VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.5% → 99.7%

Time: 13.8s

Alternatives: 28

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.5% 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, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -7.6 \cdot 10^{+34}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 100000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, F, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;{\sin B}^{-1} - \frac{\cos B \cdot x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -7.6e+34)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 100000000.0)
     (fma (/ (pow (fma 2.0 x (fma F F 2.0)) -0.5) (sin B)) F (/ (- x) (tan B)))
     (- (pow (sin B) -1.0) (/ (* (cos B) x) (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -7.6e+34) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 100000000.0) {
		tmp = fma((pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) / sin(B)), F, (-x / tan(B)));
	} else {
		tmp = pow(sin(B), -1.0) - ((cos(B) * x) / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -7.6e+34)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 100000000.0)
		tmp = fma(Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) / sin(B)), F, Float64(Float64(-x) / tan(B)));
	else
		tmp = Float64((sin(B) ^ -1.0) - Float64(Float64(cos(B) * x) / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -7.6e+34], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 100000000.0], N[(N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] * F + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[Sin[B], $MachinePrecision], -1.0], $MachinePrecision] - N[(N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -7.6000000000000003e34

   1. Initial program 65.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. associate-*r* N/A

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

      9. div-inv N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 76.7%

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

   6. Applied rewrites 76.8%

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

   7. Taylor expanded in F around -inf

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

   9. Applied rewrites 99.7%

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

   if -7.6000000000000003e34 < F < 1e8

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. associate-*r* N/A

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

      9. div-inv N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 99.7%

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

   if 1e8 < F

   1. Initial program 55.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. associate-*r* N/A

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

      9. div-inv N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 72.2%

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

   5. Taylor expanded in F around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-sin.f64 99.9

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

   7. Applied rewrites 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7.6 \cdot 10^{+34}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 100000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, F, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;{\sin B}^{-1} - \frac{\cos B \cdot x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{+43}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 135000000:\\ \;\;\;\;\mathsf{fma}\left({\left(\sin B \cdot \sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{-1}, F, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;{\sin B}^{-1} - \frac{\cos B \cdot x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -5e+43)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 135000000.0)
     (fma
      (pow (* (sin B) (sqrt (fma x 2.0 (fma F F 2.0)))) -1.0)
      F
      (/ (- x) (tan B)))
     (- (pow (sin B) -1.0) (/ (* (cos B) x) (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -5e+43) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 135000000.0) {
		tmp = fma(pow((sin(B) * sqrt(fma(x, 2.0, fma(F, F, 2.0)))), -1.0), F, (-x / tan(B)));
	} else {
		tmp = pow(sin(B), -1.0) - ((cos(B) * x) / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -5e+43)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 135000000.0)
		tmp = fma((Float64(sin(B) * sqrt(fma(x, 2.0, fma(F, F, 2.0)))) ^ -1.0), F, Float64(Float64(-x) / tan(B)));
	else
		tmp = Float64((sin(B) ^ -1.0) - Float64(Float64(cos(B) * x) / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -5e+43], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 135000000.0], N[(N[Power[N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * F + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[Sin[B], $MachinePrecision], -1.0], $MachinePrecision] - N[(N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.0000000000000004e43

   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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. associate-*r* N/A

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

      9. div-inv N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 75.7%

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

   6. Applied rewrites 75.8%

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

   7. Taylor expanded in F around -inf

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

   9. Applied rewrites 99.7%

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

   if -5.0000000000000004e43 < F < 1.35e8

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. associate-*r* N/A

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

      9. div-inv N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. div-inv N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-flip N/A

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

      7. metadata-eval N/A

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

      8. pow1/2 N/A

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

      9. lift-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-fma.f64 N/A

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

      12. associate-+l+ N/A

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

      13. +-commutative N/A

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

      14. lift-fma.f64 N/A

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

      15. lift-fma.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lift-fma.f64 N/A

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

      18. lift-fma.f64 N/A

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

      19. +-commutative N/A

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

      20. associate-+l+ N/A

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

      21. lift-fma.f64 N/A

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

      22. +-commutative N/A

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

      23. lift-fma.f64 N/A

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

   6. Applied rewrites 99.6%

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

   if 1.35e8 < F

   1. Initial program 55.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. associate-*r* N/A

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

      9. div-inv N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 72.2%

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

   5. Taylor expanded in F around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-sin.f64 99.9

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

   7. Applied rewrites 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{+43}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 135000000:\\ \;\;\;\;\mathsf{fma}\left({\left(\sin B \cdot \sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{-1}, F, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;{\sin B}^{-1} - \frac{\cos B \cdot x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{+43}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 100000000:\\ \;\;\;\;\mathsf{fma}\left({\left(\sin B \cdot \sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{-1}, F, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;{\sin B}^{-1} - \cos B \cdot \frac{x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -5e+43)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 100000000.0)
     (fma
      (pow (* (sin B) (sqrt (fma x 2.0 (fma F F 2.0)))) -1.0)
      F
      (/ (- x) (tan B)))
     (- (pow (sin B) -1.0) (* (cos B) (/ x (sin B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -5e+43) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 100000000.0) {
		tmp = fma(pow((sin(B) * sqrt(fma(x, 2.0, fma(F, F, 2.0)))), -1.0), F, (-x / tan(B)));
	} else {
		tmp = pow(sin(B), -1.0) - (cos(B) * (x / sin(B)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -5e+43)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 100000000.0)
		tmp = fma((Float64(sin(B) * sqrt(fma(x, 2.0, fma(F, F, 2.0)))) ^ -1.0), F, Float64(Float64(-x) / tan(B)));
	else
		tmp = Float64((sin(B) ^ -1.0) - Float64(cos(B) * Float64(x / sin(B))));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -5e+43], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 100000000.0], N[(N[Power[N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * F + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[Sin[B], $MachinePrecision], -1.0], $MachinePrecision] - N[(N[Cos[B], $MachinePrecision] * N[(x / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.0000000000000004e43

   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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. associate-*r* N/A

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

      9. div-inv N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 75.7%

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

   6. Applied rewrites 75.8%

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

   7. Taylor expanded in F around -inf

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

   9. Applied rewrites 99.7%

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

   if -5.0000000000000004e43 < F < 1e8

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. associate-*r* N/A

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

      9. div-inv N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. div-inv N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-flip N/A

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

      7. metadata-eval N/A

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

      8. pow1/2 N/A

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

      9. lift-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-fma.f64 N/A

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

      12. associate-+l+ N/A

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

      13. +-commutative N/A

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

      14. lift-fma.f64 N/A

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

      15. lift-fma.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lift-fma.f64 N/A

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

      18. lift-fma.f64 N/A

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

      19. +-commutative N/A

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

      20. associate-+l+ N/A

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

      21. lift-fma.f64 N/A

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

      22. +-commutative N/A

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

      23. lift-fma.f64 N/A

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

   6. Applied rewrites 99.6%

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

   if 1e8 < F

   1. Initial program 55.9%

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

   3. Taylor expanded in F around inf

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sin.f64 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{+43}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 100000000:\\ \;\;\;\;\mathsf{fma}\left({\left(\sin B \cdot \sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{-1}, F, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;{\sin B}^{-1} - \cos B \cdot \frac{x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -5 \cdot 10^{+43}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 135000000:\\ \;\;\;\;\mathsf{fma}\left({\left(\sin B \cdot \sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{-1}, F, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -5e+43)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 135000000.0)
       (fma
        (pow (* (sin B) (sqrt (fma x 2.0 (fma F F 2.0)))) -1.0)
        F
        (/ (- x) (tan B)))
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -5e+43) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 135000000.0) {
		tmp = fma(pow((sin(B) * sqrt(fma(x, 2.0, fma(F, F, 2.0)))), -1.0), F, (-x / tan(B)));
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -5e+43)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 135000000.0)
		tmp = fma((Float64(sin(B) * sqrt(fma(x, 2.0, fma(F, F, 2.0)))) ^ -1.0), F, Float64(Float64(-x) / tan(B)));
	else
		tmp = Float64(Float64(1.0 / sin(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, -5e+43], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 135000000.0], N[(N[Power[N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * F + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.0000000000000004e43

   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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. associate-*r* N/A

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

      9. div-inv N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 75.7%

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

   6. Applied rewrites 75.8%

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

   7. Taylor expanded in F around -inf

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

   9. Applied rewrites 99.7%

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

   if -5.0000000000000004e43 < F < 1.35e8

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. associate-*r* N/A

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

      9. div-inv N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. div-inv N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-flip N/A

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

      7. metadata-eval N/A

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

      8. pow1/2 N/A

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

      9. lift-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-fma.f64 N/A

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

      12. associate-+l+ N/A

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

      13. +-commutative N/A

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

      14. lift-fma.f64 N/A

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

      15. lift-fma.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lift-fma.f64 N/A

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

      18. lift-fma.f64 N/A

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

      19. +-commutative N/A

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

      20. associate-+l+ N/A

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

      21. lift-fma.f64 N/A

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

      22. +-commutative N/A

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

      23. lift-fma.f64 N/A

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

   6. Applied rewrites 99.6%

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

   if 1.35e8 < F

   1. Initial program 55.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. associate-*r* N/A

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

      9. div-inv N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 72.2%

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

   6. Applied rewrites 72.2%

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

   7. Taylor expanded in F around inf

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

   9. Applied rewrites 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{+43}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 135000000:\\ \;\;\;\;\mathsf{fma}\left({\left(\sin B \cdot \sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{-1}, F, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -7.6 \cdot 10^{+34}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1000:\\ \;\;\;\;\frac{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot F}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -7.6e+34)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1000.0)
       (- (/ (* (sqrt (pow (fma F F 2.0) -1.0)) F) (sin B)) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -7.6e+34) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1000.0) {
		tmp = ((sqrt(pow(fma(F, F, 2.0), -1.0)) * F) / sin(B)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -7.6e+34)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1000.0)
		tmp = Float64(Float64(Float64(sqrt((fma(F, F, 2.0) ^ -1.0)) * F) / sin(B)) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(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, -7.6e+34], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1000.0], N[(N[(N[(N[Sqrt[N[Power[N[(F * F + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -7.6000000000000003e34

   1. Initial program 65.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. associate-*r* N/A

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

      9. div-inv N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 76.7%

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

   6. Applied rewrites 76.8%

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

   7. Taylor expanded in F around -inf

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

   9. Applied rewrites 99.7%

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

   if -7.6000000000000003e34 < F < 1e3

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. associate-*r* N/A

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

      9. div-inv N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 99.2

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

   9. Applied rewrites 99.2%

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

   if 1e3 < F

   1. Initial program 55.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. associate-*r* N/A

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

      9. div-inv N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 72.2%

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

   6. Applied rewrites 72.2%

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

   7. Taylor expanded in F around inf

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

   9. Applied rewrites 99.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7.6 \cdot 10^{+34}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1000:\\ \;\;\;\;\frac{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot F}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{F}{\sin B}}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - \frac{x}{B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3.3 \cdot 10^{+28}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_1\\ \mathbf{elif}\;F \leq -2.1 \cdot 10^{-155}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 7 \cdot 10^{-127}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}}}{B}, F, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;F \leq 200000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ (/ F (sin B)) (sqrt (fma 2.0 x (fma F F 2.0)))) (/ x B)))
        (t_1 (/ x (tan B))))
   (if (<= F -3.3e+28)
     (- (/ -1.0 (sin B)) t_1)
     (if (<= F -2.1e-155)
       t_0
       (if (<= F 7e-127)
         (fma
          (/ (sqrt (pow (fma F F (fma 2.0 x 2.0)) -1.0)) B)
          F
          (/ (- x) (tan B)))
         (if (<= F 200000.0) t_0 (- (/ 1.0 (sin B)) t_1)))))))

C

double code(double F, double B, double x) {
	double t_0 = ((F / sin(B)) / sqrt(fma(2.0, x, fma(F, F, 2.0)))) - (x / B);
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -3.3e+28) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= -2.1e-155) {
		tmp = t_0;
	} else if (F <= 7e-127) {
		tmp = fma((sqrt(pow(fma(F, F, fma(2.0, x, 2.0)), -1.0)) / B), F, (-x / tan(B)));
	} else if (F <= 200000.0) {
		tmp = t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_1;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(Float64(F / sin(B)) / sqrt(fma(2.0, x, fma(F, F, 2.0)))) - Float64(x / B))
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -3.3e+28)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= -2.1e-155)
		tmp = t_0;
	elseif (F <= 7e-127)
		tmp = fma(Float64(sqrt((fma(F, F, fma(2.0, x, 2.0)) ^ -1.0)) / B), F, Float64(Float64(-x) / tan(B)));
	elseif (F <= 200000.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_1);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -3.3e+28], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, -2.1e-155], t$95$0, If[LessEqual[F, 7e-127], N[(N[(N[Sqrt[N[Power[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision] * F + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 200000.0], t$95$0, N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.3e28

   1. Initial program 66.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. associate-*r* N/A

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

      9. div-inv N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 77.3%

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

   6. Applied rewrites 77.4%

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

   7. Taylor expanded in F around -inf

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

   9. Applied rewrites 99.7%

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

   if -3.3e28 < F < -2.1000000000000002e-155 or 6.99999999999999979e-127 < F < 2e5

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-neg.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. lower-/.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 89.9

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

   7. Applied rewrites 89.9%

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-neg.f64 N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

   9. Applied rewrites 90.0%

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

   if -2.1000000000000002e-155 < F < 6.99999999999999979e-127

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. associate-*r* N/A

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

      9. div-inv N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in B around 0

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. unpow2 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 94.9

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

   7. Applied rewrites 94.9%

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

   if 2e5 < F

   1. Initial program 55.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. associate-*r* N/A

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

      9. div-inv N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 72.2%

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

   6. Applied rewrites 72.2%

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

   7. Taylor expanded in F around inf

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

   9. Applied rewrites 99.8%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.3 \cdot 10^{+28}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -2.1 \cdot 10^{-155}:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 7 \cdot 10^{-127}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}}}{B}, F, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;F \leq 200000:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{F}{\sin B}}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - \frac{x}{B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3.3 \cdot 10^{+28}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_1\\ \mathbf{elif}\;F \leq -2.1 \cdot 10^{-155}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 7 \cdot 10^{-127}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}}}{B}, F, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;F \leq 1.5 \cdot 10^{+154}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ (/ F (sin B)) (sqrt (fma 2.0 x (fma F F 2.0)))) (/ x B)))
        (t_1 (/ x (tan B))))
   (if (<= F -3.3e+28)
     (- (/ -1.0 (sin B)) t_1)
     (if (<= F -2.1e-155)
       t_0
       (if (<= F 7e-127)
         (fma
          (/ (sqrt (pow (fma F F (fma 2.0 x 2.0)) -1.0)) B)
          F
          (/ (- x) (tan B)))
         (if (<= F 1.5e+154)
           t_0
           (- (/ (fma (* B B) 0.16666666666666666 1.0) B) t_1)))))))

C

double code(double F, double B, double x) {
	double t_0 = ((F / sin(B)) / sqrt(fma(2.0, x, fma(F, F, 2.0)))) - (x / B);
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -3.3e+28) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= -2.1e-155) {
		tmp = t_0;
	} else if (F <= 7e-127) {
		tmp = fma((sqrt(pow(fma(F, F, fma(2.0, x, 2.0)), -1.0)) / B), F, (-x / tan(B)));
	} else if (F <= 1.5e+154) {
		tmp = t_0;
	} else {
		tmp = (fma((B * B), 0.16666666666666666, 1.0) / B) - t_1;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(Float64(F / sin(B)) / sqrt(fma(2.0, x, fma(F, F, 2.0)))) - Float64(x / B))
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -3.3e+28)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= -2.1e-155)
		tmp = t_0;
	elseif (F <= 7e-127)
		tmp = fma(Float64(sqrt((fma(F, F, fma(2.0, x, 2.0)) ^ -1.0)) / B), F, Float64(Float64(-x) / tan(B)));
	elseif (F <= 1.5e+154)
		tmp = t_0;
	else
		tmp = Float64(Float64(fma(Float64(B * B), 0.16666666666666666, 1.0) / B) - t_1);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -3.3e+28], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, -2.1e-155], t$95$0, If[LessEqual[F, 7e-127], N[(N[(N[Sqrt[N[Power[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision] * F + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.5e+154], t$95$0, N[(N[(N[(N[(B * B), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] / B), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.3e28

   1. Initial program 66.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. associate-*r* N/A

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

      9. div-inv N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 77.3%

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

   6. Applied rewrites 77.4%

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

   7. Taylor expanded in F around -inf

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

   9. Applied rewrites 99.7%

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

   if -3.3e28 < F < -2.1000000000000002e-155 or 6.99999999999999979e-127 < F < 1.50000000000000013e154

   1. Initial program 98.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-neg.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. lower-/.f64 98.4

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

   4. Applied rewrites 98.4%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 87.1

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

   7. Applied rewrites 87.1%

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-neg.f64 N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

   9. Applied rewrites 87.2%

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

   if -2.1000000000000002e-155 < F < 6.99999999999999979e-127

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. associate-*r* N/A

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

      9. div-inv N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in B around 0

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. unpow2 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 94.9

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

   7. Applied rewrites 94.9%

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

   if 1.50000000000000013e154 < F

   1. Initial program 25.9%

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

   3. Taylor expanded in B around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 28.9%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 69.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. div-inv N/A

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

      7. lift-/.f64 N/A

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

   10. Applied rewrites 69.7%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} - \frac{x}{\tan B}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.3 \cdot 10^{+28}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -2.1 \cdot 10^{-155}:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 7 \cdot 10^{-127}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}}}{B}, F, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;F \leq 1.5 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ t_1 := \frac{\frac{F}{\sin B}}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - \frac{x}{B}\\ \mathbf{if}\;F \leq -4.8 \cdot 10^{+82}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, B \cdot B, -0.16666666666666666\right), B \cdot B, 1\right) \cdot B} - t\_0\\ \mathbf{elif}\;F \leq -2.1 \cdot 10^{-155}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;F \leq 7 \cdot 10^{-127}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}}}{B}, F, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;F \leq 1.5 \cdot 10^{+154}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B)))
        (t_1 (- (/ (/ F (sin B)) (sqrt (fma 2.0 x (fma F F 2.0)))) (/ x B))))
   (if (<= F -4.8e+82)
     (-
      (/
       -1.0
       (*
        (fma
         (fma 0.008333333333333333 (* B B) -0.16666666666666666)
         (* B B)
         1.0)
        B))
      t_0)
     (if (<= F -2.1e-155)
       t_1
       (if (<= F 7e-127)
         (fma
          (/ (sqrt (pow (fma F F (fma 2.0 x 2.0)) -1.0)) B)
          F
          (/ (- x) (tan B)))
         (if (<= F 1.5e+154)
           t_1
           (- (/ (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 t_1 = ((F / sin(B)) / sqrt(fma(2.0, x, fma(F, F, 2.0)))) - (x / B);
	double tmp;
	if (F <= -4.8e+82) {
		tmp = (-1.0 / (fma(fma(0.008333333333333333, (B * B), -0.16666666666666666), (B * B), 1.0) * B)) - t_0;
	} else if (F <= -2.1e-155) {
		tmp = t_1;
	} else if (F <= 7e-127) {
		tmp = fma((sqrt(pow(fma(F, F, fma(2.0, x, 2.0)), -1.0)) / B), F, (-x / tan(B)));
	} else if (F <= 1.5e+154) {
		tmp = t_1;
	} 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))
	t_1 = Float64(Float64(Float64(F / sin(B)) / sqrt(fma(2.0, x, fma(F, F, 2.0)))) - Float64(x / B))
	tmp = 0.0
	if (F <= -4.8e+82)
		tmp = Float64(Float64(-1.0 / Float64(fma(fma(0.008333333333333333, Float64(B * B), -0.16666666666666666), Float64(B * B), 1.0) * B)) - t_0);
	elseif (F <= -2.1e-155)
		tmp = t_1;
	elseif (F <= 7e-127)
		tmp = fma(Float64(sqrt((fma(F, F, fma(2.0, x, 2.0)) ^ -1.0)) / B), F, Float64(Float64(-x) / tan(B)));
	elseif (F <= 1.5e+154)
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -4.8e+82], N[(N[(-1.0 / N[(N[(N[(0.008333333333333333 * N[(B * B), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -2.1e-155], t$95$1, If[LessEqual[F, 7e-127], N[(N[(N[Sqrt[N[Power[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision] * F + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.5e+154], t$95$1, N[(N[(N[(N[(B * B), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] / B), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -4.79999999999999996e82

   1. Initial program 56.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. associate-*r* N/A

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

      9. div-inv N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 70.6%

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

   6. Applied rewrites 70.6%

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

   7. Taylor expanded in F around -inf

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

   9. Applied rewrites 99.7%

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

   10. Taylor expanded in B around 0

       \[\leadsto \frac{-1}{\color{blue}{B \cdot \left(1 + {B}^{2} \cdot \left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right)\right)}} - \frac{x}{\tan B} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{-1}{\color{blue}{\left(1 + {B}^{2} \cdot \left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right)\right) \cdot B}} - \frac{x}{\tan B} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{-1}{\color{blue}{\left(1 + {B}^{2} \cdot \left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right)\right) \cdot B}} - \frac{x}{\tan B} \]

       3. +-commutative N/A

          \[\leadsto \frac{-1}{\color{blue}{\left({B}^{2} \cdot \left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right) + 1\right)} \cdot B} - \frac{x}{\tan B} \]

       4. *-commutative N/A

          \[\leadsto \frac{-1}{\left(\color{blue}{\left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right) \cdot {B}^{2}} + 1\right) \cdot B} - \frac{x}{\tan B} \]

       5. lower-fma.f64 N/A

          \[\leadsto \frac{-1}{\color{blue}{\mathsf{fma}\left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}, {B}^{2}, 1\right)} \cdot B} - \frac{x}{\tan B} \]

       6. sub-neg N/A

          \[\leadsto \frac{-1}{\mathsf{fma}\left(\color{blue}{\frac{1}{120} \cdot {B}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, {B}^{2}, 1\right) \cdot B} - \frac{x}{\tan B} \]

       7. metadata-eval N/A

          \[\leadsto \frac{-1}{\mathsf{fma}\left(\frac{1}{120} \cdot {B}^{2} + \color{blue}{\frac{-1}{6}}, {B}^{2}, 1\right) \cdot B} - \frac{x}{\tan B} \]

       8. lower-fma.f64 N/A

          \[\leadsto \frac{-1}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{120}, {B}^{2}, \frac{-1}{6}\right)}, {B}^{2}, 1\right) \cdot B} - \frac{x}{\tan B} \]

       9. unpow2 N/A

          \[\leadsto \frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, \color{blue}{B \cdot B}, \frac{-1}{6}\right), {B}^{2}, 1\right) \cdot B} - \frac{x}{\tan B} \]

       10. lower-*.f64 N/A

           \[\leadsto \frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, \color{blue}{B \cdot B}, \frac{-1}{6}\right), {B}^{2}, 1\right) \cdot B} - \frac{x}{\tan B} \]

       11. unpow2 N/A

           \[\leadsto \frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120}, B \cdot B, \frac{-1}{6}\right), \color{blue}{B \cdot B}, 1\right) \cdot B} - \frac{x}{\tan B} \]

       12. lower-*.f64 78.3

           \[\leadsto \frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, B \cdot B, -0.16666666666666666\right), \color{blue}{B \cdot B}, 1\right) \cdot B} - \frac{x}{\tan B} \]

   12. Applied rewrites 78.3%

       \[\leadsto \frac{-1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, B \cdot B, -0.16666666666666666\right), B \cdot B, 1\right) \cdot B}} - \frac{x}{\tan B} \]

   if -4.79999999999999996e82 < F < -2.1000000000000002e-155 or 6.99999999999999979e-127 < F < 1.50000000000000013e154

   1. Initial program 98.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-neg.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. lower-/.f64 98.6

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 85.8

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

   7. Applied rewrites 85.8%

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-neg.f64 N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

   9. Applied rewrites 85.9%

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

   if -2.1000000000000002e-155 < F < 6.99999999999999979e-127

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. associate-*r* N/A

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

      9. div-inv N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in B around 0

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. unpow2 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 94.9

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

   7. Applied rewrites 94.9%

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

   if 1.50000000000000013e154 < F

   1. Initial program 25.9%

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

   3. Taylor expanded in B around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 28.9%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 69.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. div-inv N/A

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

      7. lift-/.f64 N/A

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

   10. Applied rewrites 69.7%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} - \frac{x}{\tan B}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.8 \cdot 10^{+82}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, B \cdot B, -0.16666666666666666\right), B \cdot B, 1\right) \cdot B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -2.1 \cdot 10^{-155}:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 7 \cdot 10^{-127}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}}}{B}, F, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;F \leq 1.5 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.42:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 2.05:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, 2\right)}}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.42)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 2.05)
       (- (/ (/ F (sqrt (fma 2.0 x 2.0))) (sin B)) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.42) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 2.05) {
		tmp = ((F / sqrt(fma(2.0, x, 2.0))) / sin(B)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.42)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 2.05)
		tmp = Float64(Float64(Float64(F / sqrt(fma(2.0, x, 2.0))) / sin(B)) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(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, -1.42], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 2.05], N[(N[(N[(F / N[Sqrt[N[(2.0 * x + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.4199999999999999

   1. Initial program 69.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. associate-*r* N/A

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

      9. div-inv N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 79.0%

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

   6. Applied rewrites 79.1%

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

   7. Taylor expanded in F around -inf

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

   9. Applied rewrites 98.6%

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

   if -1.4199999999999999 < F < 2.0499999999999998

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. associate-*r* N/A

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

      9. div-inv N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in F around 0

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

      1. lower-sqrt.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 99.6

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

   9. Applied rewrites 99.6%

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

   if 2.0499999999999998 < F

   1. Initial program 58.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. associate-*r* N/A

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

      9. div-inv N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 73.6%

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

   6. Applied rewrites 73.6%

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

   7. Taylor expanded in F around inf

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

   9. Applied rewrites 97.9%

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

Alternative 10: 99.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.42:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;\frac{\sqrt{0.5} \cdot F}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.42)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.4)
       (- (/ (* (sqrt 0.5) F) (sin B)) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.42) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.4) {
		tmp = ((sqrt(0.5) * F) / sin(B)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.42d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.4d0) then
        tmp = ((sqrt(0.5d0) * f) / sin(b)) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.42) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.4) {
		tmp = ((Math.sqrt(0.5) * F) / Math.sin(B)) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.42:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.4:
		tmp = ((math.sqrt(0.5) * F) / math.sin(B)) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.42)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.4)
		tmp = Float64(Float64(Float64(sqrt(0.5) * F) / sin(B)) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.42)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.4)
		tmp = ((sqrt(0.5) * F) / sin(B)) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.42], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.4], N[(N[(N[(N[Sqrt[0.5], $MachinePrecision] * F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.4199999999999999

   1. Initial program 69.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. associate-*r* N/A

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

      9. div-inv N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 79.0%

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

   6. Applied rewrites 79.1%

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

   7. Taylor expanded in F around -inf

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

   9. Applied rewrites 98.6%

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

   if -1.4199999999999999 < F < 1.3999999999999999

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. associate-*r* N/A

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

      9. div-inv N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 99.2

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

   9. Applied rewrites 99.2%

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

   10. Taylor expanded in F around 0

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

   12. Applied rewrites 99.2%

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

   if 1.3999999999999999 < F

   1. Initial program 58.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. associate-*r* N/A

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

      9. div-inv N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 73.6%

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

   6. Applied rewrites 73.6%

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

   7. Taylor expanded in F around inf

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

   9. Applied rewrites 97.9%

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

Alternative 11: 69.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{\mathsf{fma}\left(-0.16666666666666666 \cdot B, B, 1\right) \cdot B} - \frac{x}{\tan B}\\ \mathbf{if}\;x \leq -2.7 \cdot 10^{-10}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-154}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-88}:\\ \;\;\;\;\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot \frac{F}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0
         (-
          (/ -1.0 (* (fma (* -0.16666666666666666 B) B 1.0) B))
          (/ x (tan B)))))
   (if (<= x -2.7e-10)
     t_0
     (if (<= x -3.5e-154)
       (/ (- (/ F (sqrt (fma x 2.0 (fma F F 2.0)))) x) B)
       (if (<= x 5.8e-88)
         (* (sqrt (pow (fma F F 2.0) -1.0)) (/ F (sin B)))
         t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 / (fma((-0.16666666666666666 * B), B, 1.0) * B)) - (x / tan(B));
	double tmp;
	if (x <= -2.7e-10) {
		tmp = t_0;
	} else if (x <= -3.5e-154) {
		tmp = ((F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - x) / B;
	} else if (x <= 5.8e-88) {
		tmp = sqrt(pow(fma(F, F, 2.0), -1.0)) * (F / sin(B));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 / Float64(fma(Float64(-0.16666666666666666 * B), B, 1.0) * B)) - Float64(x / tan(B)))
	tmp = 0.0
	if (x <= -2.7e-10)
		tmp = t_0;
	elseif (x <= -3.5e-154)
		tmp = Float64(Float64(Float64(F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - x) / B);
	elseif (x <= 5.8e-88)
		tmp = Float64(sqrt((fma(F, F, 2.0) ^ -1.0)) * Float64(F / sin(B)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(-1.0 / N[(N[(N[(-0.16666666666666666 * B), $MachinePrecision] * B + 1.0), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.7e-10], t$95$0, If[LessEqual[x, -3.5e-154], N[(N[(N[(F / N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[x, 5.8e-88], N[(N[Sqrt[N[Power[N[(F * F + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.7e-10 or 5.8000000000000003e-88 < x

   1. Initial program 86.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. associate-*r* N/A

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

      9. div-inv N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 95.2%

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

   6. Applied rewrites 95.2%

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

   7. Taylor expanded in F around -inf

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

   9. Applied rewrites 89.9%

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

   10. Taylor expanded in B around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. unpow2 N/A

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

       5. associate-*r* N/A

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

       6. lower-fma.f64 N/A

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

       7. lower-*.f64 92.8

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

   12. Applied rewrites 92.8%

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

   if -2.7e-10 < x < -3.5000000000000001e-154

   1. Initial program 75.4%

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

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 54.0

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

   5. Applied rewrites 54.0%

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

   7. Applied rewrites 54.1%

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

   if -3.5000000000000001e-154 < x < 5.8000000000000003e-88

   1. Initial program 75.4%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sin.f64 61.4

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

   5. Applied rewrites 61.4%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-10}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(-0.16666666666666666 \cdot B, B, 1\right) \cdot B} - \frac{x}{\tan B}\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-154}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-88}:\\ \;\;\;\;\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot \frac{F}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(-0.16666666666666666 \cdot B, B, 1\right) \cdot B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 52.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -4 \cdot 10^{-13}:\\ \;\;\;\;\left(1 + x\right) \cdot \left({\left(\left(F \cdot F\right) \cdot B\right)}^{-1} - {B}^{-1}\right)\\ \mathbf{elif}\;F \leq 10^{+45}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4e-13)
   (* (+ 1.0 x) (- (pow (* (* F F) B) -1.0) (pow B -1.0)))
   (if (<= F 1e+45)
     (/ (- (/ F (sqrt (fma x 2.0 (fma F F 2.0)))) x) B)
     (+ (- (/ x B)) (/ (fma (* B B) 0.16666666666666666 1.0) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4e-13) {
		tmp = (1.0 + x) * (pow(((F * F) * B), -1.0) - pow(B, -1.0));
	} else if (F <= 1e+45) {
		tmp = ((F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - x) / B;
	} else {
		tmp = -(x / B) + (fma((B * B), 0.16666666666666666, 1.0) / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -4e-13)
		tmp = Float64(Float64(1.0 + x) * Float64((Float64(Float64(F * F) * B) ^ -1.0) - (B ^ -1.0)));
	elseif (F <= 1e+45)
		tmp = Float64(Float64(Float64(F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - x) / B);
	else
		tmp = Float64(Float64(-Float64(x / B)) + Float64(fma(Float64(B * B), 0.16666666666666666, 1.0) / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -4e-13], N[(N[(1.0 + x), $MachinePrecision] * N[(N[Power[N[(N[(F * F), $MachinePrecision] * B), $MachinePrecision], -1.0], $MachinePrecision] - N[Power[B, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1e+45], N[(N[(N[(F / N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[((-N[(x / B), $MachinePrecision]) + N[(N[(N[(B * B), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.0000000000000001e-13

   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

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   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. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 33.1

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

   5. Applied rewrites 33.1%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 40.3%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 45.1%

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

   if -4.0000000000000001e-13 < F < 9.9999999999999993e44

   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. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 47.7

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

   5. Applied rewrites 47.7%

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

   7. Applied rewrites 47.7%

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

   if 9.9999999999999993e44 < F

   1. Initial program 50.7%

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

   3. Taylor expanded in B around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 36.9%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 63.1%

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

   9. Taylor expanded in B around 0

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

       1. lower-/.f64 56.2

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

   11. Applied rewrites 56.2%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4 \cdot 10^{-13}:\\ \;\;\;\;\left(1 + x\right) \cdot \left({\left(\left(F \cdot F\right) \cdot B\right)}^{-1} - {B}^{-1}\right)\\ \mathbf{elif}\;F \leq 10^{+45}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 52.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -4 \cdot 10^{-13}:\\ \;\;\;\;\left(1 + x\right) \cdot \left({\left(\left(F \cdot F\right) \cdot B\right)}^{-1} - {B}^{-1}\right)\\ \mathbf{elif}\;F \leq 7.2 \cdot 10^{+44}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}}, F, -x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4e-13)
   (* (+ 1.0 x) (- (pow (* (* F F) B) -1.0) (pow B -1.0)))
   (if (<= F 7.2e+44)
     (/ (fma (sqrt (pow (fma F F 2.0) -1.0)) F (- x)) B)
     (+ (- (/ x B)) (/ (fma (* B B) 0.16666666666666666 1.0) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4e-13) {
		tmp = (1.0 + x) * (pow(((F * F) * B), -1.0) - pow(B, -1.0));
	} else if (F <= 7.2e+44) {
		tmp = fma(sqrt(pow(fma(F, F, 2.0), -1.0)), F, -x) / B;
	} else {
		tmp = -(x / B) + (fma((B * B), 0.16666666666666666, 1.0) / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -4e-13)
		tmp = Float64(Float64(1.0 + x) * Float64((Float64(Float64(F * F) * B) ^ -1.0) - (B ^ -1.0)));
	elseif (F <= 7.2e+44)
		tmp = Float64(fma(sqrt((fma(F, F, 2.0) ^ -1.0)), F, Float64(-x)) / B);
	else
		tmp = Float64(Float64(-Float64(x / B)) + Float64(fma(Float64(B * B), 0.16666666666666666, 1.0) / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -4e-13], N[(N[(1.0 + x), $MachinePrecision] * N[(N[Power[N[(N[(F * F), $MachinePrecision] * B), $MachinePrecision], -1.0], $MachinePrecision] - N[Power[B, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7.2e+44], N[(N[(N[Sqrt[N[Power[N[(F * F + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * F + (-x)), $MachinePrecision] / B), $MachinePrecision], N[((-N[(x / B), $MachinePrecision]) + N[(N[(N[(B * B), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.0000000000000001e-13

   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

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   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. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 33.1

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

   5. Applied rewrites 33.1%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 40.3%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 45.1%

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

   if -4.0000000000000001e-13 < F < 7.2e44

   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. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-neg.f64 47.7

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 47.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{2 + {F}^{2}}}, F, -x\right)}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 47.7%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}, F, -x\right)}{B} \]

   if 7.2e44 < F

   1. Initial program 50.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)}}{B} \]

      2. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{\left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}}}{B} \]

      3. associate-*l* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{\left({B}^{2} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)} \cdot \frac{1}{6}}{B} \]

      4. associate-*r* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{{B}^{2} \cdot \left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}\right)}}{B} \]

      5. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}{B}} \]

   5. Applied rewrites 36.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(B \cdot B, 0.16666666666666666 \cdot F, F\right)}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1 + \frac{1}{6} \cdot {B}^{2}}{\color{blue}{B}} \]
   7. Step-by-step derivation

   8. Applied rewrites 63.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{\color{blue}{B}} \]

   9. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]
   10. Step-by-step derivation

       1. lower-/.f64 56.2

          \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} \]

   11. Applied rewrites 56.2%

       \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4 \cdot 10^{-13}:\\ \;\;\;\;\left(1 + x\right) \cdot \left({\left(\left(F \cdot F\right) \cdot B\right)}^{-1} - {B}^{-1}\right)\\ \mathbf{elif}\;F \leq 7.2 \cdot 10^{+44}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}}, F, -x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 51.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -4 \cdot 10^{-13}:\\ \;\;\;\;\left(1 + x\right) \cdot \left({\left(\left(F \cdot F\right) \cdot B\right)}^{-1} - {B}^{-1}\right)\\ \mathbf{elif}\;F \leq 70:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-1}}, F, -x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4e-13)
   (* (+ 1.0 x) (- (pow (* (* F F) B) -1.0) (pow B -1.0)))
   (if (<= F 70.0)
     (/ (fma (sqrt (pow (fma 2.0 x 2.0) -1.0)) F (- x)) B)
     (+ (- (/ x B)) (/ (fma (* B B) 0.16666666666666666 1.0) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4e-13) {
		tmp = (1.0 + x) * (pow(((F * F) * B), -1.0) - pow(B, -1.0));
	} else if (F <= 70.0) {
		tmp = fma(sqrt(pow(fma(2.0, x, 2.0), -1.0)), F, -x) / B;
	} else {
		tmp = -(x / B) + (fma((B * B), 0.16666666666666666, 1.0) / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -4e-13)
		tmp = Float64(Float64(1.0 + x) * Float64((Float64(Float64(F * F) * B) ^ -1.0) - (B ^ -1.0)));
	elseif (F <= 70.0)
		tmp = Float64(fma(sqrt((fma(2.0, x, 2.0) ^ -1.0)), F, Float64(-x)) / B);
	else
		tmp = Float64(Float64(-Float64(x / B)) + Float64(fma(Float64(B * B), 0.16666666666666666, 1.0) / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -4e-13], N[(N[(1.0 + x), $MachinePrecision] * N[(N[Power[N[(N[(F * F), $MachinePrecision] * B), $MachinePrecision], -1.0], $MachinePrecision] - N[Power[B, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 70.0], N[(N[(N[Sqrt[N[Power[N[(2.0 * x + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * F + (-x)), $MachinePrecision] / B), $MachinePrecision], N[((-N[(x / B), $MachinePrecision]) + N[(N[(N[(B * B), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.0000000000000001e-13

   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

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   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. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-neg.f64 33.1

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 33.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto -1 \cdot \frac{1 + x}{B} + \color{blue}{\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{B \cdot {F}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 40.3%

      \[\leadsto \mathsf{fma}\left(\frac{0.5}{F \cdot F}, \color{blue}{\frac{\mathsf{fma}\left(2, x, 2\right)}{B}}, \frac{-1 - x}{B}\right) \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(x \cdot \left(\frac{1}{B \cdot {F}^{2}} - \frac{1}{B}\right) + \frac{1}{B \cdot {F}^{2}}\right) - \frac{1}{\color{blue}{B}} \]
   10. Step-by-step derivation

   11. Applied rewrites 45.1%

       \[\leadsto \left(1 + x\right) \cdot \left(\frac{1}{\left(F \cdot F\right) \cdot B} - \color{blue}{\frac{1}{B}}\right) \]

   if -4.0000000000000001e-13 < F < 70

   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. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-neg.f64 47.4

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 47.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{2 + 2 \cdot x}}, F, -x\right)}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 47.4%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}}, F, -x\right)}{B} \]

   if 70 < F

   1. Initial program 57.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)}}{B} \]

      2. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{\left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}}}{B} \]

      3. associate-*l* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{\left({B}^{2} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)} \cdot \frac{1}{6}}{B} \]

      4. associate-*r* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{{B}^{2} \cdot \left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}\right)}}{B} \]

      5. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}{B}} \]

   5. Applied rewrites 39.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(B \cdot B, 0.16666666666666666 \cdot F, F\right)}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1 + \frac{1}{6} \cdot {B}^{2}}{\color{blue}{B}} \]
   7. Step-by-step derivation

   8. Applied rewrites 60.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{\color{blue}{B}} \]

   9. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]
   10. Step-by-step derivation

       1. lower-/.f64 54.7

          \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} \]

   11. Applied rewrites 54.7%

       \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4 \cdot 10^{-13}:\\ \;\;\;\;\left(1 + x\right) \cdot \left({\left(\left(F \cdot F\right) \cdot B\right)}^{-1} - {B}^{-1}\right)\\ \mathbf{elif}\;F \leq 70:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-1}}, F, -x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 44.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -4 \cdot 10^{-13}:\\ \;\;\;\;\left(1 + x\right) \cdot \left({\left(\left(F \cdot F\right) \cdot B\right)}^{-1} - {B}^{-1}\right)\\ \mathbf{elif}\;F \leq 4.05 \cdot 10^{-47}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4e-13)
   (* (+ 1.0 x) (- (pow (* (* F F) B) -1.0) (pow B -1.0)))
   (if (<= F 4.05e-47)
     (/ (- x) B)
     (+ (- (/ x B)) (/ (fma (* B B) 0.16666666666666666 1.0) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4e-13) {
		tmp = (1.0 + x) * (pow(((F * F) * B), -1.0) - pow(B, -1.0));
	} else if (F <= 4.05e-47) {
		tmp = -x / B;
	} else {
		tmp = -(x / B) + (fma((B * B), 0.16666666666666666, 1.0) / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -4e-13)
		tmp = Float64(Float64(1.0 + x) * Float64((Float64(Float64(F * F) * B) ^ -1.0) - (B ^ -1.0)));
	elseif (F <= 4.05e-47)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(-Float64(x / B)) + Float64(fma(Float64(B * B), 0.16666666666666666, 1.0) / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -4e-13], N[(N[(1.0 + x), $MachinePrecision] * N[(N[Power[N[(N[(F * F), $MachinePrecision] * B), $MachinePrecision], -1.0], $MachinePrecision] - N[Power[B, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.05e-47], N[((-x) / B), $MachinePrecision], N[((-N[(x / B), $MachinePrecision]) + N[(N[(N[(B * B), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.0000000000000001e-13

   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

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   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. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-neg.f64 33.1

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 33.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto -1 \cdot \frac{1 + x}{B} + \color{blue}{\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{B \cdot {F}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 40.3%

      \[\leadsto \mathsf{fma}\left(\frac{0.5}{F \cdot F}, \color{blue}{\frac{\mathsf{fma}\left(2, x, 2\right)}{B}}, \frac{-1 - x}{B}\right) \]

   9. Taylor expanded in x around 0

      \[\leadsto \left(x \cdot \left(\frac{1}{B \cdot {F}^{2}} - \frac{1}{B}\right) + \frac{1}{B \cdot {F}^{2}}\right) - \frac{1}{\color{blue}{B}} \]
   10. Step-by-step derivation

   11. Applied rewrites 45.1%

       \[\leadsto \left(1 + x\right) \cdot \left(\frac{1}{\left(F \cdot F\right) \cdot B} - \color{blue}{\frac{1}{B}}\right) \]

   if -4.0000000000000001e-13 < F < 4.0500000000000002e-47

   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. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-neg.f64 48.2

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 48.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 41.9%

      \[\leadsto \frac{-x}{B} \]

   if 4.0500000000000002e-47 < F

   1. Initial program 65.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)}}{B} \]

      2. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{\left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}}}{B} \]

      3. associate-*l* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{\left({B}^{2} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)} \cdot \frac{1}{6}}{B} \]

      4. associate-*r* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{{B}^{2} \cdot \left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}\right)}}{B} \]

      5. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}{B}} \]

   5. Applied rewrites 40.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(B \cdot B, 0.16666666666666666 \cdot F, F\right)}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1 + \frac{1}{6} \cdot {B}^{2}}{\color{blue}{B}} \]
   7. Step-by-step derivation

   8. Applied rewrites 52.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{\color{blue}{B}} \]

   9. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]
   10. Step-by-step derivation

       1. lower-/.f64 47.1

          \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} \]

   11. Applied rewrites 47.1%

       \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4 \cdot 10^{-13}:\\ \;\;\;\;\left(1 + x\right) \cdot \left({\left(\left(F \cdot F\right) \cdot B\right)}^{-1} - {B}^{-1}\right)\\ \mathbf{elif}\;F \leq 4.05 \cdot 10^{-47}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 74.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-10} \lor \neg \left(x \leq 6.5 \cdot 10^{-88}\right):\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(-0.16666666666666666 \cdot B, B, 1\right) \cdot B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (or (<= x -2.7e-10) (not (<= x 6.5e-88)))
   (- (/ -1.0 (* (fma (* -0.16666666666666666 B) B 1.0) B)) (/ x (tan B)))
   (- (/ (/ F (sin B)) (sqrt (fma 2.0 x (fma F F 2.0)))) (/ x B))))

C

double code(double F, double B, double x) {
	double tmp;
	if ((x <= -2.7e-10) || !(x <= 6.5e-88)) {
		tmp = (-1.0 / (fma((-0.16666666666666666 * B), B, 1.0) * B)) - (x / tan(B));
	} else {
		tmp = ((F / sin(B)) / sqrt(fma(2.0, x, fma(F, F, 2.0)))) - (x / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if ((x <= -2.7e-10) || !(x <= 6.5e-88))
		tmp = Float64(Float64(-1.0 / Float64(fma(Float64(-0.16666666666666666 * B), B, 1.0) * B)) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(Float64(F / sin(B)) / sqrt(fma(2.0, x, fma(F, F, 2.0)))) - Float64(x / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[Or[LessEqual[x, -2.7e-10], N[Not[LessEqual[x, 6.5e-88]], $MachinePrecision]], N[(N[(-1.0 / N[(N[(N[(-0.16666666666666666 * B), $MachinePrecision] * B + 1.0), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.7e-10 or 6.50000000000000006e-88 < x

   1. Initial program 86.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \cdot \frac{F}{\sin B}} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \cdot \color{blue}{\frac{F}{\sin B}} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      6. clear-num N/A

         \[\leadsto {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\frac{\sin B}{F}}} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      7. associate-/r/ N/A

         \[\leadsto {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot F\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \cdot \frac{1}{\sin B}\right) \cdot F} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      9. div-inv N/A

         \[\leadsto \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} \cdot F + \left(-x \cdot \frac{1}{\tan B}\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}, F, -x \cdot \frac{1}{\tan B}\right)} \]

   4. Applied rewrites 95.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, F, \frac{-x}{\tan B}\right)} \]

   6. Applied rewrites 95.2%

      \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B} - \frac{x}{\tan B}} \]

   7. Taylor expanded in F around -inf

      \[\leadsto \frac{\color{blue}{-1}}{\sin B} - \frac{x}{\tan B} \]
   8. Step-by-step derivation

   9. Applied rewrites 89.9%

      \[\leadsto \frac{\color{blue}{-1}}{\sin B} - \frac{x}{\tan B} \]

   10. Taylor expanded in B around 0

       \[\leadsto \frac{-1}{\color{blue}{B \cdot \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)}} - \frac{x}{\tan B} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{-1}{\color{blue}{\left(1 + \frac{-1}{6} \cdot {B}^{2}\right) \cdot B}} - \frac{x}{\tan B} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{-1}{\color{blue}{\left(1 + \frac{-1}{6} \cdot {B}^{2}\right) \cdot B}} - \frac{x}{\tan B} \]

       3. +-commutative N/A

          \[\leadsto \frac{-1}{\color{blue}{\left(\frac{-1}{6} \cdot {B}^{2} + 1\right)} \cdot B} - \frac{x}{\tan B} \]

       4. unpow2 N/A

          \[\leadsto \frac{-1}{\left(\frac{-1}{6} \cdot \color{blue}{\left(B \cdot B\right)} + 1\right) \cdot B} - \frac{x}{\tan B} \]

       5. associate-*r* N/A

          \[\leadsto \frac{-1}{\left(\color{blue}{\left(\frac{-1}{6} \cdot B\right) \cdot B} + 1\right) \cdot B} - \frac{x}{\tan B} \]

       6. lower-fma.f64 N/A

          \[\leadsto \frac{-1}{\color{blue}{\mathsf{fma}\left(\frac{-1}{6} \cdot B, B, 1\right)} \cdot B} - \frac{x}{\tan B} \]

       7. lower-*.f64 92.8

          \[\leadsto \frac{-1}{\mathsf{fma}\left(\color{blue}{-0.16666666666666666 \cdot B}, B, 1\right) \cdot B} - \frac{x}{\tan B} \]

   12. Applied rewrites 92.8%

       \[\leadsto \frac{-1}{\color{blue}{\mathsf{fma}\left(-0.16666666666666666 \cdot B, B, 1\right) \cdot B}} - \frac{x}{\tan B} \]

   if -2.7e-10 < x < 6.50000000000000006e-88

   1. Initial program 75.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \cdot \frac{F}{\sin B}} \]

      3. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \cdot \color{blue}{\frac{F}{\sin B}} \]

      4. clear-num N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\frac{\sin B}{F}}} \]

      5. un-div-inv N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\frac{\sin B}{F}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\frac{\sin B}{F}}} \]

      7. lift-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{{\color{blue}{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}}^{\left(-\frac{1}{2}\right)}}{\frac{\sin B}{F}} \]

      8. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{{\color{blue}{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)}}{\frac{\sin B}{F}} \]

      9. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{{\left(\color{blue}{2 \cdot x} + \left(F \cdot F + 2\right)\right)}^{\left(-\frac{1}{2}\right)}}{\frac{\sin B}{F}} \]

      10. lower-fma.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{{\color{blue}{\left(\mathsf{fma}\left(2, x, F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)}}{\frac{\sin B}{F}} \]

      11. lift-+.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{{\left(\mathsf{fma}\left(2, x, \color{blue}{F \cdot F + 2}\right)\right)}^{\left(-\frac{1}{2}\right)}}{\frac{\sin B}{F}} \]

      12. lift-*.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{{\left(\mathsf{fma}\left(2, x, \color{blue}{F \cdot F} + 2\right)\right)}^{\left(-\frac{1}{2}\right)}}{\frac{\sin B}{F}} \]

      13. lower-fma.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{{\left(\mathsf{fma}\left(2, x, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)\right)}^{\left(-\frac{1}{2}\right)}}{\frac{\sin B}{F}} \]

      14. lift-neg.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}}{\frac{\sin B}{F}} \]

      15. lift-/.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)}}{\frac{\sin B}{F}} \]

      16. metadata-eval N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)}}{\frac{\sin B}{F}} \]

      17. metadata-eval N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{\frac{-1}{2}}}}{\frac{\sin B}{F}} \]

      18. lower-/.f64 76.3

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\color{blue}{\frac{\sin B}{F}}} \]

   4. Applied rewrites 76.3%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\frac{\sin B}{F}}} \]

   5. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\frac{\sin B}{F}} \]
   6. Step-by-step derivation

      1. lower-/.f64 68.0

         \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\frac{\sin B}{F}} \]

   7. Applied rewrites 68.0%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\frac{\sin B}{F}} \]
   8. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}}{\frac{\sin B}{F}} \]

      2. metadata-eval N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)}}{\frac{\sin B}{F}} \]

      3. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-\frac{x}{B}\right) + \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\frac{\sin B}{F}}} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\frac{\sin B}{F}} + \left(-\frac{x}{B}\right)} \]

      5. lift-neg.f64 N/A

         \[\leadsto \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\frac{\sin B}{F}} + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{B}\right)\right)} \]

      6. unsub-neg N/A

         \[\leadsto \color{blue}{\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\frac{\sin B}{F}} - \frac{x}{B}} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\frac{\sin B}{F}} - \frac{x}{B}} \]

   9. Applied rewrites 67.2%

      \[\leadsto \color{blue}{\frac{\frac{F}{\sin B}}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - \frac{x}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-10} \lor \neg \left(x \leq 6.5 \cdot 10^{-88}\right):\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(-0.16666666666666666 \cdot B, B, 1\right) \cdot B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 57.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 4.3 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(-0.16666666666666666 \cdot B, B, 1\right) \cdot B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 4.3e-19)
   (/ (- (/ F (sqrt (fma x 2.0 (fma F F 2.0)))) x) B)
   (- (/ -1.0 (* (fma (* -0.16666666666666666 B) B 1.0) B)) (/ x (tan B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 4.3e-19) {
		tmp = ((F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - x) / B;
	} else {
		tmp = (-1.0 / (fma((-0.16666666666666666 * B), B, 1.0) * B)) - (x / tan(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 4.3e-19)
		tmp = Float64(Float64(Float64(F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - x) / B);
	else
		tmp = Float64(Float64(-1.0 / Float64(fma(Float64(-0.16666666666666666 * B), B, 1.0) * B)) - Float64(x / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 4.3e-19], N[(N[(N[(F / N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(-1.0 / N[(N[(N[(-0.16666666666666666 * B), $MachinePrecision] * B + 1.0), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 4.3e-19

   1. Initial program 78.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-neg.f64 56.4

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 56.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

   7. Applied rewrites 56.5%

      \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}} \]

   if 4.3e-19 < B

   1. Initial program 86.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \cdot \frac{F}{\sin B}} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \cdot \color{blue}{\frac{F}{\sin B}} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      6. clear-num N/A

         \[\leadsto {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\frac{\sin B}{F}}} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      7. associate-/r/ N/A

         \[\leadsto {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot F\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \cdot \frac{1}{\sin B}\right) \cdot F} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      9. div-inv N/A

         \[\leadsto \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} \cdot F + \left(-x \cdot \frac{1}{\tan B}\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}, F, -x \cdot \frac{1}{\tan B}\right)} \]

   4. Applied rewrites 86.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, F, \frac{-x}{\tan B}\right)} \]

   6. Applied rewrites 86.4%

      \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B} - \frac{x}{\tan B}} \]

   7. Taylor expanded in F around -inf

      \[\leadsto \frac{\color{blue}{-1}}{\sin B} - \frac{x}{\tan B} \]
   8. Step-by-step derivation

   9. Applied rewrites 53.6%

      \[\leadsto \frac{\color{blue}{-1}}{\sin B} - \frac{x}{\tan B} \]

   10. Taylor expanded in B around 0

       \[\leadsto \frac{-1}{\color{blue}{B \cdot \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)}} - \frac{x}{\tan B} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{-1}{\color{blue}{\left(1 + \frac{-1}{6} \cdot {B}^{2}\right) \cdot B}} - \frac{x}{\tan B} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{-1}{\color{blue}{\left(1 + \frac{-1}{6} \cdot {B}^{2}\right) \cdot B}} - \frac{x}{\tan B} \]

       3. +-commutative N/A

          \[\leadsto \frac{-1}{\color{blue}{\left(\frac{-1}{6} \cdot {B}^{2} + 1\right)} \cdot B} - \frac{x}{\tan B} \]

       4. unpow2 N/A

          \[\leadsto \frac{-1}{\left(\frac{-1}{6} \cdot \color{blue}{\left(B \cdot B\right)} + 1\right) \cdot B} - \frac{x}{\tan B} \]

       5. associate-*r* N/A

          \[\leadsto \frac{-1}{\left(\color{blue}{\left(\frac{-1}{6} \cdot B\right) \cdot B} + 1\right) \cdot B} - \frac{x}{\tan B} \]

       6. lower-fma.f64 N/A

          \[\leadsto \frac{-1}{\color{blue}{\mathsf{fma}\left(\frac{-1}{6} \cdot B, B, 1\right)} \cdot B} - \frac{x}{\tan B} \]

       7. lower-*.f64 46.3

          \[\leadsto \frac{-1}{\mathsf{fma}\left(\color{blue}{-0.16666666666666666 \cdot B}, B, 1\right) \cdot B} - \frac{x}{\tan B} \]

   12. Applied rewrites 46.3%

       \[\leadsto \frac{-1}{\color{blue}{\mathsf{fma}\left(-0.16666666666666666 \cdot B, B, 1\right) \cdot B}} - \frac{x}{\tan B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4.3 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(-0.16666666666666666 \cdot B, B, 1\right) \cdot B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 49.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 3.6 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot B - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 3.6e-7)
   (/ (- (/ F (sqrt (fma x 2.0 (fma F F 2.0)))) x) B)
   (- (* 0.16666666666666666 B) (/ x (tan B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 3.6e-7) {
		tmp = ((F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - x) / B;
	} else {
		tmp = (0.16666666666666666 * B) - (x / tan(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 3.6e-7)
		tmp = Float64(Float64(Float64(F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - x) / B);
	else
		tmp = Float64(Float64(0.16666666666666666 * B) - Float64(x / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 3.6e-7], N[(N[(N[(F / N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(0.16666666666666666 * B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 3.59999999999999994e-7

   1. Initial program 78.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-neg.f64 56.9

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 56.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

   7. Applied rewrites 56.9%

      \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}} \]

   if 3.59999999999999994e-7 < B

   1. Initial program 86.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)}}{B} \]

      2. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{\left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}}}{B} \]

      3. associate-*l* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{\left({B}^{2} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)} \cdot \frac{1}{6}}{B} \]

      4. associate-*r* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{{B}^{2} \cdot \left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}\right)}}{B} \]

      5. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}{B}} \]

   5. Applied rewrites 17.5%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(B \cdot B, 0.16666666666666666 \cdot F, F\right)}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1 + \frac{1}{6} \cdot {B}^{2}}{\color{blue}{B}} \]
   7. Step-by-step derivation

   8. Applied rewrites 19.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{\color{blue}{B}} \]

   9. Taylor expanded in B around inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{6} \cdot B \]
   10. Step-by-step derivation

   11. Applied rewrites 20.2%

       \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + 0.16666666666666666 \cdot B \]
   12. Step-by-step derivation

       1. lift-+.f64 N/A

          \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{6} \cdot B} \]

       2. +-commutative N/A

          \[\leadsto \color{blue}{\frac{1}{6} \cdot B + \left(-x \cdot \frac{1}{\tan B}\right)} \]

       3. lift-neg.f64 N/A

          \[\leadsto \frac{1}{6} \cdot B + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{1}{6} \cdot B + \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{1}{\tan B}}\right)\right) \]

       5. lift-/.f64 N/A

          \[\leadsto \frac{1}{6} \cdot B + \left(\mathsf{neg}\left(x \cdot \color{blue}{\frac{1}{\tan B}}\right)\right) \]

       6. div-inv N/A

          \[\leadsto \frac{1}{6} \cdot B + \left(\mathsf{neg}\left(\color{blue}{\frac{x}{\tan B}}\right)\right) \]

       7. lift-/.f64 N/A

          \[\leadsto \frac{1}{6} \cdot B + \left(\mathsf{neg}\left(\color{blue}{\frac{x}{\tan B}}\right)\right) \]

   13. Applied rewrites 20.2%

       \[\leadsto \color{blue}{0.16666666666666666 \cdot B - \frac{x}{\tan B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.6 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot B - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 52.5% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(0.3333333333333333, B \cdot B, -1\right)}{B} + \frac{\mathsf{fma}\left(-0.16666666666666666, B \cdot B, -1\right)}{B}\\ \mathbf{elif}\;F \leq 2.8 \cdot 10^{+139}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0021164021164021165, B \cdot B, -0.022222222222222223\right), B \cdot B, -0.3333333333333333\right), B \cdot B, 1\right)}{B} + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.35e+154)
   (+
    (/ (* x (fma 0.3333333333333333 (* B B) -1.0)) B)
    (/ (fma -0.16666666666666666 (* B B) -1.0) B))
   (if (<= F 2.8e+139)
     (/ (- (/ F (sqrt (fma x 2.0 (fma F F 2.0)))) x) B)
     (+
      (*
       (- x)
       (/
        (fma
         (fma
          (fma -0.0021164021164021165 (* B B) -0.022222222222222223)
          (* B B)
          -0.3333333333333333)
         (* B B)
         1.0)
        B))
      (/ (fma (* B B) 0.16666666666666666 1.0) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.35e+154) {
		tmp = ((x * fma(0.3333333333333333, (B * B), -1.0)) / B) + (fma(-0.16666666666666666, (B * B), -1.0) / B);
	} else if (F <= 2.8e+139) {
		tmp = ((F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - x) / B;
	} else {
		tmp = (-x * (fma(fma(fma(-0.0021164021164021165, (B * B), -0.022222222222222223), (B * B), -0.3333333333333333), (B * B), 1.0) / B)) + (fma((B * B), 0.16666666666666666, 1.0) / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.35e+154)
		tmp = Float64(Float64(Float64(x * fma(0.3333333333333333, Float64(B * B), -1.0)) / B) + Float64(fma(-0.16666666666666666, Float64(B * B), -1.0) / B));
	elseif (F <= 2.8e+139)
		tmp = Float64(Float64(Float64(F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - x) / B);
	else
		tmp = Float64(Float64(Float64(-x) * Float64(fma(fma(fma(-0.0021164021164021165, Float64(B * B), -0.022222222222222223), Float64(B * B), -0.3333333333333333), Float64(B * B), 1.0) / B)) + Float64(fma(Float64(B * B), 0.16666666666666666, 1.0) / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.35e+154], N[(N[(N[(x * N[(0.3333333333333333 * N[(B * B), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision] + N[(N[(-0.16666666666666666 * N[(B * B), $MachinePrecision] + -1.0), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.8e+139], N[(N[(N[(F / N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[((-x) * N[(N[(N[(N[(-0.0021164021164021165 * N[(B * B), $MachinePrecision] + -0.022222222222222223), $MachinePrecision] * N[(B * B), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(B * B), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.35000000000000003e154

   1. Initial program 42.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)}}{B} \]

      2. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{\left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}}}{B} \]

      3. associate-*l* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{\left({B}^{2} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)} \cdot \frac{1}{6}}{B} \]

      4. associate-*r* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{{B}^{2} \cdot \left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}\right)}}{B} \]

      5. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}{B}} \]

   5. Applied rewrites 21.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(B \cdot B, 0.16666666666666666 \cdot F, F\right)}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1 + \frac{1}{6} \cdot {B}^{2}}{\color{blue}{B}} \]
   7. Step-by-step derivation

   8. Applied rewrites 31.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{\color{blue}{B}} \]

   9. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B}} + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B}} + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       2. sub-neg N/A

          \[\leadsto \frac{\color{blue}{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) + \left(\mathsf{neg}\left(x\right)\right)}}{B} + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       3. associate-*r* N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{1}{3} \cdot {B}^{2}\right) \cdot x} + \left(\mathsf{neg}\left(x\right)\right)}{B} + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       4. mul-1-neg N/A

          \[\leadsto \frac{\left(\frac{1}{3} \cdot {B}^{2}\right) \cdot x + \color{blue}{-1 \cdot x}}{B} + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       5. distribute-rgt-out N/A

          \[\leadsto \frac{\color{blue}{x \cdot \left(\frac{1}{3} \cdot {B}^{2} + -1\right)}}{B} + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{x \cdot \left(\frac{1}{3} \cdot {B}^{2} + -1\right)}}{B} + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       7. lower-fma.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{3}, {B}^{2}, -1\right)}}{B} + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       8. unpow2 N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(\frac{1}{3}, \color{blue}{B \cdot B}, -1\right)}{B} + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       9. lower-*.f64 14.2

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(0.3333333333333333, \color{blue}{B \cdot B}, -1\right)}{B} + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} \]

   11. Applied rewrites 14.2%

       \[\leadsto \color{blue}{\frac{x \cdot \mathsf{fma}\left(0.3333333333333333, B \cdot B, -1\right)}{B}} + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} \]

   12. Taylor expanded in F around -inf

       \[\leadsto \frac{x \cdot \mathsf{fma}\left(\frac{1}{3}, B \cdot B, -1\right)}{B} + \frac{\frac{-1}{6} \cdot {B}^{2} - 1}{\color{blue}{B}} \]
   13. Step-by-step derivation

   14. Applied rewrites 43.6%

       \[\leadsto \frac{x \cdot \mathsf{fma}\left(0.3333333333333333, B \cdot B, -1\right)}{B} + \frac{\mathsf{fma}\left(-0.16666666666666666, B \cdot B, -1\right)}{\color{blue}{B}} \]

   if -1.35000000000000003e154 < F < 2.7999999999999998e139

   1. Initial program 96.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-neg.f64 48.4

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 48.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

   7. Applied rewrites 48.5%

      \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}} \]

   if 2.7999999999999998e139 < F

   1. Initial program 32.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)}}{B} \]

      2. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{\left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}}}{B} \]

      3. associate-*l* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{\left({B}^{2} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)} \cdot \frac{1}{6}}{B} \]

      4. associate-*r* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{{B}^{2} \cdot \left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}\right)}}{B} \]

      5. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}{B}} \]

   5. Applied rewrites 26.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(B \cdot B, 0.16666666666666666 \cdot F, F\right)}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1 + \frac{1}{6} \cdot {B}^{2}}{\color{blue}{B}} \]
   7. Step-by-step derivation

   8. Applied rewrites 63.8%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{\color{blue}{B}} \]

   9. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \color{blue}{\frac{1 + {B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) - \frac{1}{3}\right)}{B}}\right) + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \left(-x \cdot \color{blue}{\frac{1 + {B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) - \frac{1}{3}\right)}{B}}\right) + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       2. +-commutative N/A

          \[\leadsto \left(-x \cdot \frac{\color{blue}{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) - \frac{1}{3}\right) + 1}}{B}\right) + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       3. *-commutative N/A

          \[\leadsto \left(-x \cdot \frac{\color{blue}{\left({B}^{2} \cdot \left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) - \frac{1}{3}\right) \cdot {B}^{2}} + 1}{B}\right) + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       4. lower-fma.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\color{blue}{\mathsf{fma}\left({B}^{2} \cdot \left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) - \frac{1}{3}, {B}^{2}, 1\right)}}{B}\right) + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       5. sub-neg N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\color{blue}{{B}^{2} \cdot \left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, {B}^{2}, 1\right)}{B}\right) + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       6. *-commutative N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) \cdot {B}^{2}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), {B}^{2}, 1\right)}{B}\right) + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       7. metadata-eval N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) \cdot {B}^{2} + \color{blue}{\frac{-1}{3}}, {B}^{2}, 1\right)}{B}\right) + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       8. lower-fma.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}, {B}^{2}, \frac{-1}{3}\right)}, {B}^{2}, 1\right)}{B}\right) + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       9. sub-neg N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-2}{945} \cdot {B}^{2} + \left(\mathsf{neg}\left(\frac{1}{45}\right)\right)}, {B}^{2}, \frac{-1}{3}\right), {B}^{2}, 1\right)}{B}\right) + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       10. metadata-eval N/A

           \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{945} \cdot {B}^{2} + \color{blue}{\frac{-1}{45}}, {B}^{2}, \frac{-1}{3}\right), {B}^{2}, 1\right)}{B}\right) + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       11. lower-fma.f64 N/A

           \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-2}{945}, {B}^{2}, \frac{-1}{45}\right)}, {B}^{2}, \frac{-1}{3}\right), {B}^{2}, 1\right)}{B}\right) + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       12. unpow2 N/A

           \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{945}, \color{blue}{B \cdot B}, \frac{-1}{45}\right), {B}^{2}, \frac{-1}{3}\right), {B}^{2}, 1\right)}{B}\right) + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       13. lower-*.f64 N/A

           \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{945}, \color{blue}{B \cdot B}, \frac{-1}{45}\right), {B}^{2}, \frac{-1}{3}\right), {B}^{2}, 1\right)}{B}\right) + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       14. unpow2 N/A

           \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{945}, B \cdot B, \frac{-1}{45}\right), \color{blue}{B \cdot B}, \frac{-1}{3}\right), {B}^{2}, 1\right)}{B}\right) + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       15. lower-*.f64 N/A

           \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{945}, B \cdot B, \frac{-1}{45}\right), \color{blue}{B \cdot B}, \frac{-1}{3}\right), {B}^{2}, 1\right)}{B}\right) + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       16. unpow2 N/A

           \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-2}{945}, B \cdot B, \frac{-1}{45}\right), B \cdot B, \frac{-1}{3}\right), \color{blue}{B \cdot B}, 1\right)}{B}\right) + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       17. lower-*.f64 55.5

           \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0021164021164021165, B \cdot B, -0.022222222222222223\right), B \cdot B, -0.3333333333333333\right), \color{blue}{B \cdot B}, 1\right)}{B}\right) + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} \]

   11. Applied rewrites 55.5%

       \[\leadsto \left(-x \cdot \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0021164021164021165, B \cdot B, -0.022222222222222223\right), B \cdot B, -0.3333333333333333\right), B \cdot B, 1\right)}{B}}\right) + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(0.3333333333333333, B \cdot B, -1\right)}{B} + \frac{\mathsf{fma}\left(-0.16666666666666666, B \cdot B, -1\right)}{B}\\ \mathbf{elif}\;F \leq 2.8 \cdot 10^{+139}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0021164021164021165, B \cdot B, -0.022222222222222223\right), B \cdot B, -0.3333333333333333\right), B \cdot B, 1\right)}{B} + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 52.5% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(0.3333333333333333, B \cdot B, -1\right)}{B} + \frac{\mathsf{fma}\left(-0.16666666666666666, B \cdot B, -1\right)}{B}\\ \mathbf{elif}\;F \leq 2.8 \cdot 10^{+139}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, B \cdot B, -0.3333333333333333\right), B \cdot B, 1\right)}{B} + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.35e+154)
   (+
    (/ (* x (fma 0.3333333333333333 (* B B) -1.0)) B)
    (/ (fma -0.16666666666666666 (* B B) -1.0) B))
   (if (<= F 2.8e+139)
     (/ (- (/ F (sqrt (fma x 2.0 (fma F F 2.0)))) x) B)
     (+
      (*
       (- x)
       (/
        (fma
         (fma -0.022222222222222223 (* B B) -0.3333333333333333)
         (* B B)
         1.0)
        B))
      (/ (fma (* B B) 0.16666666666666666 1.0) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.35e+154) {
		tmp = ((x * fma(0.3333333333333333, (B * B), -1.0)) / B) + (fma(-0.16666666666666666, (B * B), -1.0) / B);
	} else if (F <= 2.8e+139) {
		tmp = ((F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - x) / B;
	} else {
		tmp = (-x * (fma(fma(-0.022222222222222223, (B * B), -0.3333333333333333), (B * B), 1.0) / B)) + (fma((B * B), 0.16666666666666666, 1.0) / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.35e+154)
		tmp = Float64(Float64(Float64(x * fma(0.3333333333333333, Float64(B * B), -1.0)) / B) + Float64(fma(-0.16666666666666666, Float64(B * B), -1.0) / B));
	elseif (F <= 2.8e+139)
		tmp = Float64(Float64(Float64(F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - x) / B);
	else
		tmp = Float64(Float64(Float64(-x) * Float64(fma(fma(-0.022222222222222223, Float64(B * B), -0.3333333333333333), Float64(B * B), 1.0) / B)) + Float64(fma(Float64(B * B), 0.16666666666666666, 1.0) / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.35e+154], N[(N[(N[(x * N[(0.3333333333333333 * N[(B * B), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision] + N[(N[(-0.16666666666666666 * N[(B * B), $MachinePrecision] + -1.0), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.8e+139], N[(N[(N[(F / N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[((-x) * N[(N[(N[(-0.022222222222222223 * N[(B * B), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(B * B), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.35000000000000003e154

   1. Initial program 42.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)}}{B} \]

      2. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{\left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}}}{B} \]

      3. associate-*l* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{\left({B}^{2} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)} \cdot \frac{1}{6}}{B} \]

      4. associate-*r* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{{B}^{2} \cdot \left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}\right)}}{B} \]

      5. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}{B}} \]

   5. Applied rewrites 21.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(B \cdot B, 0.16666666666666666 \cdot F, F\right)}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1 + \frac{1}{6} \cdot {B}^{2}}{\color{blue}{B}} \]
   7. Step-by-step derivation

   8. Applied rewrites 31.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{\color{blue}{B}} \]

   9. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B}} + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B}} + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       2. sub-neg N/A

          \[\leadsto \frac{\color{blue}{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) + \left(\mathsf{neg}\left(x\right)\right)}}{B} + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       3. associate-*r* N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{1}{3} \cdot {B}^{2}\right) \cdot x} + \left(\mathsf{neg}\left(x\right)\right)}{B} + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       4. mul-1-neg N/A

          \[\leadsto \frac{\left(\frac{1}{3} \cdot {B}^{2}\right) \cdot x + \color{blue}{-1 \cdot x}}{B} + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       5. distribute-rgt-out N/A

          \[\leadsto \frac{\color{blue}{x \cdot \left(\frac{1}{3} \cdot {B}^{2} + -1\right)}}{B} + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{x \cdot \left(\frac{1}{3} \cdot {B}^{2} + -1\right)}}{B} + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       7. lower-fma.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{3}, {B}^{2}, -1\right)}}{B} + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       8. unpow2 N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(\frac{1}{3}, \color{blue}{B \cdot B}, -1\right)}{B} + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       9. lower-*.f64 14.2

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(0.3333333333333333, \color{blue}{B \cdot B}, -1\right)}{B} + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} \]

   11. Applied rewrites 14.2%

       \[\leadsto \color{blue}{\frac{x \cdot \mathsf{fma}\left(0.3333333333333333, B \cdot B, -1\right)}{B}} + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} \]

   12. Taylor expanded in F around -inf

       \[\leadsto \frac{x \cdot \mathsf{fma}\left(\frac{1}{3}, B \cdot B, -1\right)}{B} + \frac{\frac{-1}{6} \cdot {B}^{2} - 1}{\color{blue}{B}} \]
   13. Step-by-step derivation

   14. Applied rewrites 43.6%

       \[\leadsto \frac{x \cdot \mathsf{fma}\left(0.3333333333333333, B \cdot B, -1\right)}{B} + \frac{\mathsf{fma}\left(-0.16666666666666666, B \cdot B, -1\right)}{\color{blue}{B}} \]

   if -1.35000000000000003e154 < F < 2.7999999999999998e139

   1. Initial program 96.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-neg.f64 48.4

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 48.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

   7. Applied rewrites 48.5%

      \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}} \]

   if 2.7999999999999998e139 < F

   1. Initial program 32.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)}}{B} \]

      2. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{\left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}}}{B} \]

      3. associate-*l* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{\left({B}^{2} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)} \cdot \frac{1}{6}}{B} \]

      4. associate-*r* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{{B}^{2} \cdot \left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}\right)}}{B} \]

      5. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}{B}} \]

   5. Applied rewrites 26.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(B \cdot B, 0.16666666666666666 \cdot F, F\right)}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1 + \frac{1}{6} \cdot {B}^{2}}{\color{blue}{B}} \]
   7. Step-by-step derivation

   8. Applied rewrites 63.8%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{\color{blue}{B}} \]

   9. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \color{blue}{\frac{1 + {B}^{2} \cdot \left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right)}{B}}\right) + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \left(-x \cdot \color{blue}{\frac{1 + {B}^{2} \cdot \left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right)}{B}}\right) + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       2. +-commutative N/A

          \[\leadsto \left(-x \cdot \frac{\color{blue}{{B}^{2} \cdot \left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right) + 1}}{B}\right) + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       3. *-commutative N/A

          \[\leadsto \left(-x \cdot \frac{\color{blue}{\left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right) \cdot {B}^{2}} + 1}{B}\right) + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       4. lower-fma.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}, {B}^{2}, 1\right)}}{B}\right) + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       5. sub-neg N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{45} \cdot {B}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, {B}^{2}, 1\right)}{B}\right) + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       6. metadata-eval N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\frac{-1}{45} \cdot {B}^{2} + \color{blue}{\frac{-1}{3}}, {B}^{2}, 1\right)}{B}\right) + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       7. lower-fma.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{45}, {B}^{2}, \frac{-1}{3}\right)}, {B}^{2}, 1\right)}{B}\right) + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       8. unpow2 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, \color{blue}{B \cdot B}, \frac{-1}{3}\right), {B}^{2}, 1\right)}{B}\right) + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       9. lower-*.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, \color{blue}{B \cdot B}, \frac{-1}{3}\right), {B}^{2}, 1\right)}{B}\right) + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       10. unpow2 N/A

           \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, B \cdot B, \frac{-1}{3}\right), \color{blue}{B \cdot B}, 1\right)}{B}\right) + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       11. lower-*.f64 55.2

           \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, B \cdot B, -0.3333333333333333\right), \color{blue}{B \cdot B}, 1\right)}{B}\right) + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} \]

   11. Applied rewrites 55.2%

       \[\leadsto \left(-x \cdot \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, B \cdot B, -0.3333333333333333\right), B \cdot B, 1\right)}{B}}\right) + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(0.3333333333333333, B \cdot B, -1\right)}{B} + \frac{\mathsf{fma}\left(-0.16666666666666666, B \cdot B, -1\right)}{B}\\ \mathbf{elif}\;F \leq 2.8 \cdot 10^{+139}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, B \cdot B, -0.3333333333333333\right), B \cdot B, 1\right)}{B} + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 52.5% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(0.3333333333333333, B \cdot B, -1\right)}{B} + \frac{\mathsf{fma}\left(-0.16666666666666666, B \cdot B, -1\right)}{B}\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{+143}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{\mathsf{fma}\left(\left(B \cdot B\right) \cdot x, -0.3333333333333333, x\right)}{B}\right) + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.35e+154)
   (+
    (/ (* x (fma 0.3333333333333333 (* B B) -1.0)) B)
    (/ (fma -0.16666666666666666 (* B B) -1.0) B))
   (if (<= F 4.5e+143)
     (/ (- (/ F (sqrt (fma x 2.0 (fma F F 2.0)))) x) B)
     (+
      (- (/ (fma (* (* B B) x) -0.3333333333333333 x) B))
      (/ (fma (* B B) 0.16666666666666666 1.0) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.35e+154) {
		tmp = ((x * fma(0.3333333333333333, (B * B), -1.0)) / B) + (fma(-0.16666666666666666, (B * B), -1.0) / B);
	} else if (F <= 4.5e+143) {
		tmp = ((F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - x) / B;
	} else {
		tmp = -(fma(((B * B) * x), -0.3333333333333333, x) / B) + (fma((B * B), 0.16666666666666666, 1.0) / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.35e+154)
		tmp = Float64(Float64(Float64(x * fma(0.3333333333333333, Float64(B * B), -1.0)) / B) + Float64(fma(-0.16666666666666666, Float64(B * B), -1.0) / B));
	elseif (F <= 4.5e+143)
		tmp = Float64(Float64(Float64(F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - x) / B);
	else
		tmp = Float64(Float64(-Float64(fma(Float64(Float64(B * B) * x), -0.3333333333333333, x) / B)) + Float64(fma(Float64(B * B), 0.16666666666666666, 1.0) / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.35e+154], N[(N[(N[(x * N[(0.3333333333333333 * N[(B * B), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision] + N[(N[(-0.16666666666666666 * N[(B * B), $MachinePrecision] + -1.0), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.5e+143], N[(N[(N[(F / N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[((-N[(N[(N[(N[(B * B), $MachinePrecision] * x), $MachinePrecision] * -0.3333333333333333 + x), $MachinePrecision] / B), $MachinePrecision]) + N[(N[(N[(B * B), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.35000000000000003e154

   1. Initial program 42.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)}}{B} \]

      2. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{\left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}}}{B} \]

      3. associate-*l* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{\left({B}^{2} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)} \cdot \frac{1}{6}}{B} \]

      4. associate-*r* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{{B}^{2} \cdot \left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}\right)}}{B} \]

      5. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}{B}} \]

   5. Applied rewrites 21.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(B \cdot B, 0.16666666666666666 \cdot F, F\right)}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1 + \frac{1}{6} \cdot {B}^{2}}{\color{blue}{B}} \]
   7. Step-by-step derivation

   8. Applied rewrites 31.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{\color{blue}{B}} \]

   9. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B}} + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B}} + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       2. sub-neg N/A

          \[\leadsto \frac{\color{blue}{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) + \left(\mathsf{neg}\left(x\right)\right)}}{B} + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       3. associate-*r* N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{1}{3} \cdot {B}^{2}\right) \cdot x} + \left(\mathsf{neg}\left(x\right)\right)}{B} + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       4. mul-1-neg N/A

          \[\leadsto \frac{\left(\frac{1}{3} \cdot {B}^{2}\right) \cdot x + \color{blue}{-1 \cdot x}}{B} + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       5. distribute-rgt-out N/A

          \[\leadsto \frac{\color{blue}{x \cdot \left(\frac{1}{3} \cdot {B}^{2} + -1\right)}}{B} + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{x \cdot \left(\frac{1}{3} \cdot {B}^{2} + -1\right)}}{B} + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       7. lower-fma.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{3}, {B}^{2}, -1\right)}}{B} + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       8. unpow2 N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(\frac{1}{3}, \color{blue}{B \cdot B}, -1\right)}{B} + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       9. lower-*.f64 14.2

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(0.3333333333333333, \color{blue}{B \cdot B}, -1\right)}{B} + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} \]

   11. Applied rewrites 14.2%

       \[\leadsto \color{blue}{\frac{x \cdot \mathsf{fma}\left(0.3333333333333333, B \cdot B, -1\right)}{B}} + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} \]

   12. Taylor expanded in F around -inf

       \[\leadsto \frac{x \cdot \mathsf{fma}\left(\frac{1}{3}, B \cdot B, -1\right)}{B} + \frac{\frac{-1}{6} \cdot {B}^{2} - 1}{\color{blue}{B}} \]
   13. Step-by-step derivation

   14. Applied rewrites 43.6%

       \[\leadsto \frac{x \cdot \mathsf{fma}\left(0.3333333333333333, B \cdot B, -1\right)}{B} + \frac{\mathsf{fma}\left(-0.16666666666666666, B \cdot B, -1\right)}{\color{blue}{B}} \]

   if -1.35000000000000003e154 < F < 4.4999999999999997e143

   1. Initial program 96.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-neg.f64 48.2

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 48.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

   7. Applied rewrites 48.2%

      \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}} \]

   if 4.4999999999999997e143 < F

   1. Initial program 30.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)}}{B} \]

      2. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{\left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}}}{B} \]

      3. associate-*l* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{\left({B}^{2} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)} \cdot \frac{1}{6}}{B} \]

      4. associate-*r* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{{B}^{2} \cdot \left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}\right)}}{B} \]

      5. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}{B}} \]

   5. Applied rewrites 27.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(B \cdot B, 0.16666666666666666 \cdot F, F\right)}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1 + \frac{1}{6} \cdot {B}^{2}}{\color{blue}{B}} \]
   7. Step-by-step derivation

   8. Applied rewrites 65.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{\color{blue}{B}} \]

   9. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x + \frac{-1}{3} \cdot \left({B}^{2} \cdot x\right)}{B}}\right) + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \left(-\color{blue}{\frac{x + \frac{-1}{3} \cdot \left({B}^{2} \cdot x\right)}{B}}\right) + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       2. +-commutative N/A

          \[\leadsto \left(-\frac{\color{blue}{\frac{-1}{3} \cdot \left({B}^{2} \cdot x\right) + x}}{B}\right) + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       3. *-commutative N/A

          \[\leadsto \left(-\frac{\color{blue}{\left({B}^{2} \cdot x\right) \cdot \frac{-1}{3}} + x}{B}\right) + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       4. lower-fma.f64 N/A

          \[\leadsto \left(-\frac{\color{blue}{\mathsf{fma}\left({B}^{2} \cdot x, \frac{-1}{3}, x\right)}}{B}\right) + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       5. lower-*.f64 N/A

          \[\leadsto \left(-\frac{\mathsf{fma}\left(\color{blue}{{B}^{2} \cdot x}, \frac{-1}{3}, x\right)}{B}\right) + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       6. unpow2 N/A

          \[\leadsto \left(-\frac{\mathsf{fma}\left(\color{blue}{\left(B \cdot B\right)} \cdot x, \frac{-1}{3}, x\right)}{B}\right) + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       7. lower-*.f64 56.2

          \[\leadsto \left(-\frac{\mathsf{fma}\left(\color{blue}{\left(B \cdot B\right)} \cdot x, -0.3333333333333333, x\right)}{B}\right) + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} \]

   11. Applied rewrites 56.2%

       \[\leadsto \left(-\color{blue}{\frac{\mathsf{fma}\left(\left(B \cdot B\right) \cdot x, -0.3333333333333333, x\right)}{B}}\right) + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(0.3333333333333333, B \cdot B, -1\right)}{B} + \frac{\mathsf{fma}\left(-0.16666666666666666, B \cdot B, -1\right)}{B}\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{+143}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{\mathsf{fma}\left(\left(B \cdot B\right) \cdot x, -0.3333333333333333, x\right)}{B}\right) + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 52.5% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \mathsf{fma}\left(0.3333333333333333, B \cdot B, -1\right)}{B}\\ \mathbf{if}\;F \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t\_0 + \frac{\mathsf{fma}\left(-0.16666666666666666, B \cdot B, -1\right)}{B}\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{+143}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (* x (fma 0.3333333333333333 (* B B) -1.0)) B)))
   (if (<= F -1.35e+154)
     (+ t_0 (/ (fma -0.16666666666666666 (* B B) -1.0) B))
     (if (<= F 4.5e+143)
       (/ (- (/ F (sqrt (fma x 2.0 (fma F F 2.0)))) x) B)
       (+ t_0 (/ (fma (* B B) 0.16666666666666666 1.0) B))))))

C

double code(double F, double B, double x) {
	double t_0 = (x * fma(0.3333333333333333, (B * B), -1.0)) / B;
	double tmp;
	if (F <= -1.35e+154) {
		tmp = t_0 + (fma(-0.16666666666666666, (B * B), -1.0) / B);
	} else if (F <= 4.5e+143) {
		tmp = ((F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - x) / B;
	} else {
		tmp = t_0 + (fma((B * B), 0.16666666666666666, 1.0) / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(x * fma(0.3333333333333333, Float64(B * B), -1.0)) / B)
	tmp = 0.0
	if (F <= -1.35e+154)
		tmp = Float64(t_0 + Float64(fma(-0.16666666666666666, Float64(B * B), -1.0) / B));
	elseif (F <= 4.5e+143)
		tmp = Float64(Float64(Float64(F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - x) / B);
	else
		tmp = Float64(t_0 + Float64(fma(Float64(B * B), 0.16666666666666666, 1.0) / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(x * N[(0.3333333333333333 * N[(B * B), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]}, If[LessEqual[F, -1.35e+154], N[(t$95$0 + N[(N[(-0.16666666666666666 * N[(B * B), $MachinePrecision] + -1.0), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.5e+143], N[(N[(N[(F / N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(t$95$0 + N[(N[(N[(B * B), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.35000000000000003e154

   1. Initial program 42.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)}}{B} \]

      2. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{\left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}}}{B} \]

      3. associate-*l* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{\left({B}^{2} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)} \cdot \frac{1}{6}}{B} \]

      4. associate-*r* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{{B}^{2} \cdot \left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}\right)}}{B} \]

      5. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}{B}} \]

   5. Applied rewrites 21.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(B \cdot B, 0.16666666666666666 \cdot F, F\right)}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1 + \frac{1}{6} \cdot {B}^{2}}{\color{blue}{B}} \]
   7. Step-by-step derivation

   8. Applied rewrites 31.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{\color{blue}{B}} \]

   9. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B}} + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B}} + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       2. sub-neg N/A

          \[\leadsto \frac{\color{blue}{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) + \left(\mathsf{neg}\left(x\right)\right)}}{B} + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       3. associate-*r* N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{1}{3} \cdot {B}^{2}\right) \cdot x} + \left(\mathsf{neg}\left(x\right)\right)}{B} + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       4. mul-1-neg N/A

          \[\leadsto \frac{\left(\frac{1}{3} \cdot {B}^{2}\right) \cdot x + \color{blue}{-1 \cdot x}}{B} + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       5. distribute-rgt-out N/A

          \[\leadsto \frac{\color{blue}{x \cdot \left(\frac{1}{3} \cdot {B}^{2} + -1\right)}}{B} + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{x \cdot \left(\frac{1}{3} \cdot {B}^{2} + -1\right)}}{B} + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       7. lower-fma.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{3}, {B}^{2}, -1\right)}}{B} + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       8. unpow2 N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(\frac{1}{3}, \color{blue}{B \cdot B}, -1\right)}{B} + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       9. lower-*.f64 14.2

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(0.3333333333333333, \color{blue}{B \cdot B}, -1\right)}{B} + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} \]

   11. Applied rewrites 14.2%

       \[\leadsto \color{blue}{\frac{x \cdot \mathsf{fma}\left(0.3333333333333333, B \cdot B, -1\right)}{B}} + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} \]

   12. Taylor expanded in F around -inf

       \[\leadsto \frac{x \cdot \mathsf{fma}\left(\frac{1}{3}, B \cdot B, -1\right)}{B} + \frac{\frac{-1}{6} \cdot {B}^{2} - 1}{\color{blue}{B}} \]
   13. Step-by-step derivation

   14. Applied rewrites 43.6%

       \[\leadsto \frac{x \cdot \mathsf{fma}\left(0.3333333333333333, B \cdot B, -1\right)}{B} + \frac{\mathsf{fma}\left(-0.16666666666666666, B \cdot B, -1\right)}{\color{blue}{B}} \]

   if -1.35000000000000003e154 < F < 4.4999999999999997e143

   1. Initial program 96.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-neg.f64 48.2

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 48.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

   7. Applied rewrites 48.2%

      \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}} \]

   if 4.4999999999999997e143 < F

   1. Initial program 30.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)}}{B} \]

      2. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{\left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}}}{B} \]

      3. associate-*l* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{\left({B}^{2} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)} \cdot \frac{1}{6}}{B} \]

      4. associate-*r* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{{B}^{2} \cdot \left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}\right)}}{B} \]

      5. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}{B}} \]

   5. Applied rewrites 27.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(B \cdot B, 0.16666666666666666 \cdot F, F\right)}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1 + \frac{1}{6} \cdot {B}^{2}}{\color{blue}{B}} \]
   7. Step-by-step derivation

   8. Applied rewrites 65.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{\color{blue}{B}} \]

   9. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B}} + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B}} + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       2. sub-neg N/A

          \[\leadsto \frac{\color{blue}{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) + \left(\mathsf{neg}\left(x\right)\right)}}{B} + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       3. associate-*r* N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{1}{3} \cdot {B}^{2}\right) \cdot x} + \left(\mathsf{neg}\left(x\right)\right)}{B} + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       4. mul-1-neg N/A

          \[\leadsto \frac{\left(\frac{1}{3} \cdot {B}^{2}\right) \cdot x + \color{blue}{-1 \cdot x}}{B} + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       5. distribute-rgt-out N/A

          \[\leadsto \frac{\color{blue}{x \cdot \left(\frac{1}{3} \cdot {B}^{2} + -1\right)}}{B} + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{x \cdot \left(\frac{1}{3} \cdot {B}^{2} + -1\right)}}{B} + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       7. lower-fma.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{3}, {B}^{2}, -1\right)}}{B} + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       8. unpow2 N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(\frac{1}{3}, \color{blue}{B \cdot B}, -1\right)}{B} + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       9. lower-*.f64 56.2

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(0.3333333333333333, \color{blue}{B \cdot B}, -1\right)}{B} + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} \]

   11. Applied rewrites 56.2%

       \[\leadsto \color{blue}{\frac{x \cdot \mathsf{fma}\left(0.3333333333333333, B \cdot B, -1\right)}{B}} + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(0.3333333333333333, B \cdot B, -1\right)}{B} + \frac{\mathsf{fma}\left(-0.16666666666666666, B \cdot B, -1\right)}{B}\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{+143}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(0.3333333333333333, B \cdot B, -1\right)}{B} + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 52.5% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(0.3333333333333333, B \cdot B, -1\right)}{B} + \frac{\mathsf{fma}\left(-0.16666666666666666, B \cdot B, -1\right)}{B}\\ \mathbf{elif}\;F \leq 10^{+45}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.35e+154)
   (+
    (/ (* x (fma 0.3333333333333333 (* B B) -1.0)) B)
    (/ (fma -0.16666666666666666 (* B B) -1.0) B))
   (if (<= F 1e+45)
     (/ (- (/ F (sqrt (fma x 2.0 (fma F F 2.0)))) x) B)
     (+ (- (/ x B)) (/ (fma (* B B) 0.16666666666666666 1.0) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.35e+154) {
		tmp = ((x * fma(0.3333333333333333, (B * B), -1.0)) / B) + (fma(-0.16666666666666666, (B * B), -1.0) / B);
	} else if (F <= 1e+45) {
		tmp = ((F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - x) / B;
	} else {
		tmp = -(x / B) + (fma((B * B), 0.16666666666666666, 1.0) / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.35e+154)
		tmp = Float64(Float64(Float64(x * fma(0.3333333333333333, Float64(B * B), -1.0)) / B) + Float64(fma(-0.16666666666666666, Float64(B * B), -1.0) / B));
	elseif (F <= 1e+45)
		tmp = Float64(Float64(Float64(F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - x) / B);
	else
		tmp = Float64(Float64(-Float64(x / B)) + Float64(fma(Float64(B * B), 0.16666666666666666, 1.0) / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.35e+154], N[(N[(N[(x * N[(0.3333333333333333 * N[(B * B), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision] + N[(N[(-0.16666666666666666 * N[(B * B), $MachinePrecision] + -1.0), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1e+45], N[(N[(N[(F / N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[((-N[(x / B), $MachinePrecision]) + N[(N[(N[(B * B), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.35000000000000003e154

   1. Initial program 42.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)}}{B} \]

      2. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{\left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}}}{B} \]

      3. associate-*l* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{\left({B}^{2} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)} \cdot \frac{1}{6}}{B} \]

      4. associate-*r* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{{B}^{2} \cdot \left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}\right)}}{B} \]

      5. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}{B}} \]

   5. Applied rewrites 21.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(B \cdot B, 0.16666666666666666 \cdot F, F\right)}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1 + \frac{1}{6} \cdot {B}^{2}}{\color{blue}{B}} \]
   7. Step-by-step derivation

   8. Applied rewrites 31.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{\color{blue}{B}} \]

   9. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B}} + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B}} + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       2. sub-neg N/A

          \[\leadsto \frac{\color{blue}{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) + \left(\mathsf{neg}\left(x\right)\right)}}{B} + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       3. associate-*r* N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{1}{3} \cdot {B}^{2}\right) \cdot x} + \left(\mathsf{neg}\left(x\right)\right)}{B} + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       4. mul-1-neg N/A

          \[\leadsto \frac{\left(\frac{1}{3} \cdot {B}^{2}\right) \cdot x + \color{blue}{-1 \cdot x}}{B} + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       5. distribute-rgt-out N/A

          \[\leadsto \frac{\color{blue}{x \cdot \left(\frac{1}{3} \cdot {B}^{2} + -1\right)}}{B} + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{x \cdot \left(\frac{1}{3} \cdot {B}^{2} + -1\right)}}{B} + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       7. lower-fma.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{3}, {B}^{2}, -1\right)}}{B} + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       8. unpow2 N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(\frac{1}{3}, \color{blue}{B \cdot B}, -1\right)}{B} + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]

       9. lower-*.f64 14.2

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(0.3333333333333333, \color{blue}{B \cdot B}, -1\right)}{B} + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} \]

   11. Applied rewrites 14.2%

       \[\leadsto \color{blue}{\frac{x \cdot \mathsf{fma}\left(0.3333333333333333, B \cdot B, -1\right)}{B}} + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} \]

   12. Taylor expanded in F around -inf

       \[\leadsto \frac{x \cdot \mathsf{fma}\left(\frac{1}{3}, B \cdot B, -1\right)}{B} + \frac{\frac{-1}{6} \cdot {B}^{2} - 1}{\color{blue}{B}} \]
   13. Step-by-step derivation

   14. Applied rewrites 43.6%

       \[\leadsto \frac{x \cdot \mathsf{fma}\left(0.3333333333333333, B \cdot B, -1\right)}{B} + \frac{\mathsf{fma}\left(-0.16666666666666666, B \cdot B, -1\right)}{\color{blue}{B}} \]

   if -1.35000000000000003e154 < F < 9.9999999999999993e44

   1. Initial program 96.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-neg.f64 47.3

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 47.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

   7. Applied rewrites 47.4%

      \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}} \]

   if 9.9999999999999993e44 < F

   1. Initial program 50.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)}}{B} \]

      2. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{\left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}}}{B} \]

      3. associate-*l* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{\left({B}^{2} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)} \cdot \frac{1}{6}}{B} \]

      4. associate-*r* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{{B}^{2} \cdot \left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}\right)}}{B} \]

      5. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}{B}} \]

   5. Applied rewrites 36.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(B \cdot B, 0.16666666666666666 \cdot F, F\right)}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1 + \frac{1}{6} \cdot {B}^{2}}{\color{blue}{B}} \]
   7. Step-by-step derivation

   8. Applied rewrites 63.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{\color{blue}{B}} \]

   9. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]
   10. Step-by-step derivation

       1. lower-/.f64 56.2

          \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} \]

   11. Applied rewrites 56.2%

       \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(0.3333333333333333, B \cdot B, -1\right)}{B} + \frac{\mathsf{fma}\left(-0.16666666666666666, B \cdot B, -1\right)}{B}\\ \mathbf{elif}\;F \leq 10^{+45}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 52.1% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -4 \cdot 10^{-13}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{0.5}{F}, \frac{\mathsf{fma}\left(2, x, 2\right)}{F}, -1 - x\right)}{B}\\ \mathbf{elif}\;F \leq 10^{+45}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4e-13)
   (/ (fma (/ 0.5 F) (/ (fma 2.0 x 2.0) F) (- -1.0 x)) B)
   (if (<= F 1e+45)
     (/ (- (/ F (sqrt (fma x 2.0 (fma F F 2.0)))) x) B)
     (+ (- (/ x B)) (/ (fma (* B B) 0.16666666666666666 1.0) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4e-13) {
		tmp = fma((0.5 / F), (fma(2.0, x, 2.0) / F), (-1.0 - x)) / B;
	} else if (F <= 1e+45) {
		tmp = ((F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - x) / B;
	} else {
		tmp = -(x / B) + (fma((B * B), 0.16666666666666666, 1.0) / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -4e-13)
		tmp = Float64(fma(Float64(0.5 / F), Float64(fma(2.0, x, 2.0) / F), Float64(-1.0 - x)) / B);
	elseif (F <= 1e+45)
		tmp = Float64(Float64(Float64(F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - x) / B);
	else
		tmp = Float64(Float64(-Float64(x / B)) + Float64(fma(Float64(B * B), 0.16666666666666666, 1.0) / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -4e-13], N[(N[(N[(0.5 / F), $MachinePrecision] * N[(N[(2.0 * x + 2.0), $MachinePrecision] / F), $MachinePrecision] + N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1e+45], N[(N[(N[(F / N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[((-N[(x / B), $MachinePrecision]) + N[(N[(N[(B * B), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.0000000000000001e-13

   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

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   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. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-neg.f64 33.1

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 33.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}} - \left(1 + x\right)}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 45.2%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{0.5}{F}, \frac{\mathsf{fma}\left(2, x, 2\right)}{F}, -1 - x\right)}{B} \]

   if -4.0000000000000001e-13 < F < 9.9999999999999993e44

   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. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-neg.f64 47.7

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 47.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

   7. Applied rewrites 47.7%

      \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}} \]

   if 9.9999999999999993e44 < F

   1. Initial program 50.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)}}{B} \]

      2. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{\left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}}}{B} \]

      3. associate-*l* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{\left({B}^{2} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)} \cdot \frac{1}{6}}{B} \]

      4. associate-*r* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{{B}^{2} \cdot \left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}\right)}}{B} \]

      5. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}{B}} \]

   5. Applied rewrites 36.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(B \cdot B, 0.16666666666666666 \cdot F, F\right)}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1 + \frac{1}{6} \cdot {B}^{2}}{\color{blue}{B}} \]
   7. Step-by-step derivation

   8. Applied rewrites 63.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{\color{blue}{B}} \]

   9. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]
   10. Step-by-step derivation

       1. lower-/.f64 56.2

          \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} \]

   11. Applied rewrites 56.2%

       \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4 \cdot 10^{-13}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{0.5}{F}, \frac{\mathsf{fma}\left(2, x, 2\right)}{F}, -1 - x\right)}{B}\\ \mathbf{elif}\;F \leq 10^{+45}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 44.8% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.6 \cdot 10^{-40}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 4.05 \cdot 10^{-47}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.6e-40)
   (/ (- -1.0 x) B)
   (if (<= F 4.05e-47)
     (/ (- x) B)
     (+ (- (/ x B)) (/ (fma (* B B) 0.16666666666666666 1.0) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.6e-40) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 4.05e-47) {
		tmp = -x / B;
	} else {
		tmp = -(x / B) + (fma((B * B), 0.16666666666666666, 1.0) / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.6e-40)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 4.05e-47)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(-Float64(x / B)) + Float64(fma(Float64(B * B), 0.16666666666666666, 1.0) / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.6e-40], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 4.05e-47], N[((-x) / B), $MachinePrecision], N[((-N[(x / B), $MachinePrecision]) + N[(N[(N[(B * B), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.6000000000000001e-40

   1. Initial program 72.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-neg.f64 34.1

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 34.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 \cdot \left(1 + x\right)}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 44.0%

      \[\leadsto \frac{-1 - x}{B} \]

   if -2.6000000000000001e-40 < F < 4.0500000000000002e-47

   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. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-neg.f64 48.4

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 48.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 42.7%

      \[\leadsto \frac{-x}{B} \]

   if 4.0500000000000002e-47 < F

   1. Initial program 65.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)}}{B} \]

      2. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{\left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}}}{B} \]

      3. associate-*l* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{\left({B}^{2} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)} \cdot \frac{1}{6}}{B} \]

      4. associate-*r* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + \color{blue}{{B}^{2} \cdot \left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}\right)}}{B} \]

      5. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}{B}} \]

   5. Applied rewrites 40.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(B \cdot B, 0.16666666666666666 \cdot F, F\right)}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1 + \frac{1}{6} \cdot {B}^{2}}{\color{blue}{B}} \]
   7. Step-by-step derivation

   8. Applied rewrites 52.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{\color{blue}{B}} \]

   9. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} \]
   10. Step-by-step derivation

       1. lower-/.f64 47.1

          \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} \]

   11. Applied rewrites 47.1%

       \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.6 \cdot 10^{-40}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 4.05 \cdot 10^{-47}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 44.8% accurate, 13.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.6 \cdot 10^{-40}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 4.05 \cdot 10^{-47}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.6e-40)
   (/ (- -1.0 x) B)
   (if (<= F 4.05e-47) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.6e-40) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 4.05e-47) {
		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 <= (-2.6d-40)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 4.05d-47) 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 <= -2.6e-40) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 4.05e-47) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.6e-40:
		tmp = (-1.0 - x) / B
	elif F <= 4.05e-47:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.6e-40)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 4.05e-47)
		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 <= -2.6e-40)
		tmp = (-1.0 - x) / B;
	elseif (F <= 4.05e-47)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.6e-40], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 4.05e-47], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.6000000000000001e-40

   1. Initial program 72.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-neg.f64 34.1

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 34.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 \cdot \left(1 + x\right)}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 44.0%

      \[\leadsto \frac{-1 - x}{B} \]

   if -2.6000000000000001e-40 < F < 4.0500000000000002e-47

   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. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-neg.f64 48.4

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 48.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 42.7%

      \[\leadsto \frac{-x}{B} \]

   if 4.0500000000000002e-47 < F

   1. Initial program 65.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-neg.f64 39.0

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 39.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 46.6%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.6 \cdot 10^{-40}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 4.05 \cdot 10^{-47}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 37.7% accurate, 17.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.6 \cdot 10^{-40}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.6e-40) (/ (- -1.0 x) B) (/ (- x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.6e-40) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-2.6d-40)) then
        tmp = ((-1.0d0) - x) / b
    else
        tmp = -x / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.6e-40) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.6e-40:
		tmp = (-1.0 - x) / B
	else:
		tmp = -x / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.6e-40)
		tmp = Float64(Float64(-1.0 - x) / B);
	else
		tmp = Float64(Float64(-x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.6e-40)
		tmp = (-1.0 - x) / B;
	else
		tmp = -x / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.6e-40], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[((-x) / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -2.6000000000000001e-40

   1. Initial program 72.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-neg.f64 34.1

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 34.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 \cdot \left(1 + x\right)}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 44.0%

      \[\leadsto \frac{-1 - x}{B} \]

   if -2.6000000000000001e-40 < 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. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. lower-neg.f64 44.4

          \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

   5. Applied rewrites 44.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 33.7%

      \[\leadsto \frac{-x}{B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.6 \cdot 10^{-40}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 28: 30.0% accurate, 26.3× speedup

Math

\[\begin{array}{l} \\ \frac{-x}{B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (/ (- x) B))

C

double code(double F, double B, double x) {
	return -x / B;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = -x / b
end function

Java

public static double code(double F, double B, double x) {
	return -x / B;
}

Python

def code(F, B, x):
	return -x / B

Julia

function code(F, B, x)
	return Float64(Float64(-x) / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = -x / B;
end

Wolfram

code[F_, B_, x_] := N[((-x) / B), $MachinePrecision]

Derivation

1. Initial program 80.6%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
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. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}}{B} \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

   7. associate-+r+ N/A

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

   8. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

   9. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

   10. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

   11. +-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

   12. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}}, F, \mathsf{neg}\left(x\right)\right)}{B} \]

   13. lower-neg.f64 40.9

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, \color{blue}{-x}\right)}{B} \]

5. Applied rewrites 40.9%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)}{B}} \]

6. Taylor expanded in F around 0

   \[\leadsto \frac{-1 \cdot x}{B} \]
7. Step-by-step derivation

8. Applied rewrites 28.5%

   \[\leadsto \frac{-x}{B} \]

9. Final simplification 28.5%

   \[\leadsto \frac{-x}{B} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024318 
(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))))))