VandenBroeck and Keller, Equation (23)

Percentage Accurate: 77.0% → 99.7%

Time: 12.2s

Alternatives: 19

Speedup: 1.4×

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: 77.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2 \cdot 10^{+22}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 150000000:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(F, F, 2\right)}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -2e+22)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 150000000.0)
       (- (/ F (* (sin B) (sqrt (fma F F 2.0)))) 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 <= -2e+22) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 150000000.0) {
		tmp = (F / (sin(B) * sqrt(fma(F, F, 2.0)))) - 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 <= -2e+22)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 150000000.0)
		tmp = Float64(Float64(F / Float64(sin(B) * sqrt(fma(F, F, 2.0)))) - 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, -2e+22], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 150000000.0], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(F * F + 2.0), $MachinePrecision]], $MachinePrecision]), $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 < -2e22

   1. Initial program 47.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 68.1%

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

   6. Applied rewrites 68.1%

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

   7. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 99.7

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

   9. Applied rewrites 99.7%

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

   if -2e22 < F < 1.5e8

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around 0

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

      1. lower-sqrt.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. lower-fma.f64 99.7

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

   9. Applied rewrites 99.7%

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

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/l/ N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 99.7

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

   11. Applied rewrites 99.7%

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

   if 1.5e8 < F

   1. Initial program 62.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 79.6%

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

   6. Applied rewrites 79.7%

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

   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. Add Preprocessing

Alternative 2: 91.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -8 \cdot 10^{-37}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 280000:\\ \;\;\;\;\sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}} \cdot \frac{F}{B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -8e-37)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 280000.0)
       (- (* (sqrt (pow (fma F F (fma 2.0 x 2.0)) -1.0)) (/ F 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 <= -8e-37) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 280000.0) {
		tmp = (sqrt(pow(fma(F, F, fma(2.0, x, 2.0)), -1.0)) * (F / 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 <= -8e-37)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 280000.0)
		tmp = Float64(Float64(sqrt((fma(F, F, fma(2.0, x, 2.0)) ^ -1.0)) * Float64(F / 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, -8e-37], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 280000.0], N[(N[(N[Sqrt[N[Power[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * N[(F / 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 < -8.00000000000000053e-37

   1. Initial program 51.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 70.3%

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

   6. Applied rewrites 70.4%

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

   7. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 96.6

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

   9. Applied rewrites 96.6%

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

   if -8.00000000000000053e-37 < F < 2.8e5

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. unpow2 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 87.5

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

   9. Applied rewrites 87.5%

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

   if 2.8e5 < F

   1. Initial program 62.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 79.6%

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

   6. Applied rewrites 79.7%

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

   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 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -8 \cdot 10^{-37}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 280000:\\ \;\;\;\;\sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -8 \cdot 10^{-37}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1.15 \cdot 10^{-164}:\\ \;\;\;\;\sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}} \cdot \frac{F}{B} - t\_0\\ \mathbf{elif}\;F \leq 6 \cdot 10^{+153}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{-1}{B}, \frac{-x}{\tan B}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -8e-37)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.15e-164)
       (- (* (sqrt (pow (fma F F (fma 2.0 x 2.0)) -1.0)) (/ F B)) t_0)
       (if (<= F 6e+153)
         (- (/ (/ F (sqrt (fma F F 2.0))) (sin B)) (/ x B))
         (fma -1.0 (/ -1.0 B) (/ (- x) (tan B))))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -8e-37) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.15e-164) {
		tmp = (sqrt(pow(fma(F, F, fma(2.0, x, 2.0)), -1.0)) * (F / B)) - t_0;
	} else if (F <= 6e+153) {
		tmp = ((F / sqrt(fma(F, F, 2.0))) / sin(B)) - (x / B);
	} else {
		tmp = fma(-1.0, (-1.0 / B), (-x / tan(B)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -8e-37)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.15e-164)
		tmp = Float64(Float64(sqrt((fma(F, F, fma(2.0, x, 2.0)) ^ -1.0)) * Float64(F / B)) - t_0);
	elseif (F <= 6e+153)
		tmp = Float64(Float64(Float64(F / sqrt(fma(F, F, 2.0))) / sin(B)) - Float64(x / B));
	else
		tmp = fma(-1.0, Float64(-1.0 / B), Float64(Float64(-x) / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -8e-37], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.15e-164], N[(N[(N[Sqrt[N[Power[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 6e+153], N[(N[(N[(F / N[Sqrt[N[(F * F + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(-1.0 / B), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -8.00000000000000053e-37

   1. Initial program 51.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 70.3%

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

   6. Applied rewrites 70.4%

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

   7. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 96.6

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

   9. Applied rewrites 96.6%

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

   if -8.00000000000000053e-37 < F < 1.14999999999999993e-164

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. unpow2 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 91.2

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

   9. Applied rewrites 91.2%

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

   if 1.14999999999999993e-164 < F < 6.00000000000000037e153

   1. Initial program 86.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 99.5%

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around 0

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

      1. lower-sqrt.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. lower-fma.f64 99.7

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

   9. Applied rewrites 99.7%

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

   10. Taylor expanded in B around 0

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

       1. lower-/.f64 89.6

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

   12. Applied rewrites 89.6%

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

   if 6.00000000000000037e153 < F

   1. Initial program 48.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 56.7%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 56.7

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

   7. Applied rewrites 56.7%

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

   8. Taylor expanded in F around inf

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

   10. Applied rewrites 84.8%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -8 \cdot 10^{-37}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.15 \cdot 10^{-164}:\\ \;\;\;\;\sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 6 \cdot 10^{+153}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{-1}{B}, \frac{-x}{\tan B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.9 \cdot 10^{+63}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.15 \cdot 10^{-164}:\\ \;\;\;\;\sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 6 \cdot 10^{+153}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{-1}{B}, \frac{-x}{\tan B}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.9e+63)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 1.15e-164)
     (- (* (sqrt (pow (fma F F (fma 2.0 x 2.0)) -1.0)) (/ F B)) (/ x (tan B)))
     (if (<= F 6e+153)
       (- (/ (/ F (sqrt (fma F F 2.0))) (sin B)) (/ x B))
       (fma -1.0 (/ -1.0 B) (/ (- x) (tan B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.9e+63) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 1.15e-164) {
		tmp = (sqrt(pow(fma(F, F, fma(2.0, x, 2.0)), -1.0)) * (F / B)) - (x / tan(B));
	} else if (F <= 6e+153) {
		tmp = ((F / sqrt(fma(F, F, 2.0))) / sin(B)) - (x / B);
	} else {
		tmp = fma(-1.0, (-1.0 / B), (-x / tan(B)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.9e+63)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 1.15e-164)
		tmp = Float64(Float64(sqrt((fma(F, F, fma(2.0, x, 2.0)) ^ -1.0)) * Float64(F / B)) - Float64(x / tan(B)));
	elseif (F <= 6e+153)
		tmp = Float64(Float64(Float64(F / sqrt(fma(F, F, 2.0))) / sin(B)) - Float64(x / B));
	else
		tmp = fma(-1.0, Float64(-1.0 / B), Float64(Float64(-x) / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.9e+63], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.15e-164], N[(N[(N[Sqrt[N[Power[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 6e+153], N[(N[(N[(F / N[Sqrt[N[(F * F + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(-1.0 / B), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.9000000000000001e63

   1. Initial program 43.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 65.6%

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

   6. Applied rewrites 65.6%

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

   7. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 99.8

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

   9. Applied rewrites 99.8%

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

   10. Taylor expanded in B around 0

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

       1. lower-/.f64 79.1

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

   12. Applied rewrites 79.1%

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

   if -1.9000000000000001e63 < F < 1.14999999999999993e-164

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. unpow2 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 87.1

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

   9. Applied rewrites 87.1%

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

   if 1.14999999999999993e-164 < F < 6.00000000000000037e153

   1. Initial program 86.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 99.5%

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around 0

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

      1. lower-sqrt.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. lower-fma.f64 99.7

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

   9. Applied rewrites 99.7%

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

   10. Taylor expanded in B around 0

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

       1. lower-/.f64 89.6

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

   12. Applied rewrites 89.6%

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

   if 6.00000000000000037e153 < F

   1. Initial program 48.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 56.7%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 56.7

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

   7. Applied rewrites 56.7%

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

   8. Taylor expanded in F around inf

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

   10. Applied rewrites 84.8%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.9 \cdot 10^{+63}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.15 \cdot 10^{-164}:\\ \;\;\;\;\sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 6 \cdot 10^{+153}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{-1}{B}, \frac{-x}{\tan B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 9.5 \cdot 10^{-272}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(-0.16666666666666666, B \cdot B, 1\right) \cdot B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 6 \cdot 10^{+153}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{-1}{B}, \frac{-x}{\tan B}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F 9.5e-272)
   (- (/ -1.0 (* (fma -0.16666666666666666 (* B B) 1.0) B)) (/ x (tan B)))
   (if (<= F 6e+153)
     (- (/ (/ F (sqrt (fma F F 2.0))) (sin B)) (/ x B))
     (fma -1.0 (/ -1.0 B) (/ (- x) (tan B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= 9.5e-272) {
		tmp = (-1.0 / (fma(-0.16666666666666666, (B * B), 1.0) * B)) - (x / tan(B));
	} else if (F <= 6e+153) {
		tmp = ((F / sqrt(fma(F, F, 2.0))) / sin(B)) - (x / B);
	} else {
		tmp = fma(-1.0, (-1.0 / B), (-x / tan(B)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= 9.5e-272)
		tmp = Float64(Float64(-1.0 / Float64(fma(-0.16666666666666666, Float64(B * B), 1.0) * B)) - Float64(x / tan(B)));
	elseif (F <= 6e+153)
		tmp = Float64(Float64(Float64(F / sqrt(fma(F, F, 2.0))) / sin(B)) - Float64(x / B));
	else
		tmp = fma(-1.0, Float64(-1.0 / B), Float64(Float64(-x) / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, 9.5e-272], N[(N[(-1.0 / N[(N[(-0.16666666666666666 * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 6e+153], N[(N[(N[(F / N[Sqrt[N[(F * F + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(-1.0 / B), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < 9.50000000000000024e-272

   1. Initial program 71.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 82.8%

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

   6. Applied rewrites 82.9%

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

   7. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 71.8

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

   9. Applied rewrites 71.8%

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

   10. Taylor expanded in B around 0

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

   12. Applied rewrites 70.9%

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

   if 9.50000000000000024e-272 < F < 6.00000000000000037e153

   1. Initial program 89.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in x around 0

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

      1. lower-sqrt.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. lower-fma.f64 99.8

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

   9. Applied rewrites 99.8%

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

   10. Taylor expanded in B around 0

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

       1. lower-/.f64 83.6

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

   12. Applied rewrites 83.6%

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

   if 6.00000000000000037e153 < F

   1. Initial program 48.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 56.7%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 56.7

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

   7. Applied rewrites 56.7%

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

   8. Taylor expanded in F around inf

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

   10. Applied rewrites 84.8%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 9.5 \cdot 10^{-272}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(-0.16666666666666666, B \cdot B, 1\right) \cdot B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 6 \cdot 10^{+153}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{-1}{B}, \frac{-x}{\tan B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 9.5 \cdot 10^{-272}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(-0.16666666666666666, B \cdot B, 1\right) \cdot B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.45 \cdot 10^{-11}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{F}{\sin B} \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{-1}{B}, \frac{-x}{\tan B}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F 9.5e-272)
   (- (/ -1.0 (* (fma -0.16666666666666666 (* B B) 1.0) B)) (/ x (tan B)))
   (if (<= F 2.45e-11)
     (+ (- (/ x B)) (* (/ F (sin B)) (sqrt 0.5)))
     (fma -1.0 (/ -1.0 B) (/ (- x) (tan B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= 9.5e-272) {
		tmp = (-1.0 / (fma(-0.16666666666666666, (B * B), 1.0) * B)) - (x / tan(B));
	} else if (F <= 2.45e-11) {
		tmp = -(x / B) + ((F / sin(B)) * sqrt(0.5));
	} else {
		tmp = fma(-1.0, (-1.0 / B), (-x / tan(B)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= 9.5e-272)
		tmp = Float64(Float64(-1.0 / Float64(fma(-0.16666666666666666, Float64(B * B), 1.0) * B)) - Float64(x / tan(B)));
	elseif (F <= 2.45e-11)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(Float64(F / sin(B)) * sqrt(0.5)));
	else
		tmp = fma(-1.0, Float64(-1.0 / B), Float64(Float64(-x) / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, 9.5e-272], N[(N[(-1.0 / N[(N[(-0.16666666666666666 * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.45e-11], N[((-N[(x / B), $MachinePrecision]) + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(-1.0 / B), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < 9.50000000000000024e-272

   1. Initial program 71.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 82.8%

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

   6. Applied rewrites 82.9%

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

   7. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 71.8

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

   9. Applied rewrites 71.8%

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

   10. Taylor expanded in B around 0

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

   12. Applied rewrites 70.9%

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

   if 9.50000000000000024e-272 < F < 2.4499999999999999e-11

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in B around 0

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

      1. lower-/.f64 79.5

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

   8. Applied rewrites 79.5%

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

   9. Taylor expanded in F around 0

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

   11. Applied rewrites 79.5%

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

   if 2.4499999999999999e-11 < F

   1. Initial program 63.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 80.6%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 64.2

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

   7. Applied rewrites 64.2%

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

   8. Taylor expanded in F around inf

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

   10. Applied rewrites 74.9%

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

Alternative 7: 50.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.0235:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \frac{\frac{\mathsf{fma}\left(2, x, 2\right)}{F}}{F}, -1 - x\right)}{B}\\ \mathbf{elif}\;F \leq 2.7 \cdot 10^{+181}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}}, F, -x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.0235)
   (/ (fma 0.5 (/ (/ (fma 2.0 x 2.0) F) F) (- -1.0 x)) B)
   (if (<= F 2.7e+181)
     (/ (fma (sqrt (pow (fma F F (fma 2.0 x 2.0)) -1.0)) F (- x)) B)
     (/
      (- (fma (fma 0.3333333333333333 x 0.16666666666666666) (* B B) 1.0) x)
      B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.0235) {
		tmp = fma(0.5, ((fma(2.0, x, 2.0) / F) / F), (-1.0 - x)) / B;
	} else if (F <= 2.7e+181) {
		tmp = fma(sqrt(pow(fma(F, F, fma(2.0, x, 2.0)), -1.0)), F, -x) / B;
	} else {
		tmp = (fma(fma(0.3333333333333333, x, 0.16666666666666666), (B * B), 1.0) - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.0235)
		tmp = Float64(fma(0.5, Float64(Float64(fma(2.0, x, 2.0) / F) / F), Float64(-1.0 - x)) / B);
	elseif (F <= 2.7e+181)
		tmp = Float64(fma(sqrt((fma(F, F, fma(2.0, x, 2.0)) ^ -1.0)), F, Float64(-x)) / B);
	else
		tmp = Float64(Float64(fma(fma(0.3333333333333333, x, 0.16666666666666666), Float64(B * B), 1.0) - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.0235], N[(N[(0.5 * N[(N[(N[(2.0 * x + 2.0), $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision] + N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 2.7e+181], N[(N[(N[Sqrt[N[Power[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * F + (-x)), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(N[(0.3333333333333333 * x + 0.16666666666666666), $MachinePrecision] * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.0235

   1. Initial program 48.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 36.7

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

   5. Applied rewrites 36.7%

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

   6. Taylor expanded in 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 52.3%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{0.5}{F}, \frac{\mathsf{fma}\left(2, x, 2\right)}{F}, -1 - x\right)}{B} \]
   9. Step-by-step derivation

   10. Applied rewrites 52.3%

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

   if -0.0235 < F < 2.70000000000000007e181

   1. Initial program 93.8%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. sub-neg N/A

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

      3. *-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 50.9

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

   5. Applied rewrites 50.9%

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

   if 2.70000000000000007e181 < F

   1. Initial program 41.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 51.2%

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

   5. Taylor expanded in B around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 15.6%

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

   8. Taylor expanded in F around inf

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

   10. Applied rewrites 48.3%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.0235:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \frac{\frac{\mathsf{fma}\left(2, x, 2\right)}{F}}{F}, -1 - x\right)}{B}\\ \mathbf{elif}\;F \leq 2.7 \cdot 10^{+181}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}}, F, -x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 42.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.1 \cdot 10^{-94}:\\ \;\;\;\;\frac{{\left(\frac{-1 + x}{1 - x \cdot x}\right)}^{-1}}{B}\\ \mathbf{elif}\;F \leq 4 \cdot 10^{-79}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 550000:\\ \;\;\;\;\frac{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot F}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.1e-94)
   (/ (pow (/ (+ -1.0 x) (- 1.0 (* x x))) -1.0) B)
   (if (<= F 4e-79)
     (/ (- x) B)
     (if (<= F 550000.0)
       (/ (* (sqrt (pow (fma F F 2.0) -1.0)) F) B)
       (/
        (- (fma (fma 0.3333333333333333 x 0.16666666666666666) (* B B) 1.0) x)
        B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.1e-94) {
		tmp = pow(((-1.0 + x) / (1.0 - (x * x))), -1.0) / B;
	} else if (F <= 4e-79) {
		tmp = -x / B;
	} else if (F <= 550000.0) {
		tmp = (sqrt(pow(fma(F, F, 2.0), -1.0)) * F) / B;
	} else {
		tmp = (fma(fma(0.3333333333333333, x, 0.16666666666666666), (B * B), 1.0) - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.1e-94)
		tmp = Float64((Float64(Float64(-1.0 + x) / Float64(1.0 - Float64(x * x))) ^ -1.0) / B);
	elseif (F <= 4e-79)
		tmp = Float64(Float64(-x) / B);
	elseif (F <= 550000.0)
		tmp = Float64(Float64(sqrt((fma(F, F, 2.0) ^ -1.0)) * F) / B);
	else
		tmp = Float64(Float64(fma(fma(0.3333333333333333, x, 0.16666666666666666), Float64(B * B), 1.0) - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.1e-94], N[(N[Power[N[(N[(-1.0 + x), $MachinePrecision] / N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 4e-79], N[((-x) / B), $MachinePrecision], If[LessEqual[F, 550000.0], N[(N[(N[Sqrt[N[Power[N[(F * F + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * F), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(N[(0.3333333333333333 * x + 0.16666666666666666), $MachinePrecision] * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.1000000000000001e-94

   1. Initial program 56.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 36.8

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

   5. Applied rewrites 36.8%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 46.2%

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

   10. Applied rewrites 46.2%

       \[\leadsto \frac{\frac{1}{\frac{-1 + x}{1 - x \cdot x}}}{B} \]

   if -2.1000000000000001e-94 < F < 4e-79

   1. Initial program 99.6%

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

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   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 50.4

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

   5. Applied rewrites 50.4%

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

   6. Taylor expanded in F around 0

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

   8. Applied rewrites 43.8%

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

   if 4e-79 < F < 5.5e5

   1. Initial program 99.3%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. 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%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 45.7%

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

   if 5.5e5 < F

   1. Initial program 62.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 79.6%

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

   5. Taylor expanded in B around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 38.5%

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

   8. Taylor expanded in F around inf

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

   10. Applied rewrites 51.6%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B} \]
3. Recombined 4 regimes into one program.

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.1 \cdot 10^{-94}:\\ \;\;\;\;\frac{{\left(\frac{-1 + x}{1 - x \cdot x}\right)}^{-1}}{B}\\ \mathbf{elif}\;F \leq 4 \cdot 10^{-79}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 550000:\\ \;\;\;\;\frac{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot F}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 43.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.1 \cdot 10^{-94}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 4 \cdot 10^{-79}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 550000:\\ \;\;\;\;\frac{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot F}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.1e-94)
   (/ (- -1.0 x) B)
   (if (<= F 4e-79)
     (/ (- x) B)
     (if (<= F 550000.0)
       (/ (* (sqrt (pow (fma F F 2.0) -1.0)) F) B)
       (/
        (- (fma (fma 0.3333333333333333 x 0.16666666666666666) (* B B) 1.0) x)
        B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.1e-94) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 4e-79) {
		tmp = -x / B;
	} else if (F <= 550000.0) {
		tmp = (sqrt(pow(fma(F, F, 2.0), -1.0)) * F) / B;
	} else {
		tmp = (fma(fma(0.3333333333333333, x, 0.16666666666666666), (B * B), 1.0) - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.1e-94)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 4e-79)
		tmp = Float64(Float64(-x) / B);
	elseif (F <= 550000.0)
		tmp = Float64(Float64(sqrt((fma(F, F, 2.0) ^ -1.0)) * F) / B);
	else
		tmp = Float64(Float64(fma(fma(0.3333333333333333, x, 0.16666666666666666), Float64(B * B), 1.0) - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.1e-94], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 4e-79], N[((-x) / B), $MachinePrecision], If[LessEqual[F, 550000.0], N[(N[(N[Sqrt[N[Power[N[(F * F + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * F), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(N[(0.3333333333333333 * x + 0.16666666666666666), $MachinePrecision] * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.1000000000000001e-94

   1. Initial program 56.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 36.8

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

   5. Applied rewrites 36.8%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 46.2%

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

   if -2.1000000000000001e-94 < F < 4e-79

   1. Initial program 99.6%

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

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   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 50.4

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

   5. Applied rewrites 50.4%

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

   6. Taylor expanded in F around 0

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

   8. Applied rewrites 43.8%

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

   if 4e-79 < F < 5.5e5

   1. Initial program 99.3%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. 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%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 45.7%

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

   if 5.5e5 < F

   1. Initial program 62.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 79.6%

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

   5. Taylor expanded in B around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 38.5%

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

   8. Taylor expanded in F around inf

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

   10. Applied rewrites 51.6%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B} \]
3. Recombined 4 regimes into one program.

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.1 \cdot 10^{-94}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 4 \cdot 10^{-79}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 550000:\\ \;\;\;\;\frac{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot F}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 43.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.1 \cdot 10^{-94}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 4 \cdot 10^{-79}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 550000:\\ \;\;\;\;\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.1e-94)
   (/ (- -1.0 x) B)
   (if (<= F 4e-79)
     (/ (- x) B)
     (if (<= F 550000.0)
       (* (sqrt (pow (fma F F 2.0) -1.0)) (/ F B))
       (/
        (- (fma (fma 0.3333333333333333 x 0.16666666666666666) (* B B) 1.0) x)
        B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.1e-94) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 4e-79) {
		tmp = -x / B;
	} else if (F <= 550000.0) {
		tmp = sqrt(pow(fma(F, F, 2.0), -1.0)) * (F / B);
	} else {
		tmp = (fma(fma(0.3333333333333333, x, 0.16666666666666666), (B * B), 1.0) - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.1e-94)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 4e-79)
		tmp = Float64(Float64(-x) / B);
	elseif (F <= 550000.0)
		tmp = Float64(sqrt((fma(F, F, 2.0) ^ -1.0)) * Float64(F / B));
	else
		tmp = Float64(Float64(fma(fma(0.3333333333333333, x, 0.16666666666666666), Float64(B * B), 1.0) - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.1e-94], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 4e-79], N[((-x) / B), $MachinePrecision], If[LessEqual[F, 550000.0], N[(N[Sqrt[N[Power[N[(F * F + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.3333333333333333 * x + 0.16666666666666666), $MachinePrecision] * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.1000000000000001e-94

   1. Initial program 56.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 36.8

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

   5. Applied rewrites 36.8%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 46.2%

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

   if -2.1000000000000001e-94 < F < 4e-79

   1. Initial program 99.6%

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

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   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 50.4

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

   5. Applied rewrites 50.4%

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

   6. Taylor expanded in F around 0

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

   8. Applied rewrites 43.8%

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

   if 4e-79 < F < 5.5e5

   1. Initial program 99.3%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. 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%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 45.6%

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

   if 5.5e5 < F

   1. Initial program 62.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 79.6%

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

   5. Taylor expanded in B around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 38.5%

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

   8. Taylor expanded in F around inf

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

   10. Applied rewrites 51.6%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B} \]
3. Recombined 4 regimes into one program.

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.1 \cdot 10^{-94}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 4 \cdot 10^{-79}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 550000:\\ \;\;\;\;\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 49.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -8 \cdot 10^{-37}:\\ \;\;\;\;\frac{{\left(\frac{-1 + x}{1 - x \cdot x}\right)}^{-1}}{B}\\ \mathbf{elif}\;F \leq 1.88:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-1}}, F, -x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -8e-37)
   (/ (pow (/ (+ -1.0 x) (- 1.0 (* x x))) -1.0) B)
   (if (<= F 1.88)
     (/ (fma (sqrt (pow (fma 2.0 x 2.0) -1.0)) F (- x)) B)
     (/
      (- (fma (fma 0.3333333333333333 x 0.16666666666666666) (* B B) 1.0) x)
      B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -8e-37) {
		tmp = pow(((-1.0 + x) / (1.0 - (x * x))), -1.0) / B;
	} else if (F <= 1.88) {
		tmp = fma(sqrt(pow(fma(2.0, x, 2.0), -1.0)), F, -x) / B;
	} else {
		tmp = (fma(fma(0.3333333333333333, x, 0.16666666666666666), (B * B), 1.0) - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -8e-37)
		tmp = Float64((Float64(Float64(-1.0 + x) / Float64(1.0 - Float64(x * x))) ^ -1.0) / B);
	elseif (F <= 1.88)
		tmp = Float64(fma(sqrt((fma(2.0, x, 2.0) ^ -1.0)), F, Float64(-x)) / B);
	else
		tmp = Float64(Float64(fma(fma(0.3333333333333333, x, 0.16666666666666666), Float64(B * B), 1.0) - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -8e-37], N[(N[Power[N[(N[(-1.0 + x), $MachinePrecision] / N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.88], N[(N[(N[Sqrt[N[Power[N[(2.0 * x + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * F + (-x)), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(N[(0.3333333333333333 * x + 0.16666666666666666), $MachinePrecision] * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -8.00000000000000053e-37

   1. Initial program 51.3%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. sub-neg N/A

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

      3. *-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 36.3

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

   5. Applied rewrites 36.3%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 51.1%

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

   10. Applied rewrites 51.1%

       \[\leadsto \frac{\frac{1}{\frac{-1 + x}{1 - x \cdot x}}}{B} \]

   if -8.00000000000000053e-37 < F < 1.8799999999999999

   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 50.2

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

   5. Applied rewrites 50.2%

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

   6. Taylor expanded in F around 0

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

   8. Applied rewrites 50.0%

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

   if 1.8799999999999999 < F

   1. Initial program 62.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 79.9%

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

   5. Taylor expanded in B around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 39.5%

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

   8. Taylor expanded in F around inf

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

   10. Applied rewrites 52.0%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -8 \cdot 10^{-37}:\\ \;\;\;\;\frac{{\left(\frac{-1 + x}{1 - x \cdot x}\right)}^{-1}}{B}\\ \mathbf{elif}\;F \leq 1.88:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-1}}, F, -x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 56.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 3.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(-0.16666666666666666, B \cdot B, 1\right) \cdot B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 3.2e-9)
   (/ (- (/ 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 <= 3.2e-9) {
		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 <= 3.2e-9)
		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(-0.16666666666666666, Float64(B * B), 1.0) * B)) - Float64(x / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 3.2e-9], 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[(-0.16666666666666666 * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 3.20000000000000012e-9

   1. Initial program 70.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 54.5

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

   5. Applied rewrites 54.5%

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

   7. Applied rewrites 54.5%

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

   if 3.20000000000000012e-9 < B

   1. Initial program 89.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 89.6%

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

   6. Applied rewrites 89.7%

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

   7. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 62.0

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

   9. Applied rewrites 62.0%

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

   10. Taylor expanded in B around 0

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

   12. Applied rewrites 66.1%

       \[\leadsto \frac{-1}{\mathsf{fma}\left(-0.16666666666666666, B \cdot B, 1\right) \cdot \color{blue}{B}} - \frac{x}{\tan B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(-0.16666666666666666, B \cdot B, 1\right) \cdot B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 49.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -9 \cdot 10^{+170}:\\ \;\;\;\;\frac{{\left(\frac{-1 + x}{1 - x \cdot x}\right)}^{-1}}{B}\\ \mathbf{elif}\;F \leq 2.7 \cdot 10^{+181}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -9e+170)
   (/ (pow (/ (+ -1.0 x) (- 1.0 (* x x))) -1.0) B)
   (if (<= F 2.7e+181)
     (/ (- (/ F (sqrt (fma x 2.0 (fma F F 2.0)))) x) B)
     (/
      (- (fma (fma 0.3333333333333333 x 0.16666666666666666) (* B B) 1.0) x)
      B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -9e+170) {
		tmp = pow(((-1.0 + x) / (1.0 - (x * x))), -1.0) / B;
	} else if (F <= 2.7e+181) {
		tmp = ((F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - x) / B;
	} else {
		tmp = (fma(fma(0.3333333333333333, x, 0.16666666666666666), (B * B), 1.0) - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -9e+170)
		tmp = Float64((Float64(Float64(-1.0 + x) / Float64(1.0 - Float64(x * x))) ^ -1.0) / B);
	elseif (F <= 2.7e+181)
		tmp = Float64(Float64(Float64(F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - x) / B);
	else
		tmp = Float64(Float64(fma(fma(0.3333333333333333, x, 0.16666666666666666), Float64(B * B), 1.0) - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -9e+170], N[(N[Power[N[(N[(-1.0 + x), $MachinePrecision] / N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 2.7e+181], 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[(0.3333333333333333 * x + 0.16666666666666666), $MachinePrecision] * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -9.00000000000000044e170

   1. Initial program 26.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 21.2

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

   5. Applied rewrites 21.2%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 50.5%

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

   10. Applied rewrites 50.5%

       \[\leadsto \frac{\frac{1}{\frac{-1 + x}{1 - x \cdot x}}}{B} \]

   if -9.00000000000000044e170 < F < 2.70000000000000007e181

   1. Initial program 89.8%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. sub-neg N/A

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

      3. *-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 51.6

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

   5. Applied rewrites 51.6%

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

   7. Applied rewrites 51.6%

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

   if 2.70000000000000007e181 < F

   1. Initial program 41.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 51.2%

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

   5. Taylor expanded in B around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 15.6%

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

   8. Taylor expanded in F around inf

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

   10. Applied rewrites 48.3%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -9 \cdot 10^{+170}:\\ \;\;\;\;\frac{{\left(\frac{-1 + x}{1 - x \cdot x}\right)}^{-1}}{B}\\ \mathbf{elif}\;F \leq 2.7 \cdot 10^{+181}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 55.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 3.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{-1}{B}, \frac{-x}{\tan B}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 3.5e-9)
   (/ (- (/ F (sqrt (fma x 2.0 (fma F F 2.0)))) x) B)
   (fma -1.0 (/ -1.0 B) (/ (- x) (tan B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 3.5e-9) {
		tmp = ((F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - x) / B;
	} else {
		tmp = fma(-1.0, (-1.0 / B), (-x / tan(B)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 3.5e-9)
		tmp = Float64(Float64(Float64(F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - x) / B);
	else
		tmp = fma(-1.0, Float64(-1.0 / B), Float64(Float64(-x) / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 3.5e-9], N[(N[(N[(F / N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(-1.0 * N[(-1.0 / B), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 3.4999999999999999e-9

   1. Initial program 70.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 54.5

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

   5. Applied rewrites 54.5%

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

   7. Applied rewrites 54.5%

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

   if 3.4999999999999999e-9 < B

   1. Initial program 89.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 89.6%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 69.4

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

   7. Applied rewrites 69.4%

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

   8. Taylor expanded in F around inf

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

   10. Applied rewrites 59.5%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{-1}{B}, \frac{-x}{\tan B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 55.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 3.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 3.2e-9)
   (/ (- (/ F (sqrt (fma x 2.0 (fma F F 2.0)))) x) B)
   (- (/ -1.0 B) (/ x (tan B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 3.2e-9) {
		tmp = ((F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - x) / B;
	} else {
		tmp = (-1.0 / B) - (x / tan(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 3.2e-9)
		tmp = Float64(Float64(Float64(F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - x) / B);
	else
		tmp = Float64(Float64(-1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 3.2e-9], 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 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 3.20000000000000012e-9

   1. Initial program 70.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 54.5

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

   5. Applied rewrites 54.5%

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

   7. Applied rewrites 54.5%

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

   if 3.20000000000000012e-9 < B

   1. Initial program 89.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 89.6%

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

   6. Applied rewrites 89.7%

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

   7. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 62.0

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

   9. Applied rewrites 62.0%

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

   10. Taylor expanded in B around 0

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

   12. Applied rewrites 59.4%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 50.7% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.0235:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \frac{\frac{\mathsf{fma}\left(2, x, 2\right)}{F}}{F}, -1 - x\right)}{B}\\ \mathbf{elif}\;F \leq 2.7 \cdot 10^{+181}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.0235)
   (/ (fma 0.5 (/ (/ (fma 2.0 x 2.0) F) F) (- -1.0 x)) B)
   (if (<= F 2.7e+181)
     (/ (- (/ F (sqrt (fma x 2.0 (fma F F 2.0)))) x) B)
     (/
      (- (fma (fma 0.3333333333333333 x 0.16666666666666666) (* B B) 1.0) x)
      B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.0235) {
		tmp = fma(0.5, ((fma(2.0, x, 2.0) / F) / F), (-1.0 - x)) / B;
	} else if (F <= 2.7e+181) {
		tmp = ((F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - x) / B;
	} else {
		tmp = (fma(fma(0.3333333333333333, x, 0.16666666666666666), (B * B), 1.0) - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.0235)
		tmp = Float64(fma(0.5, Float64(Float64(fma(2.0, x, 2.0) / F) / F), Float64(-1.0 - x)) / B);
	elseif (F <= 2.7e+181)
		tmp = Float64(Float64(Float64(F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - x) / B);
	else
		tmp = Float64(Float64(fma(fma(0.3333333333333333, x, 0.16666666666666666), Float64(B * B), 1.0) - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.0235], N[(N[(0.5 * N[(N[(N[(2.0 * x + 2.0), $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision] + N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 2.7e+181], 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[(0.3333333333333333 * x + 0.16666666666666666), $MachinePrecision] * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.0235

   1. Initial program 48.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 36.7

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

   5. Applied rewrites 36.7%

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

   6. Taylor expanded in 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 52.3%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{0.5}{F}, \frac{\mathsf{fma}\left(2, x, 2\right)}{F}, -1 - x\right)}{B} \]
   9. Step-by-step derivation

   10. Applied rewrites 52.3%

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

   if -0.0235 < F < 2.70000000000000007e181

   1. Initial program 93.8%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. sub-neg N/A

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

      3. *-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 50.9

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

   5. Applied rewrites 50.9%

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

   7. Applied rewrites 50.9%

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

   if 2.70000000000000007e181 < F

   1. Initial program 41.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 51.2%

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

   5. Taylor expanded in B around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 15.6%

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

   8. Taylor expanded in F around inf

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

   10. Applied rewrites 48.3%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.0235:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \frac{\frac{\mathsf{fma}\left(2, x, 2\right)}{F}}{F}, -1 - x\right)}{B}\\ \mathbf{elif}\;F \leq 2.7 \cdot 10^{+181}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 43.2% accurate, 13.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.1 \cdot 10^{-94}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.95 \cdot 10^{-80}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.1e-94)
   (/ (- -1.0 x) B)
   (if (<= F 1.95e-80) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.1e-94) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.95e-80) {
		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 <= (-2.1d-94)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 1.95d-80) 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 <= -2.1e-94) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.95e-80) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.1e-94:
		tmp = (-1.0 - x) / B
	elif F <= 1.95e-80:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.1e-94)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 1.95e-80)
		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 <= -2.1e-94)
		tmp = (-1.0 - x) / B;
	elseif (F <= 1.95e-80)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.1e-94], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.95e-80], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.1000000000000001e-94

   1. Initial program 56.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 36.8

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

   5. Applied rewrites 36.8%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 46.2%

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

   if -2.1000000000000001e-94 < F < 1.9499999999999999e-80

   1. Initial program 99.6%

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

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   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 49.8

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

   5. Applied rewrites 49.8%

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

   6. Taylor expanded in F around 0

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

   8. Applied rewrites 43.1%

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

   if 1.9499999999999999e-80 < F

   1. Initial program 71.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 43.4

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

   5. Applied rewrites 43.4%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 45.0%

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

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.1 \cdot 10^{-94}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.95 \cdot 10^{-80}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 36.4% accurate, 17.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.1 \cdot 10^{-94}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.1e-94) (/ (- -1.0 x) B) (/ (- x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.1e-94) {
		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 <= (-2.1d-94)) 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 <= -2.1e-94) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.1e-94:
		tmp = (-1.0 - x) / B
	else:
		tmp = -x / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.1e-94)
		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 <= -2.1e-94)
		tmp = (-1.0 - x) / B;
	else
		tmp = -x / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.1e-94], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[((-x) / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -2.1000000000000001e-94

   1. Initial program 56.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 36.8

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

   5. Applied rewrites 36.8%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 46.2%

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

   if -2.1000000000000001e-94 < F

   1. Initial program 85.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 46.7

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

   5. Applied rewrites 46.7%

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

   6. Taylor expanded in F around 0

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

   8. Applied rewrites 32.3%

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

4. Final simplification 37.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.1 \cdot 10^{-94}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 28.8% accurate, 26.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.7%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
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 43.0

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

5. Applied rewrites 43.0%

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

6. Taylor expanded in F around 0

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

8. Applied rewrites 29.2%

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

9. Final simplification 29.2%

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

Reproduce

herbie shell --seed 2024326 
(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))))))