VandenBroeck and Keller, Equation (23)

Percentage Accurate: 77.2% → 99.7%

Time: 12.3s

Alternatives: 21

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: 77.2% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -7000000000:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 120000000:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -7000000000.0)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 120000000.0)
       (- (/ F (* (sin B) (sqrt (fma x 2.0 (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 <= -7000000000.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 120000000.0) {
		tmp = (F / (sin(B) * sqrt(fma(x, 2.0, 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 <= -7000000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 120000000.0)
		tmp = Float64(Float64(F / Float64(sin(B) * sqrt(fma(x, 2.0, 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, -7000000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 120000000.0], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $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 < -7e9

   1. Initial program 70.9%

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

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 78.4%

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

   6. Applied rewrites 78.5%

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

   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 99.9%

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

   if -7e9 < F < 1.2e8

   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. frac-2neg N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

   6. Applied rewrites 99.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 99.6

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 99.6

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

   8. Applied rewrites 99.6%

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

   if 1.2e8 < F

   1. Initial program 52.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 73.8%

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

   6. Applied rewrites 73.9%

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

   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: 76.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ t_1 := \frac{F}{\sin B}\\ t_2 := t\_0 + t\_1 \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-1}{2}\right)}\\ \mathbf{if}\;t\_2 \leq -15:\\ \;\;\;\;\mathsf{fma}\left(-F, \frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}}}{-B}, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;t\_2 \leq 5:\\ \;\;\;\;\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+257}:\\ \;\;\;\;\frac{\frac{F}{B}}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + {B}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B))))
        (t_1 (/ F (sin B)))
        (t_2 (+ t_0 (* t_1 (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (/ -1.0 2.0))))))
   (if (<= t_2 -15.0)
     (fma
      (- F)
      (/ (sqrt (pow (+ (fma 2.0 x (* F F)) 2.0) -1.0)) (- B))
      (/ (- x) (tan B)))
     (if (<= t_2 5.0)
       (* (sqrt (pow (fma F F 2.0) -1.0)) t_1)
       (if (<= t_2 1e+257)
         (- (/ (/ F B) (sqrt (fma x 2.0 (fma F F 2.0)))) (/ x (tan B)))
         (+ t_0 (pow B -1.0)))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double t_1 = F / sin(B);
	double t_2 = t_0 + (t_1 * pow((((F * F) + 2.0) + (2.0 * x)), (-1.0 / 2.0)));
	double tmp;
	if (t_2 <= -15.0) {
		tmp = fma(-F, (sqrt(pow((fma(2.0, x, (F * F)) + 2.0), -1.0)) / -B), (-x / tan(B)));
	} else if (t_2 <= 5.0) {
		tmp = sqrt(pow(fma(F, F, 2.0), -1.0)) * t_1;
	} else if (t_2 <= 1e+257) {
		tmp = ((F / B) / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - (x / tan(B));
	} else {
		tmp = t_0 + pow(B, -1.0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	t_1 = Float64(F / sin(B))
	t_2 = Float64(t_0 + Float64(t_1 * (Float64(Float64(Float64(F * F) + 2.0) + Float64(2.0 * x)) ^ Float64(-1.0 / 2.0))))
	tmp = 0.0
	if (t_2 <= -15.0)
		tmp = fma(Float64(-F), Float64(sqrt((Float64(fma(2.0, x, Float64(F * F)) + 2.0) ^ -1.0)) / Float64(-B)), Float64(Float64(-x) / tan(B)));
	elseif (t_2 <= 5.0)
		tmp = Float64(sqrt((fma(F, F, 2.0) ^ -1.0)) * t_1);
	elseif (t_2 <= 1e+257)
		tmp = Float64(Float64(Float64(F / B) / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - Float64(x / tan(B)));
	else
		tmp = Float64(t_0 + (B ^ -1.0));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 + N[(t$95$1 * 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]}, If[LessEqual[t$95$2, -15.0], N[((-F) * N[(N[Sqrt[N[Power[N[(N[(2.0 * x + N[(F * F), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] / (-B)), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5.0], N[(N[Sqrt[N[Power[N[(F * F + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 1e+257], N[(N[(N[(F / B), $MachinePrecision] / N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[Power[B, -1.0], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < -15

   1. Initial program 95.5%

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

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in B around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-lft-identity N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 99.6

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

   7. Applied rewrites 99.6%

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

   if -15 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < 5

   1. Initial program 76.3%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sin.f64 54.0

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

   5. Applied rewrites 54.0%

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

   if 5 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < 1.00000000000000003e257

   1. Initial program 99.5%

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

   3. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 97.5

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

   5. Applied rewrites 97.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 97.7

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

   7. Applied rewrites 97.8%

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

   if 1.00000000000000003e257 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))))))

   1. Initial program 35.1%

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

   3. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 35.1

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

   5. Applied rewrites 35.1%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 81.7%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-1}{2}\right)} \leq -15:\\ \;\;\;\;\mathsf{fma}\left(-F, \frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}}}{-B}, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-1}{2}\right)} \leq 5:\\ \;\;\;\;\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot \frac{F}{\sin B}\\ \mathbf{elif}\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-1}{2}\right)} \leq 10^{+257}:\\ \;\;\;\;\frac{\frac{F}{B}}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + {B}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ t_1 := \frac{\frac{F}{B}}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - \frac{x}{\tan B}\\ t_2 := \frac{F}{\sin B}\\ t_3 := t\_0 + t\_2 \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-1}{2}\right)}\\ \mathbf{if}\;t\_3 \leq -15:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 5:\\ \;\;\;\;\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot t\_2\\ \mathbf{elif}\;t\_3 \leq 10^{+257}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 + {B}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B))))
        (t_1 (- (/ (/ F B) (sqrt (fma x 2.0 (fma F F 2.0)))) (/ x (tan B))))
        (t_2 (/ F (sin B)))
        (t_3 (+ t_0 (* t_2 (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (/ -1.0 2.0))))))
   (if (<= t_3 -15.0)
     t_1
     (if (<= t_3 5.0)
       (* (sqrt (pow (fma F F 2.0) -1.0)) t_2)
       (if (<= t_3 1e+257) t_1 (+ t_0 (pow B -1.0)))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double t_1 = ((F / B) / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - (x / tan(B));
	double t_2 = F / sin(B);
	double t_3 = t_0 + (t_2 * pow((((F * F) + 2.0) + (2.0 * x)), (-1.0 / 2.0)));
	double tmp;
	if (t_3 <= -15.0) {
		tmp = t_1;
	} else if (t_3 <= 5.0) {
		tmp = sqrt(pow(fma(F, F, 2.0), -1.0)) * t_2;
	} else if (t_3 <= 1e+257) {
		tmp = t_1;
	} else {
		tmp = t_0 + pow(B, -1.0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	t_1 = Float64(Float64(Float64(F / B) / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - Float64(x / tan(B)))
	t_2 = Float64(F / sin(B))
	t_3 = Float64(t_0 + Float64(t_2 * (Float64(Float64(Float64(F * F) + 2.0) + Float64(2.0 * x)) ^ Float64(-1.0 / 2.0))))
	tmp = 0.0
	if (t_3 <= -15.0)
		tmp = t_1;
	elseif (t_3 <= 5.0)
		tmp = Float64(sqrt((fma(F, F, 2.0) ^ -1.0)) * t_2);
	elseif (t_3 <= 1e+257)
		tmp = t_1;
	else
		tmp = Float64(t_0 + (B ^ -1.0));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(F / B), $MachinePrecision] / N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 + N[(t$95$2 * 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]}, If[LessEqual[t$95$3, -15.0], t$95$1, If[LessEqual[t$95$3, 5.0], N[(N[Sqrt[N[Power[N[(F * F + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$3, 1e+257], t$95$1, N[(t$95$0 + N[Power[B, -1.0], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < -15 or 5 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < 1.00000000000000003e257

   1. Initial program 97.3%

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

   3. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 96.4

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

   5. Applied rewrites 96.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 96.6

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

   7. Applied rewrites 96.6%

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

   if -15 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < 5

   1. Initial program 76.3%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sin.f64 54.0

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

   5. Applied rewrites 54.0%

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

   if 1.00000000000000003e257 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))))))

   1. Initial program 35.1%

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

   3. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 35.1

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

   5. Applied rewrites 35.1%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 81.7%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-1}{2}\right)} \leq -15:\\ \;\;\;\;\frac{\frac{F}{B}}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - \frac{x}{\tan B}\\ \mathbf{elif}\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-1}{2}\right)} \leq 5:\\ \;\;\;\;\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot \frac{F}{\sin B}\\ \mathbf{elif}\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-1}{2}\right)} \leq 10^{+257}:\\ \;\;\;\;\frac{\frac{F}{B}}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + {B}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 56.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.33:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right), \sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}} \cdot F, \mathsf{fma}\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(0.00205026455026455, B \cdot B, 0.019444444444444445\right) \cdot F\right) \cdot \sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}}, B \cdot B, 0.3333333333333333 \cdot x\right), B \cdot B, -x\right)\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B} \cdot {F}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 0.33)
   (/
    (fma
     (fma (* B B) 0.16666666666666666 1.0)
     (* (sqrt (pow (fma F F (fma 2.0 x 2.0)) -1.0)) F)
     (fma
      (fma
       (*
        (* (fma 0.00205026455026455 (* B B) 0.019444444444444445) F)
        (sqrt (pow (fma F F 2.0) -1.0)))
       (* B B)
       (* 0.3333333333333333 x))
      (* B B)
      (- x)))
    B)
   (+ (* x (/ -1.0 (tan B))) (* (/ F B) (pow F -1.0)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 0.33) {
		tmp = fma(fma((B * B), 0.16666666666666666, 1.0), (sqrt(pow(fma(F, F, fma(2.0, x, 2.0)), -1.0)) * F), fma(fma(((fma(0.00205026455026455, (B * B), 0.019444444444444445) * F) * sqrt(pow(fma(F, F, 2.0), -1.0))), (B * B), (0.3333333333333333 * x)), (B * B), -x)) / B;
	} else {
		tmp = (x * (-1.0 / tan(B))) + ((F / B) * pow(F, -1.0));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 0.33)
		tmp = Float64(fma(fma(Float64(B * B), 0.16666666666666666, 1.0), Float64(sqrt((fma(F, F, fma(2.0, x, 2.0)) ^ -1.0)) * F), fma(fma(Float64(Float64(fma(0.00205026455026455, Float64(B * B), 0.019444444444444445) * F) * sqrt((fma(F, F, 2.0) ^ -1.0))), Float64(B * B), Float64(0.3333333333333333 * x)), Float64(B * B), Float64(-x))) / B);
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(Float64(F / B) * (F ^ -1.0)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 0.33], N[(N[(N[(N[(B * B), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * N[(N[Sqrt[N[Power[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * F), $MachinePrecision] + N[(N[(N[(N[(N[(0.00205026455026455 * N[(B * B), $MachinePrecision] + 0.019444444444444445), $MachinePrecision] * F), $MachinePrecision] * N[Sqrt[N[Power[N[(F * F + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(B * B), $MachinePrecision] + N[(0.3333333333333333 * x), $MachinePrecision]), $MachinePrecision] * N[(B * B), $MachinePrecision] + (-x)), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(F / B), $MachinePrecision] * N[Power[F, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 0.330000000000000016

   1. Initial program 77.6%

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

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \left(\frac{1}{3} \cdot x + {B}^{2} \cdot \left(-1 \cdot \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \left(\frac{-1}{36} \cdot F + \frac{1}{120} \cdot F\right)\right) + \left(\frac{-1}{9} \cdot x + \left(\frac{2}{15} \cdot x + {B}^{2} \cdot \left(-1 \cdot \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \left(\frac{-1}{5040} \cdot F + \left(\frac{1}{720} \cdot F + \frac{1}{6} \cdot \left(\frac{-1}{36} \cdot F + \frac{1}{120} \cdot F\right)\right)\right)\right) + \left(\frac{-1}{3} \cdot \left(\frac{-1}{9} \cdot x + \frac{2}{15} \cdot x\right) + \left(\frac{-2}{45} \cdot x + \frac{17}{315} \cdot x\right)\right)\right)\right)\right)\right)\right)\right)\right) - x}{B}} \]

   5. Applied rewrites 61.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right), \sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.022222222222222223, x, \mathsf{fma}\left(\left(x \cdot 0.0021164021164021165\right) \cdot B, B, \sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \left(0.019444444444444445 \cdot F - \left(\left(F \cdot -0.00205026455026455\right) \cdot B\right) \cdot B\right)\right)\right), B \cdot B, 0.3333333333333333 \cdot x\right), B \cdot B, -x\right)\right)}{B}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right), \sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{\frac{1}{2 + {F}^{2}}} \cdot \left(\frac{7}{360} \cdot F - \frac{-31}{15120} \cdot \left({B}^{2} \cdot F\right)\right), B \cdot B, \frac{1}{3} \cdot x\right), B \cdot B, -x\right)\right)}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 61.4%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right), \sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F, \mathsf{fma}\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(0.00205026455026455, B \cdot B, 0.019444444444444445\right) \cdot F\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}, B \cdot B, 0.3333333333333333 \cdot x\right), B \cdot B, -x\right)\right)}{B} \]

   if 0.330000000000000016 < B

   1. Initial program 82.7%

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

   3. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 57.6

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

   5. Applied rewrites 57.6%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 56.2%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 0.33:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right), \sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}} \cdot F, \mathsf{fma}\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(0.00205026455026455, B \cdot B, 0.019444444444444445\right) \cdot F\right) \cdot \sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}}, B \cdot B, 0.3333333333333333 \cdot x\right), B \cdot B, -x\right)\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B} \cdot {F}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 55.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.33:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right), \sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}} \cdot F, \mathsf{fma}\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(0.00205026455026455, B \cdot B, 0.019444444444444445\right) \cdot F\right) \cdot \sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}}, B \cdot B, 0.3333333333333333 \cdot x\right), B \cdot B, -x\right)\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 0.33)
   (/
    (fma
     (fma (* B B) 0.16666666666666666 1.0)
     (* (sqrt (pow (fma F F (fma 2.0 x 2.0)) -1.0)) F)
     (fma
      (fma
       (*
        (* (fma 0.00205026455026455 (* B B) 0.019444444444444445) F)
        (sqrt (pow (fma F F 2.0) -1.0)))
       (* B B)
       (* 0.3333333333333333 x))
      (* B B)
      (- x)))
    B)
   (+ (* x (/ -1.0 (tan B))) (/ -1.0 B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 0.33) {
		tmp = fma(fma((B * B), 0.16666666666666666, 1.0), (sqrt(pow(fma(F, F, fma(2.0, x, 2.0)), -1.0)) * F), fma(fma(((fma(0.00205026455026455, (B * B), 0.019444444444444445) * F) * sqrt(pow(fma(F, F, 2.0), -1.0))), (B * B), (0.3333333333333333 * x)), (B * B), -x)) / B;
	} else {
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 0.33)
		tmp = Float64(fma(fma(Float64(B * B), 0.16666666666666666, 1.0), Float64(sqrt((fma(F, F, fma(2.0, x, 2.0)) ^ -1.0)) * F), fma(fma(Float64(Float64(fma(0.00205026455026455, Float64(B * B), 0.019444444444444445) * F) * sqrt((fma(F, F, 2.0) ^ -1.0))), Float64(B * B), Float64(0.3333333333333333 * x)), Float64(B * B), Float64(-x))) / B);
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(-1.0 / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 0.33], N[(N[(N[(N[(B * B), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * N[(N[Sqrt[N[Power[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * F), $MachinePrecision] + N[(N[(N[(N[(N[(0.00205026455026455 * N[(B * B), $MachinePrecision] + 0.019444444444444445), $MachinePrecision] * F), $MachinePrecision] * N[Sqrt[N[Power[N[(F * F + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(B * B), $MachinePrecision] + N[(0.3333333333333333 * x), $MachinePrecision]), $MachinePrecision] * N[(B * B), $MachinePrecision] + (-x)), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 0.330000000000000016

   1. Initial program 77.6%

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

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \left(\frac{1}{3} \cdot x + {B}^{2} \cdot \left(-1 \cdot \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \left(\frac{-1}{36} \cdot F + \frac{1}{120} \cdot F\right)\right) + \left(\frac{-1}{9} \cdot x + \left(\frac{2}{15} \cdot x + {B}^{2} \cdot \left(-1 \cdot \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \left(\frac{-1}{5040} \cdot F + \left(\frac{1}{720} \cdot F + \frac{1}{6} \cdot \left(\frac{-1}{36} \cdot F + \frac{1}{120} \cdot F\right)\right)\right)\right) + \left(\frac{-1}{3} \cdot \left(\frac{-1}{9} \cdot x + \frac{2}{15} \cdot x\right) + \left(\frac{-2}{45} \cdot x + \frac{17}{315} \cdot x\right)\right)\right)\right)\right)\right)\right)\right)\right) - x}{B}} \]

   5. Applied rewrites 61.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right), \sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.022222222222222223, x, \mathsf{fma}\left(\left(x \cdot 0.0021164021164021165\right) \cdot B, B, \sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \left(0.019444444444444445 \cdot F - \left(\left(F \cdot -0.00205026455026455\right) \cdot B\right) \cdot B\right)\right)\right), B \cdot B, 0.3333333333333333 \cdot x\right), B \cdot B, -x\right)\right)}{B}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right), \sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{\frac{1}{2 + {F}^{2}}} \cdot \left(\frac{7}{360} \cdot F - \frac{-31}{15120} \cdot \left({B}^{2} \cdot F\right)\right), B \cdot B, \frac{1}{3} \cdot x\right), B \cdot B, -x\right)\right)}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 61.4%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right), \sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F, \mathsf{fma}\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(0.00205026455026455, B \cdot B, 0.019444444444444445\right) \cdot F\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}, B \cdot B, 0.3333333333333333 \cdot x\right), B \cdot B, -x\right)\right)}{B} \]

   if 0.330000000000000016 < B

   1. Initial program 82.7%

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

   3. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 57.6

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

   5. Applied rewrites 57.6%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 56.3%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 0.33:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right), \sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}} \cdot F, \mathsf{fma}\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(0.00205026455026455, B \cdot B, 0.019444444444444445\right) \cdot F\right) \cdot \sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}}, B \cdot B, 0.3333333333333333 \cdot x\right), B \cdot B, -x\right)\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 55.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}}\\ \mathbf{if}\;B \leq 0.33:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, \mathsf{fma}\left(\left(B \cdot B\right) \cdot F, 0.019444444444444445, 0.16666666666666666 \cdot F\right), x \cdot \mathsf{fma}\left(0.022222222222222223, B \cdot B, 0.3333333333333333\right)\right), B \cdot B, \mathsf{fma}\left(t\_0, F, -x\right)\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (sqrt (pow (fma F F (fma 2.0 x 2.0)) -1.0))))
   (if (<= B 0.33)
     (/
      (fma
       (fma
        t_0
        (fma (* (* B B) F) 0.019444444444444445 (* 0.16666666666666666 F))
        (* x (fma 0.022222222222222223 (* B B) 0.3333333333333333)))
       (* B B)
       (fma t_0 F (- x)))
      B)
     (+ (* x (/ -1.0 (tan B))) (/ -1.0 B)))))

C

double code(double F, double B, double x) {
	double t_0 = sqrt(pow(fma(F, F, fma(2.0, x, 2.0)), -1.0));
	double tmp;
	if (B <= 0.33) {
		tmp = fma(fma(t_0, fma(((B * B) * F), 0.019444444444444445, (0.16666666666666666 * F)), (x * fma(0.022222222222222223, (B * B), 0.3333333333333333))), (B * B), fma(t_0, F, -x)) / B;
	} else {
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = sqrt((fma(F, F, fma(2.0, x, 2.0)) ^ -1.0))
	tmp = 0.0
	if (B <= 0.33)
		tmp = Float64(fma(fma(t_0, fma(Float64(Float64(B * B) * F), 0.019444444444444445, Float64(0.16666666666666666 * F)), Float64(x * fma(0.022222222222222223, Float64(B * B), 0.3333333333333333))), Float64(B * B), fma(t_0, F, Float64(-x))) / B);
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(-1.0 / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[Sqrt[N[Power[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[B, 0.33], N[(N[(N[(t$95$0 * N[(N[(N[(B * B), $MachinePrecision] * F), $MachinePrecision] * 0.019444444444444445 + N[(0.16666666666666666 * F), $MachinePrecision]), $MachinePrecision] + N[(x * N[(0.022222222222222223 * N[(B * B), $MachinePrecision] + 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(B * B), $MachinePrecision] + N[(t$95$0 * F + (-x)), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 0.330000000000000016

   1. Initial program 77.6%

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

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \left(\frac{1}{3} \cdot x + {B}^{2} \cdot \left(-1 \cdot \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \left(\frac{-1}{36} \cdot F + \frac{1}{120} \cdot F\right)\right) + \left(\frac{-1}{9} \cdot x + \frac{2}{15} \cdot x\right)\right)\right)\right)\right) - x}{B}} \]

   5. Applied rewrites 61.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, \mathsf{fma}\left(\left(B \cdot B\right) \cdot F, 0.019444444444444445, 0.16666666666666666 \cdot F\right), x \cdot \mathsf{fma}\left(0.022222222222222223, B \cdot B, 0.3333333333333333\right)\right), B \cdot B, \mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, F, -x\right)\right)}{B}} \]

   if 0.330000000000000016 < B

   1. Initial program 82.7%

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

   3. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 57.6

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

   5. Applied rewrites 57.6%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 56.3%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 0.33:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}}, \mathsf{fma}\left(\left(B \cdot B\right) \cdot F, 0.019444444444444445, 0.16666666666666666 \cdot F\right), x \cdot \mathsf{fma}\left(0.022222222222222223, B \cdot B, 0.3333333333333333\right)\right), B \cdot B, \mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}}, F, -x\right)\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 55.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}}\\ \mathbf{if}\;B \leq 0.196:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, F, \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot F, t\_0, 0.3333333333333333 \cdot x\right), B \cdot B, -x\right)\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (sqrt (pow (+ (fma 2.0 x (* F F)) 2.0) -1.0))))
   (if (<= B 0.196)
     (/
      (fma
       t_0
       F
       (fma
        (fma (* 0.16666666666666666 F) t_0 (* 0.3333333333333333 x))
        (* B B)
        (- x)))
      B)
     (+ (* x (/ -1.0 (tan B))) (/ -1.0 B)))))

C

double code(double F, double B, double x) {
	double t_0 = sqrt(pow((fma(2.0, x, (F * F)) + 2.0), -1.0));
	double tmp;
	if (B <= 0.196) {
		tmp = fma(t_0, F, fma(fma((0.16666666666666666 * F), t_0, (0.3333333333333333 * x)), (B * B), -x)) / B;
	} else {
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = sqrt((Float64(fma(2.0, x, Float64(F * F)) + 2.0) ^ -1.0))
	tmp = 0.0
	if (B <= 0.196)
		tmp = Float64(fma(t_0, F, fma(fma(Float64(0.16666666666666666 * F), t_0, Float64(0.3333333333333333 * x)), Float64(B * B), Float64(-x))) / B);
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(-1.0 / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[Sqrt[N[Power[N[(N[(2.0 * x + N[(F * F), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[B, 0.196], N[(N[(t$95$0 * F + N[(N[(N[(0.16666666666666666 * F), $MachinePrecision] * t$95$0 + N[(0.3333333333333333 * x), $MachinePrecision]), $MachinePrecision] * N[(B * B), $MachinePrecision] + (-x)), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 0.19600000000000001

   1. Initial program 77.6%

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

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 88.1%

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

   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 61.5%

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

   if 0.19600000000000001 < B

   1. Initial program 82.7%

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

   3. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 57.6

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

   5. Applied rewrites 57.6%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 56.3%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 0.196:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}}, F, \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot F, \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}}, 0.3333333333333333 \cdot x\right), B \cdot B, -x\right)\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.85 \cdot 10^{-18}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq -2.5 \cdot 10^{-59}:\\ \;\;\;\;\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot \frac{F}{\sin B}\\ \mathbf{elif}\;F \leq 47000:\\ \;\;\;\;\mathsf{fma}\left(-F, \frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}}}{-B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.85e-18)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F -2.5e-59)
       (* (sqrt (pow (fma F F 2.0) -1.0)) (/ F (sin B)))
       (if (<= F 47000.0)
         (fma
          (- F)
          (/ (sqrt (pow (+ (fma 2.0 x (* F F)) 2.0) -1.0)) (- B))
          (/ (- x) (tan B)))
         (- (/ 1.0 (sin B)) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.85e-18) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= -2.5e-59) {
		tmp = sqrt(pow(fma(F, F, 2.0), -1.0)) * (F / sin(B));
	} else if (F <= 47000.0) {
		tmp = fma(-F, (sqrt(pow((fma(2.0, x, (F * F)) + 2.0), -1.0)) / -B), (-x / tan(B)));
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.85e-18)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= -2.5e-59)
		tmp = Float64(sqrt((fma(F, F, 2.0) ^ -1.0)) * Float64(F / sin(B)));
	elseif (F <= 47000.0)
		tmp = fma(Float64(-F), Float64(sqrt((Float64(fma(2.0, x, Float64(F * F)) + 2.0) ^ -1.0)) / Float64(-B)), Float64(Float64(-x) / tan(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.85e-18], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -2.5e-59], N[(N[Sqrt[N[Power[N[(F * F + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 47000.0], N[((-F) * N[(N[Sqrt[N[Power[N[(N[(2.0 * x + N[(F * F), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] / (-B)), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.8500000000000002e-18

   1. Initial program 72.5%

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

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 79.6%

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

   6. Applied rewrites 79.7%

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

   7. Taylor expanded in F around -inf

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

   9. Applied rewrites 98.7%

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

   if -1.8500000000000002e-18 < F < -2.5000000000000001e-59

   1. Initial program 99.3%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sin.f64 89.5

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

   5. Applied rewrites 89.5%

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

   if -2.5000000000000001e-59 < F < 47000

   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. frac-2neg N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in B around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-lft-identity N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 84.9

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

   7. Applied rewrites 84.9%

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

   if 47000 < F

   1. Initial program 52.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 73.8%

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

   6. Applied rewrites 73.9%

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

   7. Taylor expanded in F around inf

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

   9. Applied rewrites 99.8%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.85 \cdot 10^{-18}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -2.5 \cdot 10^{-59}:\\ \;\;\;\;\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot \frac{F}{\sin B}\\ \mathbf{elif}\;F \leq 47000:\\ \;\;\;\;\mathsf{fma}\left(-F, \frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}}}{-B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot \frac{F}{\sin B}\\ \mathbf{if}\;F \leq -1.85 \cdot 10^{-18}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -2.5 \cdot 10^{-59}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 2.4 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(-F, \frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}}}{-B}, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{+30}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + {B}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (sqrt (pow (fma F F 2.0) -1.0)) (/ F (sin B)))))
   (if (<= F -1.85e-18)
     (- (/ -1.0 (sin B)) (/ x (tan B)))
     (if (<= F -2.5e-59)
       t_0
       (if (<= F 2.4e-11)
         (fma
          (- F)
          (/ (sqrt (pow (+ (fma 2.0 x (* F F)) 2.0) -1.0)) (- B))
          (/ (- x) (tan B)))
         (if (<= F 4.5e+30) t_0 (+ (* x (/ -1.0 (tan B))) (pow B -1.0))))))))

C

double code(double F, double B, double x) {
	double t_0 = sqrt(pow(fma(F, F, 2.0), -1.0)) * (F / sin(B));
	double tmp;
	if (F <= -1.85e-18) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= -2.5e-59) {
		tmp = t_0;
	} else if (F <= 2.4e-11) {
		tmp = fma(-F, (sqrt(pow((fma(2.0, x, (F * F)) + 2.0), -1.0)) / -B), (-x / tan(B)));
	} else if (F <= 4.5e+30) {
		tmp = t_0;
	} else {
		tmp = (x * (-1.0 / tan(B))) + pow(B, -1.0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(sqrt((fma(F, F, 2.0) ^ -1.0)) * Float64(F / sin(B)))
	tmp = 0.0
	if (F <= -1.85e-18)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= -2.5e-59)
		tmp = t_0;
	elseif (F <= 2.4e-11)
		tmp = fma(Float64(-F), Float64(sqrt((Float64(fma(2.0, x, Float64(F * F)) + 2.0) ^ -1.0)) / Float64(-B)), Float64(Float64(-x) / tan(B)));
	elseif (F <= 4.5e+30)
		tmp = t_0;
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + (B ^ -1.0));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[Sqrt[N[Power[N[(F * F + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.85e-18], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -2.5e-59], t$95$0, If[LessEqual[F, 2.4e-11], N[((-F) * N[(N[Sqrt[N[Power[N[(N[(2.0 * x + N[(F * F), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] / (-B)), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.5e+30], t$95$0, N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[B, -1.0], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.8500000000000002e-18

   1. Initial program 72.5%

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

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 79.6%

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

   6. Applied rewrites 79.7%

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

   7. Taylor expanded in F around -inf

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

   9. Applied rewrites 98.7%

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

   if -1.8500000000000002e-18 < F < -2.5000000000000001e-59 or 2.4000000000000001e-11 < F < 4.49999999999999995e30

   1. Initial program 99.1%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sin.f64 93.2

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

   5. Applied rewrites 93.2%

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

   if -2.5000000000000001e-59 < F < 2.4000000000000001e-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. 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. frac-2neg N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in B around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-lft-identity N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 86.3

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

   7. Applied rewrites 86.3%

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

   if 4.49999999999999995e30 < F

   1. Initial program 49.9%

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

   3. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 41.2

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

   5. Applied rewrites 41.2%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 76.7%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.85 \cdot 10^{-18}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -2.5 \cdot 10^{-59}:\\ \;\;\;\;\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot \frac{F}{\sin B}\\ \mathbf{elif}\;F \leq 2.4 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(-F, \frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}}}{-B}, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{+30}:\\ \;\;\;\;\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot \frac{F}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + {B}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 90.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.85 \cdot 10^{-18}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq -4.2 \cdot 10^{-116}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F}{\sin B}\\ \mathbf{elif}\;F \leq 47000:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.85e-18)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F -4.2e-116)
       (+ (- (/ x B)) (/ (* (pow (fma x 2.0 (fma F F 2.0)) -0.5) F) (sin B)))
       (if (<= F 47000.0)
         (+
          (* x (/ -1.0 (tan B)))
          (* (/ F B) (sqrt (pow (fma F F (fma 2.0 x 2.0)) -1.0))))
         (- (/ 1.0 (sin B)) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.85e-18) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= -4.2e-116) {
		tmp = -(x / B) + ((pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) * F) / sin(B));
	} else if (F <= 47000.0) {
		tmp = (x * (-1.0 / tan(B))) + ((F / B) * sqrt(pow(fma(F, F, fma(2.0, x, 2.0)), -1.0)));
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.85e-18)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= -4.2e-116)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) * F) / sin(B)));
	elseif (F <= 47000.0)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(Float64(F / B) * sqrt((fma(F, F, fma(2.0, x, 2.0)) ^ -1.0))));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.85e-18], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -4.2e-116], N[((-N[(x / B), $MachinePrecision]) + N[(N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 47000.0], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(F / B), $MachinePrecision] * N[Sqrt[N[Power[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.8500000000000002e-18

   1. Initial program 72.5%

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

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 79.6%

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

   6. Applied rewrites 79.7%

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

   7. Taylor expanded in F around -inf

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

   9. Applied rewrites 98.7%

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

   if -1.8500000000000002e-18 < F < -4.1999999999999998e-116

   1. Initial program 99.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 88.0

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

   7. Applied rewrites 88.0%

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

   if -4.1999999999999998e-116 < F < 47000

   1. Initial program 99.6%

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

   3. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 87.9

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

   5. Applied rewrites 87.9%

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

   if 47000 < F

   1. Initial program 52.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 73.8%

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

   6. Applied rewrites 73.9%

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

   7. Taylor expanded in F around inf

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

   9. Applied rewrites 99.8%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.85 \cdot 10^{-18}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -4.2 \cdot 10^{-116}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F}{\sin B}\\ \mathbf{elif}\;F \leq 47000:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 49.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -230000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.16666666666666666\right), B \cdot B, -1\right) - x}{B}\\ \mathbf{elif}\;F \leq 1.42:\\ \;\;\;\;\mathsf{fma}\left(\frac{F}{B}, \sqrt{{\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-1}}, \frac{-x}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - {\left(F \cdot F\right)}^{-1}\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -230000000000.0)
   (/
    (- (fma (fma 0.3333333333333333 x -0.16666666666666666) (* B B) -1.0) x)
    B)
   (if (<= F 1.42)
     (fma (/ F B) (sqrt (pow (fma 2.0 x 2.0) -1.0)) (/ (- x) B))
     (/ (- (- 1.0 (pow (* F F) -1.0)) x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -230000000000.0) {
		tmp = (fma(fma(0.3333333333333333, x, -0.16666666666666666), (B * B), -1.0) - x) / B;
	} else if (F <= 1.42) {
		tmp = fma((F / B), sqrt(pow(fma(2.0, x, 2.0), -1.0)), (-x / B));
	} else {
		tmp = ((1.0 - pow((F * F), -1.0)) - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -230000000000.0)
		tmp = Float64(Float64(fma(fma(0.3333333333333333, x, -0.16666666666666666), Float64(B * B), -1.0) - x) / B);
	elseif (F <= 1.42)
		tmp = fma(Float64(F / B), sqrt((fma(2.0, x, 2.0) ^ -1.0)), Float64(Float64(-x) / B));
	else
		tmp = Float64(Float64(Float64(1.0 - (Float64(F * F) ^ -1.0)) - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -230000000000.0], N[(N[(N[(N[(0.3333333333333333 * x + -0.16666666666666666), $MachinePrecision] * N[(B * B), $MachinePrecision] + -1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.42], N[(N[(F / B), $MachinePrecision] * N[Sqrt[N[Power[N[(2.0 * x + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] + N[((-x) / B), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[Power[N[(F * F), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.3e11

   1. Initial program 70.5%

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

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \left(\frac{1}{3} \cdot x + {B}^{2} \cdot \left(-1 \cdot \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \left(\frac{-1}{36} \cdot F + \frac{1}{120} \cdot F\right)\right) + \left(\frac{-1}{9} \cdot x + \left(\frac{2}{15} \cdot x + {B}^{2} \cdot \left(-1 \cdot \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \left(\frac{-1}{5040} \cdot F + \left(\frac{1}{720} \cdot F + \frac{1}{6} \cdot \left(\frac{-1}{36} \cdot F + \frac{1}{120} \cdot F\right)\right)\right)\right) + \left(\frac{-1}{3} \cdot \left(\frac{-1}{9} \cdot x + \frac{2}{15} \cdot x\right) + \left(\frac{-2}{45} \cdot x + \frac{17}{315} \cdot x\right)\right)\right)\right)\right)\right)\right)\right)\right) - x}{B}} \]

   5. Applied rewrites 45.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right), \sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.022222222222222223, x, \mathsf{fma}\left(\left(x \cdot 0.0021164021164021165\right) \cdot B, B, \sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \left(0.019444444444444445 \cdot F - \left(\left(F \cdot -0.00205026455026455\right) \cdot B\right) \cdot B\right)\right)\right), B \cdot B, 0.3333333333333333 \cdot x\right), B \cdot B, -x\right)\right)}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{\left(-1 \cdot \left(1 + \frac{1}{6} \cdot {B}^{2}\right) + {B}^{2} \cdot \left(\frac{1}{3} \cdot x + {B}^{2} \cdot \left(-1 \cdot \left(\frac{7}{360} + \frac{31}{15120} \cdot {B}^{2}\right) + \left(\frac{2}{945} \cdot \left({B}^{2} \cdot x\right) + \frac{1}{45} \cdot x\right)\right)\right)\right) - x}{B} \]

   8. Applied rewrites 56.0%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0021164021164021165, B \cdot B, 0.022222222222222223\right), x, -\mathsf{fma}\left(0.00205026455026455, B \cdot B, 0.019444444444444445\right)\right), B \cdot B, 0.3333333333333333 \cdot x\right), B \cdot B, -\mathsf{fma}\left(0.16666666666666666, B \cdot B, 1\right)\right) - x}{B} \]

   9. Taylor expanded in B around 0

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

   11. Applied rewrites 56.4%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.16666666666666666\right), B \cdot B, -1\right) - x}{B} \]

   if -2.3e11 < F < 1.4199999999999999

   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 51.8

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

   5. Applied rewrites 51.8%

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

   6. Taylor expanded in F around 0

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

   8. Applied rewrites 51.8%

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

   if 1.4199999999999999 < F

   1. Initial program 53.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 40.7

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

   5. Applied rewrites 40.7%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 53.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 53.1%

       \[\leadsto \frac{\left(1 - \frac{1}{F \cdot F}\right) - x}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -230000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.16666666666666666\right), B \cdot B, -1\right) - x}{B}\\ \mathbf{elif}\;F \leq 1.42:\\ \;\;\;\;\mathsf{fma}\left(\frac{F}{B}, \sqrt{{\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-1}}, \frac{-x}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - {\left(F \cdot F\right)}^{-1}\right) - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 50.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.16666666666666666, B \cdot B, 1\right)\\ \mathbf{if}\;F \leq -6.8 \cdot 10^{+44}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-\mathsf{fma}\left(0.00205026455026455, B \cdot B, 0.019444444444444445\right), B \cdot B, 0.3333333333333333 \cdot x\right), B \cdot B, -t\_0\right) - x}{B}\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{+34}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}}, F, -x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(t\_0 + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0021164021164021165, B \cdot B, 0.022222222222222223\right), x, 0.019444444444444445\right) - -0.00205026455026455 \cdot \left(B \cdot B\right), B \cdot B, 0.3333333333333333 \cdot x\right) \cdot \left(B \cdot B\right)\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (fma 0.16666666666666666 (* B B) 1.0)))
   (if (<= F -6.8e+44)
     (/
      (-
       (fma
        (fma
         (- (fma 0.00205026455026455 (* B B) 0.019444444444444445))
         (* B B)
         (* 0.3333333333333333 x))
        (* B B)
        (- t_0))
       x)
      B)
     (if (<= F 1.8e+34)
       (/ (fma (sqrt (pow (fma F F (fma 2.0 x 2.0)) -1.0)) F (- x)) B)
       (/
        (-
         (+
          t_0
          (*
           (fma
            (-
             (fma
              (fma 0.0021164021164021165 (* B B) 0.022222222222222223)
              x
              0.019444444444444445)
             (* -0.00205026455026455 (* B B)))
            (* B B)
            (* 0.3333333333333333 x))
           (* B B)))
         x)
        B)))))

C

double code(double F, double B, double x) {
	double t_0 = fma(0.16666666666666666, (B * B), 1.0);
	double tmp;
	if (F <= -6.8e+44) {
		tmp = (fma(fma(-fma(0.00205026455026455, (B * B), 0.019444444444444445), (B * B), (0.3333333333333333 * x)), (B * B), -t_0) - x) / B;
	} else if (F <= 1.8e+34) {
		tmp = fma(sqrt(pow(fma(F, F, fma(2.0, x, 2.0)), -1.0)), F, -x) / B;
	} else {
		tmp = ((t_0 + (fma((fma(fma(0.0021164021164021165, (B * B), 0.022222222222222223), x, 0.019444444444444445) - (-0.00205026455026455 * (B * B))), (B * B), (0.3333333333333333 * x)) * (B * B))) - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = fma(0.16666666666666666, Float64(B * B), 1.0)
	tmp = 0.0
	if (F <= -6.8e+44)
		tmp = Float64(Float64(fma(fma(Float64(-fma(0.00205026455026455, Float64(B * B), 0.019444444444444445)), Float64(B * B), Float64(0.3333333333333333 * x)), Float64(B * B), Float64(-t_0)) - x) / B);
	elseif (F <= 1.8e+34)
		tmp = Float64(fma(sqrt((fma(F, F, fma(2.0, x, 2.0)) ^ -1.0)), F, Float64(-x)) / B);
	else
		tmp = Float64(Float64(Float64(t_0 + Float64(fma(Float64(fma(fma(0.0021164021164021165, Float64(B * B), 0.022222222222222223), x, 0.019444444444444445) - Float64(-0.00205026455026455 * Float64(B * B))), Float64(B * B), Float64(0.3333333333333333 * x)) * Float64(B * B))) - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(0.16666666666666666 * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[F, -6.8e+44], N[(N[(N[(N[((-N[(0.00205026455026455 * N[(B * B), $MachinePrecision] + 0.019444444444444445), $MachinePrecision]) * N[(B * B), $MachinePrecision] + N[(0.3333333333333333 * x), $MachinePrecision]), $MachinePrecision] * N[(B * B), $MachinePrecision] + (-t$95$0)), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.8e+34], 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[(t$95$0 + N[(N[(N[(N[(N[(0.0021164021164021165 * N[(B * B), $MachinePrecision] + 0.022222222222222223), $MachinePrecision] * x + 0.019444444444444445), $MachinePrecision] - N[(-0.00205026455026455 * N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(B * B), $MachinePrecision] + N[(0.3333333333333333 * x), $MachinePrecision]), $MachinePrecision] * N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -6.8e44

   1. Initial program 66.0%

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

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \left(\frac{1}{3} \cdot x + {B}^{2} \cdot \left(-1 \cdot \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \left(\frac{-1}{36} \cdot F + \frac{1}{120} \cdot F\right)\right) + \left(\frac{-1}{9} \cdot x + \left(\frac{2}{15} \cdot x + {B}^{2} \cdot \left(-1 \cdot \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \left(\frac{-1}{5040} \cdot F + \left(\frac{1}{720} \cdot F + \frac{1}{6} \cdot \left(\frac{-1}{36} \cdot F + \frac{1}{120} \cdot F\right)\right)\right)\right) + \left(\frac{-1}{3} \cdot \left(\frac{-1}{9} \cdot x + \frac{2}{15} \cdot x\right) + \left(\frac{-2}{45} \cdot x + \frac{17}{315} \cdot x\right)\right)\right)\right)\right)\right)\right)\right)\right) - x}{B}} \]

   5. Applied rewrites 47.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right), \sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.022222222222222223, x, \mathsf{fma}\left(\left(x \cdot 0.0021164021164021165\right) \cdot B, B, \sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \left(0.019444444444444445 \cdot F - \left(\left(F \cdot -0.00205026455026455\right) \cdot B\right) \cdot B\right)\right)\right), B \cdot B, 0.3333333333333333 \cdot x\right), B \cdot B, -x\right)\right)}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{\left(-1 \cdot \left(1 + \frac{1}{6} \cdot {B}^{2}\right) + {B}^{2} \cdot \left(\frac{1}{3} \cdot x + {B}^{2} \cdot \left(-1 \cdot \left(\frac{7}{360} + \frac{31}{15120} \cdot {B}^{2}\right) + \left(\frac{2}{945} \cdot \left({B}^{2} \cdot x\right) + \frac{1}{45} \cdot x\right)\right)\right)\right) - x}{B} \]

   8. Applied rewrites 59.4%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0021164021164021165, B \cdot B, 0.022222222222222223\right), x, -\mathsf{fma}\left(0.00205026455026455, B \cdot B, 0.019444444444444445\right)\right), B \cdot B, 0.3333333333333333 \cdot x\right), B \cdot B, -\mathsf{fma}\left(0.16666666666666666, B \cdot B, 1\right)\right) - x}{B} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-1 \cdot \left(\frac{7}{360} + \frac{31}{15120} \cdot {B}^{2}\right), B \cdot B, \frac{1}{3} \cdot x\right), B \cdot B, -\mathsf{fma}\left(\frac{1}{6}, B \cdot B, 1\right)\right) - x}{B} \]
   10. Step-by-step derivation

   11. Applied rewrites 59.8%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-\mathsf{fma}\left(0.00205026455026455, B \cdot B, 0.019444444444444445\right), B \cdot B, 0.3333333333333333 \cdot x\right), B \cdot B, -\mathsf{fma}\left(0.16666666666666666, B \cdot B, 1\right)\right) - x}{B} \]

   if -6.8e44 < F < 1.8e34

   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 50.4

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

   5. Applied rewrites 50.4%

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

   if 1.8e34 < F

   1. Initial program 49.9%

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

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \left(\frac{1}{3} \cdot x + {B}^{2} \cdot \left(-1 \cdot \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \left(\frac{-1}{36} \cdot F + \frac{1}{120} \cdot F\right)\right) + \left(\frac{-1}{9} \cdot x + \left(\frac{2}{15} \cdot x + {B}^{2} \cdot \left(-1 \cdot \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \left(\frac{-1}{5040} \cdot F + \left(\frac{1}{720} \cdot F + \frac{1}{6} \cdot \left(\frac{-1}{36} \cdot F + \frac{1}{120} \cdot F\right)\right)\right)\right) + \left(\frac{-1}{3} \cdot \left(\frac{-1}{9} \cdot x + \frac{2}{15} \cdot x\right) + \left(\frac{-2}{45} \cdot x + \frac{17}{315} \cdot x\right)\right)\right)\right)\right)\right)\right)\right)\right) - x}{B}} \]

   5. Applied rewrites 40.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right), \sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.022222222222222223, x, \mathsf{fma}\left(\left(x \cdot 0.0021164021164021165\right) \cdot B, B, \sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \left(0.019444444444444445 \cdot F - \left(\left(F \cdot -0.00205026455026455\right) \cdot B\right) \cdot B\right)\right)\right), B \cdot B, 0.3333333333333333 \cdot x\right), B \cdot B, -x\right)\right)}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{\left(1 + \left(\frac{1}{6} \cdot {B}^{2} + {B}^{2} \cdot \left(\frac{1}{3} \cdot x + {B}^{2} \cdot \left(\left(\frac{7}{360} + \left(\frac{2}{945} \cdot \left({B}^{2} \cdot x\right) + \frac{1}{45} \cdot x\right)\right) - \frac{-31}{15120} \cdot {B}^{2}\right)\right)\right)\right) - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 53.7%

      \[\leadsto \frac{\left(\mathsf{fma}\left(0.16666666666666666, B \cdot B, 1\right) + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0021164021164021165, B \cdot B, 0.022222222222222223\right), x, 0.019444444444444445\right) - -0.00205026455026455 \cdot \left(B \cdot B\right), B \cdot B, 0.3333333333333333 \cdot x\right) \cdot \left(B \cdot B\right)\right) - x}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6.8 \cdot 10^{+44}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-\mathsf{fma}\left(0.00205026455026455, B \cdot B, 0.019444444444444445\right), B \cdot B, 0.3333333333333333 \cdot x\right), B \cdot B, -\mathsf{fma}\left(0.16666666666666666, B \cdot B, 1\right)\right) - x}{B}\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{+34}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}}, F, -x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(0.16666666666666666, B \cdot B, 1\right) + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0021164021164021165, B \cdot B, 0.022222222222222223\right), x, 0.019444444444444445\right) - -0.00205026455026455 \cdot \left(B \cdot B\right), B \cdot B, 0.3333333333333333 \cdot x\right) \cdot \left(B \cdot B\right)\right) - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 50.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -6.8 \cdot 10^{+44}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-\mathsf{fma}\left(0.00205026455026455, B \cdot B, 0.019444444444444445\right), B \cdot B, 0.3333333333333333 \cdot x\right), B \cdot B, -\mathsf{fma}\left(0.16666666666666666, B \cdot B, 1\right)\right) - x}{B}\\ \mathbf{elif}\;F \leq 50000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}}, F, -x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -6.8e+44)
   (/
    (-
     (fma
      (fma
       (- (fma 0.00205026455026455 (* B B) 0.019444444444444445))
       (* B B)
       (* 0.3333333333333333 x))
      (* B B)
      (- (fma 0.16666666666666666 (* B B) 1.0)))
     x)
    B)
   (if (<= F 50000.0)
     (/ (fma (sqrt (pow (fma F F (fma 2.0 x 2.0)) -1.0)) F (- x)) B)
     (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -6.8e+44) {
		tmp = (fma(fma(-fma(0.00205026455026455, (B * B), 0.019444444444444445), (B * B), (0.3333333333333333 * x)), (B * B), -fma(0.16666666666666666, (B * B), 1.0)) - x) / B;
	} else if (F <= 50000.0) {
		tmp = fma(sqrt(pow(fma(F, F, fma(2.0, x, 2.0)), -1.0)), F, -x) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -6.8e+44)
		tmp = Float64(Float64(fma(fma(Float64(-fma(0.00205026455026455, Float64(B * B), 0.019444444444444445)), Float64(B * B), Float64(0.3333333333333333 * x)), Float64(B * B), Float64(-fma(0.16666666666666666, Float64(B * B), 1.0))) - x) / B);
	elseif (F <= 50000.0)
		tmp = Float64(fma(sqrt((fma(F, F, fma(2.0, x, 2.0)) ^ -1.0)), F, Float64(-x)) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -6.8e+44], N[(N[(N[(N[((-N[(0.00205026455026455 * N[(B * B), $MachinePrecision] + 0.019444444444444445), $MachinePrecision]) * N[(B * B), $MachinePrecision] + N[(0.3333333333333333 * x), $MachinePrecision]), $MachinePrecision] * N[(B * B), $MachinePrecision] + (-N[(0.16666666666666666 * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision])), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 50000.0], 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[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -6.8e44

   1. Initial program 66.0%

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

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \left(\frac{1}{3} \cdot x + {B}^{2} \cdot \left(-1 \cdot \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \left(\frac{-1}{36} \cdot F + \frac{1}{120} \cdot F\right)\right) + \left(\frac{-1}{9} \cdot x + \left(\frac{2}{15} \cdot x + {B}^{2} \cdot \left(-1 \cdot \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \left(\frac{-1}{5040} \cdot F + \left(\frac{1}{720} \cdot F + \frac{1}{6} \cdot \left(\frac{-1}{36} \cdot F + \frac{1}{120} \cdot F\right)\right)\right)\right) + \left(\frac{-1}{3} \cdot \left(\frac{-1}{9} \cdot x + \frac{2}{15} \cdot x\right) + \left(\frac{-2}{45} \cdot x + \frac{17}{315} \cdot x\right)\right)\right)\right)\right)\right)\right)\right)\right) - x}{B}} \]

   5. Applied rewrites 47.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right), \sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.022222222222222223, x, \mathsf{fma}\left(\left(x \cdot 0.0021164021164021165\right) \cdot B, B, \sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \left(0.019444444444444445 \cdot F - \left(\left(F \cdot -0.00205026455026455\right) \cdot B\right) \cdot B\right)\right)\right), B \cdot B, 0.3333333333333333 \cdot x\right), B \cdot B, -x\right)\right)}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{\left(-1 \cdot \left(1 + \frac{1}{6} \cdot {B}^{2}\right) + {B}^{2} \cdot \left(\frac{1}{3} \cdot x + {B}^{2} \cdot \left(-1 \cdot \left(\frac{7}{360} + \frac{31}{15120} \cdot {B}^{2}\right) + \left(\frac{2}{945} \cdot \left({B}^{2} \cdot x\right) + \frac{1}{45} \cdot x\right)\right)\right)\right) - x}{B} \]

   8. Applied rewrites 59.4%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0021164021164021165, B \cdot B, 0.022222222222222223\right), x, -\mathsf{fma}\left(0.00205026455026455, B \cdot B, 0.019444444444444445\right)\right), B \cdot B, 0.3333333333333333 \cdot x\right), B \cdot B, -\mathsf{fma}\left(0.16666666666666666, B \cdot B, 1\right)\right) - x}{B} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-1 \cdot \left(\frac{7}{360} + \frac{31}{15120} \cdot {B}^{2}\right), B \cdot B, \frac{1}{3} \cdot x\right), B \cdot B, -\mathsf{fma}\left(\frac{1}{6}, B \cdot B, 1\right)\right) - x}{B} \]
   10. Step-by-step derivation

   11. Applied rewrites 59.8%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-\mathsf{fma}\left(0.00205026455026455, B \cdot B, 0.019444444444444445\right), B \cdot B, 0.3333333333333333 \cdot x\right), B \cdot B, -\mathsf{fma}\left(0.16666666666666666, B \cdot B, 1\right)\right) - x}{B} \]

   if -6.8e44 < F < 5e4

   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 51.0

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

   5. Applied rewrites 51.0%

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

   if 5e4 < F

   1. Initial program 52.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 39.9

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

   5. Applied rewrites 39.9%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 52.4%

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

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6.8 \cdot 10^{+44}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-\mathsf{fma}\left(0.00205026455026455, B \cdot B, 0.019444444444444445\right), B \cdot B, 0.3333333333333333 \cdot x\right), B \cdot B, -\mathsf{fma}\left(0.16666666666666666, B \cdot B, 1\right)\right) - x}{B}\\ \mathbf{elif}\;F \leq 50000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}}, F, -x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 50.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -4.5 \cdot 10^{+59}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, \mathsf{fma}\left(\mathsf{fma}\left(0.022222222222222223, x, \mathsf{fma}\left(\mathsf{fma}\left(0.0021164021164021165, x, -0.00205026455026455\right), B \cdot B, -0.019444444444444445\right)\right), B \cdot B, -0.16666666666666666\right)\right), B \cdot B, -1\right) - x}{B}\\ \mathbf{elif}\;F \leq 50000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}}, F, -x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4.5e+59)
   (/
    (-
     (fma
      (fma
       0.3333333333333333
       x
       (fma
        (fma
         0.022222222222222223
         x
         (fma
          (fma 0.0021164021164021165 x -0.00205026455026455)
          (* B B)
          -0.019444444444444445))
        (* B B)
        -0.16666666666666666))
      (* B B)
      -1.0)
     x)
    B)
   (if (<= F 50000.0)
     (/ (fma (sqrt (pow (fma F F (fma 2.0 x 2.0)) -1.0)) F (- x)) B)
     (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.5e+59) {
		tmp = (fma(fma(0.3333333333333333, x, fma(fma(0.022222222222222223, x, fma(fma(0.0021164021164021165, x, -0.00205026455026455), (B * B), -0.019444444444444445)), (B * B), -0.16666666666666666)), (B * B), -1.0) - x) / B;
	} else if (F <= 50000.0) {
		tmp = fma(sqrt(pow(fma(F, F, fma(2.0, x, 2.0)), -1.0)), F, -x) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -4.5e+59)
		tmp = Float64(Float64(fma(fma(0.3333333333333333, x, fma(fma(0.022222222222222223, x, fma(fma(0.0021164021164021165, x, -0.00205026455026455), Float64(B * B), -0.019444444444444445)), Float64(B * B), -0.16666666666666666)), Float64(B * B), -1.0) - x) / B);
	elseif (F <= 50000.0)
		tmp = Float64(fma(sqrt((fma(F, F, fma(2.0, x, 2.0)) ^ -1.0)), F, Float64(-x)) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -4.5e+59], N[(N[(N[(N[(0.3333333333333333 * x + N[(N[(0.022222222222222223 * x + N[(N[(0.0021164021164021165 * x + -0.00205026455026455), $MachinePrecision] * N[(B * B), $MachinePrecision] + -0.019444444444444445), $MachinePrecision]), $MachinePrecision] * N[(B * B), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(B * B), $MachinePrecision] + -1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 50000.0], 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[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.49999999999999959e59

   1. Initial program 64.1%

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

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \left(\frac{1}{3} \cdot x + {B}^{2} \cdot \left(-1 \cdot \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \left(\frac{-1}{36} \cdot F + \frac{1}{120} \cdot F\right)\right) + \left(\frac{-1}{9} \cdot x + \left(\frac{2}{15} \cdot x + {B}^{2} \cdot \left(-1 \cdot \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \left(\frac{-1}{5040} \cdot F + \left(\frac{1}{720} \cdot F + \frac{1}{6} \cdot \left(\frac{-1}{36} \cdot F + \frac{1}{120} \cdot F\right)\right)\right)\right) + \left(\frac{-1}{3} \cdot \left(\frac{-1}{9} \cdot x + \frac{2}{15} \cdot x\right) + \left(\frac{-2}{45} \cdot x + \frac{17}{315} \cdot x\right)\right)\right)\right)\right)\right)\right)\right)\right) - x}{B}} \]

   5. Applied rewrites 48.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right), \sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.022222222222222223, x, \mathsf{fma}\left(\left(x \cdot 0.0021164021164021165\right) \cdot B, B, \sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \left(0.019444444444444445 \cdot F - \left(\left(F \cdot -0.00205026455026455\right) \cdot B\right) \cdot B\right)\right)\right), B \cdot B, 0.3333333333333333 \cdot x\right), B \cdot B, -x\right)\right)}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{\left(-1 \cdot \left(1 + \frac{1}{6} \cdot {B}^{2}\right) + {B}^{2} \cdot \left(\frac{1}{3} \cdot x + {B}^{2} \cdot \left(-1 \cdot \left(\frac{7}{360} + \frac{31}{15120} \cdot {B}^{2}\right) + \left(\frac{2}{945} \cdot \left({B}^{2} \cdot x\right) + \frac{1}{45} \cdot x\right)\right)\right)\right) - x}{B} \]

   8. Applied rewrites 60.7%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0021164021164021165, B \cdot B, 0.022222222222222223\right), x, -\mathsf{fma}\left(0.00205026455026455, B \cdot B, 0.019444444444444445\right)\right), B \cdot B, 0.3333333333333333 \cdot x\right), B \cdot B, -\mathsf{fma}\left(0.16666666666666666, B \cdot B, 1\right)\right) - x}{B} \]

   9. Taylor expanded in B around 0

      \[\leadsto \frac{\left({B}^{2} \cdot \left(\left(\frac{1}{3} \cdot x + {B}^{2} \cdot \left(\left(\frac{1}{45} \cdot x + {B}^{2} \cdot \left(\frac{2}{945} \cdot x - \frac{31}{15120}\right)\right) - \frac{7}{360}\right)\right) - \frac{1}{6}\right) - 1\right) - x}{B} \]
   10. Step-by-step derivation

   11. Applied rewrites 61.0%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, \mathsf{fma}\left(\mathsf{fma}\left(0.022222222222222223, x, \mathsf{fma}\left(\mathsf{fma}\left(0.0021164021164021165, x, -0.00205026455026455\right), B \cdot B, -0.019444444444444445\right)\right), B \cdot B, -0.16666666666666666\right)\right), B \cdot B, -1\right) - x}{B} \]

   if -4.49999999999999959e59 < F < 5e4

   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 50.7

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

   5. Applied rewrites 50.7%

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

   if 5e4 < F

   1. Initial program 52.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 39.9

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

   5. Applied rewrites 39.9%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 52.4%

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

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.5 \cdot 10^{+59}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, \mathsf{fma}\left(\mathsf{fma}\left(0.022222222222222223, x, \mathsf{fma}\left(\mathsf{fma}\left(0.0021164021164021165, x, -0.00205026455026455\right), B \cdot B, -0.019444444444444445\right)\right), B \cdot B, -0.16666666666666666\right)\right), B \cdot B, -1\right) - x}{B}\\ \mathbf{elif}\;F \leq 50000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}}, F, -x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 49.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -230000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.16666666666666666\right), B \cdot B, -1\right) - x}{B}\\ \mathbf{elif}\;F \leq 1.42:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-1}}, F, -x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - {\left(F \cdot F\right)}^{-1}\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -230000000000.0)
   (/
    (- (fma (fma 0.3333333333333333 x -0.16666666666666666) (* B B) -1.0) x)
    B)
   (if (<= F 1.42)
     (/ (fma (sqrt (pow (fma 2.0 x 2.0) -1.0)) F (- x)) B)
     (/ (- (- 1.0 (pow (* F F) -1.0)) x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -230000000000.0) {
		tmp = (fma(fma(0.3333333333333333, x, -0.16666666666666666), (B * B), -1.0) - x) / B;
	} else if (F <= 1.42) {
		tmp = fma(sqrt(pow(fma(2.0, x, 2.0), -1.0)), F, -x) / B;
	} else {
		tmp = ((1.0 - pow((F * F), -1.0)) - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -230000000000.0)
		tmp = Float64(Float64(fma(fma(0.3333333333333333, x, -0.16666666666666666), Float64(B * B), -1.0) - x) / B);
	elseif (F <= 1.42)
		tmp = Float64(fma(sqrt((fma(2.0, x, 2.0) ^ -1.0)), F, Float64(-x)) / B);
	else
		tmp = Float64(Float64(Float64(1.0 - (Float64(F * F) ^ -1.0)) - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -230000000000.0], N[(N[(N[(N[(0.3333333333333333 * x + -0.16666666666666666), $MachinePrecision] * N[(B * B), $MachinePrecision] + -1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.42], 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[(1.0 - N[Power[N[(F * F), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.3e11

   1. Initial program 70.5%

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

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \left(\frac{1}{3} \cdot x + {B}^{2} \cdot \left(-1 \cdot \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \left(\frac{-1}{36} \cdot F + \frac{1}{120} \cdot F\right)\right) + \left(\frac{-1}{9} \cdot x + \left(\frac{2}{15} \cdot x + {B}^{2} \cdot \left(-1 \cdot \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \left(\frac{-1}{5040} \cdot F + \left(\frac{1}{720} \cdot F + \frac{1}{6} \cdot \left(\frac{-1}{36} \cdot F + \frac{1}{120} \cdot F\right)\right)\right)\right) + \left(\frac{-1}{3} \cdot \left(\frac{-1}{9} \cdot x + \frac{2}{15} \cdot x\right) + \left(\frac{-2}{45} \cdot x + \frac{17}{315} \cdot x\right)\right)\right)\right)\right)\right)\right)\right)\right) - x}{B}} \]

   5. Applied rewrites 45.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right), \sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.022222222222222223, x, \mathsf{fma}\left(\left(x \cdot 0.0021164021164021165\right) \cdot B, B, \sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \left(0.019444444444444445 \cdot F - \left(\left(F \cdot -0.00205026455026455\right) \cdot B\right) \cdot B\right)\right)\right), B \cdot B, 0.3333333333333333 \cdot x\right), B \cdot B, -x\right)\right)}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{\left(-1 \cdot \left(1 + \frac{1}{6} \cdot {B}^{2}\right) + {B}^{2} \cdot \left(\frac{1}{3} \cdot x + {B}^{2} \cdot \left(-1 \cdot \left(\frac{7}{360} + \frac{31}{15120} \cdot {B}^{2}\right) + \left(\frac{2}{945} \cdot \left({B}^{2} \cdot x\right) + \frac{1}{45} \cdot x\right)\right)\right)\right) - x}{B} \]

   8. Applied rewrites 56.0%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0021164021164021165, B \cdot B, 0.022222222222222223\right), x, -\mathsf{fma}\left(0.00205026455026455, B \cdot B, 0.019444444444444445\right)\right), B \cdot B, 0.3333333333333333 \cdot x\right), B \cdot B, -\mathsf{fma}\left(0.16666666666666666, B \cdot B, 1\right)\right) - x}{B} \]

   9. Taylor expanded in B around 0

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

   11. Applied rewrites 56.4%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.16666666666666666\right), B \cdot B, -1\right) - x}{B} \]

   if -2.3e11 < F < 1.4199999999999999

   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 51.8

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

   5. Applied rewrites 51.8%

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

   6. Taylor expanded in F around 0

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

   8. Applied rewrites 51.8%

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

   if 1.4199999999999999 < F

   1. Initial program 53.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 40.7

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

   5. Applied rewrites 40.7%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 53.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 53.1%

       \[\leadsto \frac{\left(1 - \frac{1}{F \cdot F}\right) - x}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -230000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.16666666666666666\right), B \cdot B, -1\right) - x}{B}\\ \mathbf{elif}\;F \leq 1.42:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-1}}, F, -x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - {\left(F \cdot F\right)}^{-1}\right) - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 49.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -230000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.16666666666666666\right), B \cdot B, -1\right) - x}{B}\\ \mathbf{elif}\;F \leq 1.42:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{0.5}, F, -x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - {\left(F \cdot F\right)}^{-1}\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -230000000000.0)
   (/
    (- (fma (fma 0.3333333333333333 x -0.16666666666666666) (* B B) -1.0) x)
    B)
   (if (<= F 1.42)
     (/ (fma (sqrt 0.5) F (- x)) B)
     (/ (- (- 1.0 (pow (* F F) -1.0)) x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -230000000000.0) {
		tmp = (fma(fma(0.3333333333333333, x, -0.16666666666666666), (B * B), -1.0) - x) / B;
	} else if (F <= 1.42) {
		tmp = fma(sqrt(0.5), F, -x) / B;
	} else {
		tmp = ((1.0 - pow((F * F), -1.0)) - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -230000000000.0)
		tmp = Float64(Float64(fma(fma(0.3333333333333333, x, -0.16666666666666666), Float64(B * B), -1.0) - x) / B);
	elseif (F <= 1.42)
		tmp = Float64(fma(sqrt(0.5), F, Float64(-x)) / B);
	else
		tmp = Float64(Float64(Float64(1.0 - (Float64(F * F) ^ -1.0)) - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -230000000000.0], N[(N[(N[(N[(0.3333333333333333 * x + -0.16666666666666666), $MachinePrecision] * N[(B * B), $MachinePrecision] + -1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.42], N[(N[(N[Sqrt[0.5], $MachinePrecision] * F + (-x)), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(1.0 - N[Power[N[(F * F), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.3e11

   1. Initial program 70.5%

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

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \left(\frac{1}{3} \cdot x + {B}^{2} \cdot \left(-1 \cdot \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \left(\frac{-1}{36} \cdot F + \frac{1}{120} \cdot F\right)\right) + \left(\frac{-1}{9} \cdot x + \left(\frac{2}{15} \cdot x + {B}^{2} \cdot \left(-1 \cdot \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \left(\frac{-1}{5040} \cdot F + \left(\frac{1}{720} \cdot F + \frac{1}{6} \cdot \left(\frac{-1}{36} \cdot F + \frac{1}{120} \cdot F\right)\right)\right)\right) + \left(\frac{-1}{3} \cdot \left(\frac{-1}{9} \cdot x + \frac{2}{15} \cdot x\right) + \left(\frac{-2}{45} \cdot x + \frac{17}{315} \cdot x\right)\right)\right)\right)\right)\right)\right)\right)\right) - x}{B}} \]

   5. Applied rewrites 45.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right), \sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.022222222222222223, x, \mathsf{fma}\left(\left(x \cdot 0.0021164021164021165\right) \cdot B, B, \sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \left(0.019444444444444445 \cdot F - \left(\left(F \cdot -0.00205026455026455\right) \cdot B\right) \cdot B\right)\right)\right), B \cdot B, 0.3333333333333333 \cdot x\right), B \cdot B, -x\right)\right)}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{\left(-1 \cdot \left(1 + \frac{1}{6} \cdot {B}^{2}\right) + {B}^{2} \cdot \left(\frac{1}{3} \cdot x + {B}^{2} \cdot \left(-1 \cdot \left(\frac{7}{360} + \frac{31}{15120} \cdot {B}^{2}\right) + \left(\frac{2}{945} \cdot \left({B}^{2} \cdot x\right) + \frac{1}{45} \cdot x\right)\right)\right)\right) - x}{B} \]

   8. Applied rewrites 56.0%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0021164021164021165, B \cdot B, 0.022222222222222223\right), x, -\mathsf{fma}\left(0.00205026455026455, B \cdot B, 0.019444444444444445\right)\right), B \cdot B, 0.3333333333333333 \cdot x\right), B \cdot B, -\mathsf{fma}\left(0.16666666666666666, B \cdot B, 1\right)\right) - x}{B} \]

   9. Taylor expanded in B around 0

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

   11. Applied rewrites 56.4%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.16666666666666666\right), B \cdot B, -1\right) - x}{B} \]

   if -2.3e11 < F < 1.4199999999999999

   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 51.8

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

   5. Applied rewrites 51.8%

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

   6. Taylor expanded in F around 0

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

   8. Applied rewrites 51.8%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 51.8%

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

   if 1.4199999999999999 < F

   1. Initial program 53.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 40.7

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

   5. Applied rewrites 40.7%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 53.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 53.1%

       \[\leadsto \frac{\left(1 - \frac{1}{F \cdot F}\right) - x}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -230000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.16666666666666666\right), B \cdot B, -1\right) - x}{B}\\ \mathbf{elif}\;F \leq 1.42:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{0.5}, F, -x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - {\left(F \cdot F\right)}^{-1}\right) - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 49.5% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -230000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.16666666666666666\right), B \cdot B, -1\right) - x}{B}\\ \mathbf{elif}\;F \leq 2.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{0.5}, F, -x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -230000000000.0)
   (/
    (- (fma (fma 0.3333333333333333 x -0.16666666666666666) (* B B) -1.0) x)
    B)
   (if (<= F 2.5e-12) (/ (fma (sqrt 0.5) F (- x)) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -230000000000.0) {
		tmp = (fma(fma(0.3333333333333333, x, -0.16666666666666666), (B * B), -1.0) - x) / B;
	} else if (F <= 2.5e-12) {
		tmp = fma(sqrt(0.5), F, -x) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -230000000000.0)
		tmp = Float64(Float64(fma(fma(0.3333333333333333, x, -0.16666666666666666), Float64(B * B), -1.0) - x) / B);
	elseif (F <= 2.5e-12)
		tmp = Float64(fma(sqrt(0.5), F, Float64(-x)) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -230000000000.0], N[(N[(N[(N[(0.3333333333333333 * x + -0.16666666666666666), $MachinePrecision] * N[(B * B), $MachinePrecision] + -1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 2.5e-12], N[(N[(N[Sqrt[0.5], $MachinePrecision] * F + (-x)), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.3e11

   1. Initial program 70.5%

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

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \left(\frac{1}{3} \cdot x + {B}^{2} \cdot \left(-1 \cdot \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \left(\frac{-1}{36} \cdot F + \frac{1}{120} \cdot F\right)\right) + \left(\frac{-1}{9} \cdot x + \left(\frac{2}{15} \cdot x + {B}^{2} \cdot \left(-1 \cdot \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \left(\frac{-1}{5040} \cdot F + \left(\frac{1}{720} \cdot F + \frac{1}{6} \cdot \left(\frac{-1}{36} \cdot F + \frac{1}{120} \cdot F\right)\right)\right)\right) + \left(\frac{-1}{3} \cdot \left(\frac{-1}{9} \cdot x + \frac{2}{15} \cdot x\right) + \left(\frac{-2}{45} \cdot x + \frac{17}{315} \cdot x\right)\right)\right)\right)\right)\right)\right)\right)\right) - x}{B}} \]

   5. Applied rewrites 45.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right), \sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.022222222222222223, x, \mathsf{fma}\left(\left(x \cdot 0.0021164021164021165\right) \cdot B, B, \sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \left(0.019444444444444445 \cdot F - \left(\left(F \cdot -0.00205026455026455\right) \cdot B\right) \cdot B\right)\right)\right), B \cdot B, 0.3333333333333333 \cdot x\right), B \cdot B, -x\right)\right)}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{\left(-1 \cdot \left(1 + \frac{1}{6} \cdot {B}^{2}\right) + {B}^{2} \cdot \left(\frac{1}{3} \cdot x + {B}^{2} \cdot \left(-1 \cdot \left(\frac{7}{360} + \frac{31}{15120} \cdot {B}^{2}\right) + \left(\frac{2}{945} \cdot \left({B}^{2} \cdot x\right) + \frac{1}{45} \cdot x\right)\right)\right)\right) - x}{B} \]

   8. Applied rewrites 56.0%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0021164021164021165, B \cdot B, 0.022222222222222223\right), x, -\mathsf{fma}\left(0.00205026455026455, B \cdot B, 0.019444444444444445\right)\right), B \cdot B, 0.3333333333333333 \cdot x\right), B \cdot B, -\mathsf{fma}\left(0.16666666666666666, B \cdot B, 1\right)\right) - x}{B} \]

   9. Taylor expanded in B around 0

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

   11. Applied rewrites 56.4%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.16666666666666666\right), B \cdot B, -1\right) - x}{B} \]

   if -2.3e11 < F < 2.49999999999999985e-12

   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 52.1

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

   5. Applied rewrites 52.1%

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

   6. Taylor expanded in F around 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 52.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 52.1%

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

   if 2.49999999999999985e-12 < F

   1. Initial program 55.2%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 40.6

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

   5. Applied rewrites 40.6%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 52.0%

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

Alternative 18: 49.2% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.9 \cdot 10^{-23}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{0.5}, F, -x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.9e-23)
   (/ (- -1.0 x) B)
   (if (<= F 2.5e-12) (/ (fma (sqrt 0.5) F (- x)) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.9e-23) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 2.5e-12) {
		tmp = fma(sqrt(0.5), F, -x) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.9e-23)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 2.5e-12)
		tmp = Float64(fma(sqrt(0.5), F, Float64(-x)) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.9e-23], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 2.5e-12], N[(N[(N[Sqrt[0.5], $MachinePrecision] * F + (-x)), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.90000000000000006e-23

   1. Initial program 72.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 44.6

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

   5. Applied rewrites 44.6%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 54.1%

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

   if -1.90000000000000006e-23 < F < 2.49999999999999985e-12

   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 53.1

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

   5. Applied rewrites 53.1%

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

   6. Taylor expanded in F around 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 53.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 53.1%

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

   if 2.49999999999999985e-12 < F

   1. Initial program 55.2%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 40.6

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

   5. Applied rewrites 40.6%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 52.0%

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

Alternative 19: 42.5% accurate, 13.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -4 \cdot 10^{-60}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 5.4 \cdot 10^{-24}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4e-60)
   (/ (- -1.0 x) B)
   (if (<= F 5.4e-24) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4e-60) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 5.4e-24) {
		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 <= (-4d-60)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 5.4d-24) 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 <= -4e-60) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 5.4e-24) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -4e-60:
		tmp = (-1.0 - x) / B
	elif F <= 5.4e-24:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -4e-60)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 5.4e-24)
		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 <= -4e-60)
		tmp = (-1.0 - x) / B;
	elseif (F <= 5.4e-24)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -4e-60], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 5.4e-24], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.9999999999999999e-60

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 45.1

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

   5. Applied rewrites 45.1%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 48.8%

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

   if -3.9999999999999999e-60 < F < 5.40000000000000014e-24

   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 53.1

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

   5. Applied rewrites 53.1%

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

   6. Taylor expanded in F around 0

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

   8. Applied rewrites 42.5%

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

   if 5.40000000000000014e-24 < F

   1. Initial program 56.9%

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

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   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 41.6

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

   5. Applied rewrites 41.6%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 51.5%

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

Alternative 20: 35.7% accurate, 17.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -4 \cdot 10^{-60}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4e-60) (/ (- -1.0 x) B) (/ (- x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4e-60) {
		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 <= (-4d-60)) 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 <= -4e-60) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -4e-60:
		tmp = (-1.0 - x) / B
	else:
		tmp = -x / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -4e-60)
		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 <= -4e-60)
		tmp = (-1.0 - x) / B;
	else
		tmp = -x / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -4e-60], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[((-x) / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -3.9999999999999999e-60

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 45.1

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

   5. Applied rewrites 45.1%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 48.8%

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

   if -3.9999999999999999e-60 < F

   1. Initial program 80.2%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 47.9

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

   5. Applied rewrites 47.9%

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

   6. Taylor expanded in F around 0

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

   8. Applied rewrites 32.7%

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

Alternative 21: 29.0% accurate, 26.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.9%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
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 47.0

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

5. Applied rewrites 47.0%

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

6. Taylor expanded in F around 0

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

8. Applied rewrites 29.8%

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

Reproduce

herbie shell --seed 2024314 
(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))))))