.0 / B);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = -x / B;
	tmp = 0.0;
	if (x <= -8.2e-89)
		tmp = t_0;
	elseif (x <= 1.05e-280)
		tmp = 1.0 / B;
	elseif (x <= 6.2e-129)
		tmp = -1.0 / B;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-x) / B), $MachinePrecision]}, If[LessEqual[x, -8.2e-89], t$95$0, If[LessEqual[x, 1.05e-280], N[(1.0 / B), $MachinePrecision], If[LessEqual[x, 6.2e-129], N[(-1.0 / B), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.1999999999999997e-89 or 6.2000000000000001e-129 < x

   1. Initial program 80.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   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. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, \mathsf{neg}\left(x\right)\right)}}{B} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      14. lower-neg.f64 52.6

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 52.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot x}{B}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(x\right)}}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(x\right)}{B}} \]

      4. lower-neg.f64 45.7

         \[\leadsto \frac{\color{blue}{-x}}{B} \]

   8. Applied rewrites 45.7%

      \[\leadsto \color{blue}{\frac{-x}{B}} \]

   if -8.1999999999999997e-89 < x < 1.05e-280

   1. Initial program 69.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, \mathsf{neg}\left(x\right)\right)}}{B} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      14. lower-neg.f64 34.9

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 34.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{B}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{F}{B}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{F}{B}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}} \cdot \frac{F}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{2 + {F}^{2}}}} \cdot \frac{F}{B} \]

      5. +-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{{F}^{2} + 2}}} \cdot \frac{F}{B} \]

      6. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}} \cdot \frac{F}{B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2\right)}}} \cdot \frac{F}{B} \]

      8. lower-/.f64 24.2

         \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \color{blue}{\frac{F}{B}} \]

   8. Applied rewrites 24.2%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{B}} \]

   9. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{B}} \]
   10. Step-by-step derivation

       1. lower-/.f64 24.4

          \[\leadsto \color{blue}{\frac{1}{B}} \]

   11. Applied rewrites 24.4%

       \[\leadsto \color{blue}{\frac{1}{B}} \]

   if 1.05e-280 < x < 6.2000000000000001e-129

   1. Initial program 60.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, \mathsf{neg}\left(x\right)\right)}}{B} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      14. lower-neg.f64 43.5

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 43.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{B}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{F}{B}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{F}{B}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}} \cdot \frac{F}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{2 + {F}^{2}}}} \cdot \frac{F}{B} \]

      5. +-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{{F}^{2} + 2}}} \cdot \frac{F}{B} \]

      6. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}} \cdot \frac{F}{B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2\right)}}} \cdot \frac{F}{B} \]

      8. lower-/.f64 29.4

         \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \color{blue}{\frac{F}{B}} \]

   8. Applied rewrites 29.4%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{B}} \]

   9. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{B}} \]
   10. Step-by-step derivation

       1. lower-/.f64 34.8

          \[\leadsto \color{blue}{\frac{-1}{B}} \]

   11. Applied rewrites 34.8%

       \[\leadsto \color{blue}{\frac{-1}{B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 25: 44.8% accurate, 13.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -6.2 \cdot 10^{-54}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{-38}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -6.2e-54)
   (/ (- -1.0 x) B)
   (if (<= F 4.5e-38) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -6.2e-54) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 4.5e-38) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-6.2d-54)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 4.5d-38) then
        tmp = -x / b
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -6.2e-54) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 4.5e-38) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -6.2e-54:
		tmp = (-1.0 - x) / B
	elif F <= 4.5e-38:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -6.2e-54)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 4.5e-38)
		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 <= -6.2e-54)
		tmp = (-1.0 - x) / B;
	elseif (F <= 4.5e-38)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -6.2e-54], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 4.5e-38], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -6.20000000000000008e-54

   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. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, \mathsf{neg}\left(x\right)\right)}}{B} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      14. lower-neg.f64 46.2

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 46.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\left(1 + x\right)\right)}}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(1 + x\right)\right)}{B}} \]

      4. distribute-neg-in N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-1} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      6. unsub-neg N/A

         \[\leadsto \frac{\color{blue}{-1 - x}}{B} \]

      7. lower--.f64 52.4

         \[\leadsto \frac{\color{blue}{-1 - x}}{B} \]

   8. Applied rewrites 52.4%

      \[\leadsto \color{blue}{\frac{-1 - x}{B}} \]

   if -6.20000000000000008e-54 < F < 4.50000000000000009e-38

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   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. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, \mathsf{neg}\left(x\right)\right)}}{B} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      14. lower-neg.f64 47.4

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 47.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot x}{B}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(x\right)}}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(x\right)}{B}} \]

      4. lower-neg.f64 39.1

         \[\leadsto \frac{\color{blue}{-x}}{B} \]

   8. Applied rewrites 39.1%

      \[\leadsto \color{blue}{\frac{-x}{B}} \]

   if 4.50000000000000009e-38 < F

   1. Initial program 69.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, \mathsf{neg}\left(x\right)\right)}}{B} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      14. lower-neg.f64 47.6

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 47.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

      2. lower--.f64 51.9

         \[\leadsto \frac{\color{blue}{1 - x}}{B} \]

   8. Applied rewrites 51.9%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 26: 17.5% accurate, 20.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 3000000:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (if (<= F 3000000.0) (/ -1.0 B) (/ 1.0 B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= 3000000.0) {
		tmp = -1.0 / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= 3000000.0d0) then
        tmp = (-1.0d0) / b
    else
        tmp = 1.0d0 / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= 3000000.0) {
		tmp = -1.0 / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= 3000000.0:
		tmp = -1.0 / B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= 3000000.0)
		tmp = Float64(-1.0 / B);
	else
		tmp = Float64(1.0 / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= 3000000.0)
		tmp = -1.0 / B;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, 3000000.0], N[(-1.0 / B), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 3e6

   1. Initial program 79.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, \mathsf{neg}\left(x\right)\right)}}{B} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      14. lower-neg.f64 47.0

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 47.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{B}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{F}{B}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{F}{B}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}} \cdot \frac{F}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{2 + {F}^{2}}}} \cdot \frac{F}{B} \]

      5. +-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{{F}^{2} + 2}}} \cdot \frac{F}{B} \]

      6. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}} \cdot \frac{F}{B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2\right)}}} \cdot \frac{F}{B} \]

      8. lower-/.f64 13.2

         \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \color{blue}{\frac{F}{B}} \]

   8. Applied rewrites 13.2%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{B}} \]

   9. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{B}} \]
   10. Step-by-step derivation

       1. lower-/.f64 14.5

          \[\leadsto \color{blue}{\frac{-1}{B}} \]

   11. Applied rewrites 14.5%

       \[\leadsto \color{blue}{\frac{-1}{B}} \]

   if 3e6 < F

   1. Initial program 63.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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. sub-neg N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, \mathsf{neg}\left(x\right)\right)}}{B} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      14. lower-neg.f64 47.5

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 47.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{B}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{F}{B}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{F}{B}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}} \cdot \frac{F}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{2 + {F}^{2}}}} \cdot \frac{F}{B} \]

      5. +-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{{F}^{2} + 2}}} \cdot \frac{F}{B} \]

      6. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}} \cdot \frac{F}{B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2\right)}}} \cdot \frac{F}{B} \]

      8. lower-/.f64 18.6

         \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \color{blue}{\frac{F}{B}} \]

   8. Applied rewrites 18.6%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{B}} \]

   9. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{B}} \]
   10. Step-by-step derivation

       1. lower-/.f64 32.3

          \[\leadsto \color{blue}{\frac{1}{B}} \]

   11. Applied rewrites 32.3%

       \[\leadsto \color{blue}{\frac{1}{B}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 27: 10.3% accurate, 30.7× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (/ -1.0 B))

C

double code(double F, double B, double x) {
	return -1.0 / B;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = (-1.0d0) / b
end function

Java

public static double code(double F, double B, double x) {
	return -1.0 / B;
}

Python

def code(F, B, x):
	return -1.0 / B

Julia

function code(F, B, x)
	return Float64(-1.0 / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = -1.0 / B;
end

Wolfram

code[F_, B_, x_] := N[(-1.0 / B), $MachinePrecision]

Derivation

1. Initial program 75.0%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
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. sub-neg N/A

      \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, \mathsf{neg}\left(x\right)\right)}}{B} \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

   6. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

   7. associate-+l+ N/A

      \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

   8. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

   9. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

   10. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

   11. +-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

   12. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

   13. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

   14. lower-neg.f64 47.1

       \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{-x}\right)}{B} \]

5. Applied rewrites 47.1%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{B}} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{F}{B}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{F}{B}} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}} \cdot \frac{F}{B} \]

   4. lower-/.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{1}{2 + {F}^{2}}}} \cdot \frac{F}{B} \]

   5. +-commutative N/A

      \[\leadsto \sqrt{\frac{1}{\color{blue}{{F}^{2} + 2}}} \cdot \frac{F}{B} \]

   6. unpow2 N/A

      \[\leadsto \sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}} \cdot \frac{F}{B} \]

   7. lower-fma.f64 N/A

      \[\leadsto \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2\right)}}} \cdot \frac{F}{B} \]

   8. lower-/.f64 14.7

      \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \color{blue}{\frac{F}{B}} \]

8. Applied rewrites 14.7%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{B}} \]

9. Taylor expanded in F around -inf

   \[\leadsto \color{blue}{\frac{-1}{B}} \]
10. Step-by-step derivation

    1. lower-/.f64 11.0

       \[\leadsto \color{blue}{\frac{-1}{B}} \]

11. Applied rewrites 11.0%

    \[\leadsto \color{blue}{\frac{-1}{B}} \]
12. Add Preprocessing

Alternative 28: 3.0% accurate, 61.3× speedup

Math

\[\begin{array}{l} \\ B \cdot -0.16666666666666666 \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (* B -0.16666666666666666))

C

double code(double F, double B, double x) {
	return B * -0.16666666666666666;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = b * (-0.16666666666666666d0)
end function

Java

public static double code(double F, double B, double x) {
	return B * -0.16666666666666666;
}

Python

def code(F, B, x):
	return B * -0.16666666666666666

Julia

function code(F, B, x)
	return Float64(B * -0.16666666666666666)
end

MATLAB

function tmp = code(F, B, x)
	tmp = B * -0.16666666666666666;
end

Wolfram

code[F_, B_, x_] := N[(B * -0.16666666666666666), $MachinePrecision]

Derivation

1. Initial program 75.0%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
2. Add Preprocessing

3. Taylor expanded in B around 0

   \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{\frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B}} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)}}{B} \]

   2. *-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{\left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}}}{B} \]

   3. associate-*l* N/A

      \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{\left({B}^{2} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)} \cdot \frac{1}{6}}{B} \]

   4. associate-*r* N/A

      \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{{B}^{2} \cdot \left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}\right)}}{B} \]

   5. *-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}}{B} \]

   6. lower-/.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}{B}} \]

5. Applied rewrites 54.3%

   \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \mathsf{fma}\left(0.16666666666666666, \left(B \cdot B\right) \cdot F, F\right)}{B}} \]

6. Taylor expanded in B around inf

   \[\leadsto \color{blue}{\frac{1}{6} \cdot \left(\left(B \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)} \]
7. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot \left(B \cdot F\right)\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot \left(B \cdot F\right)\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot \left(B \cdot F\right)\right)} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \]

   4. *-commutative N/A

      \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\left(F \cdot B\right)}\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \]

   5. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\left(F \cdot B\right)}\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(F \cdot B\right)\right) \cdot \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

   7. lower-/.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(F \cdot B\right)\right) \cdot \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

   8. lower-+.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(F \cdot B\right)\right) \cdot \sqrt{\frac{1}{\color{blue}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

   9. +-commutative N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(F \cdot B\right)\right) \cdot \sqrt{\frac{1}{2 + \color{blue}{\left({F}^{2} + 2 \cdot x\right)}}} \]

   10. unpow2 N/A

       \[\leadsto \left(\frac{1}{6} \cdot \left(F \cdot B\right)\right) \cdot \sqrt{\frac{1}{2 + \left(\color{blue}{F \cdot F} + 2 \cdot x\right)}} \]

   11. lower-fma.f64 N/A

       \[\leadsto \left(\frac{1}{6} \cdot \left(F \cdot B\right)\right) \cdot \sqrt{\frac{1}{2 + \color{blue}{\mathsf{fma}\left(F, F, 2 \cdot x\right)}}} \]

   12. lower-*.f64 2.6

       \[\leadsto \left(0.16666666666666666 \cdot \left(F \cdot B\right)\right) \cdot \sqrt{\frac{1}{2 + \mathsf{fma}\left(F, F, \color{blue}{2 \cdot x}\right)}} \]

8. Applied rewrites 2.6%

   \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot \left(F \cdot B\right)\right) \cdot \sqrt{\frac{1}{2 + \mathsf{fma}\left(F, F, 2 \cdot x\right)}}} \]

9. Taylor expanded in F around -inf

   \[\leadsto \color{blue}{\frac{-1}{6} \cdot B} \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \color{blue}{B \cdot \frac{-1}{6}} \]

    2. lower-*.f64 2.9

       \[\leadsto \color{blue}{B \cdot -0.16666666666666666} \]

11. Applied rewrites 2.9%

    \[\leadsto \color{blue}{B \cdot -0.16666666666666666} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(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))))))