VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.1% → 99.7%

Time: 17.9s

Alternatives: 26

Speedup: 2.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := -\frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2 \cdot 10^{+68}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, -1, t\_1\right)\\ \mathbf{elif}\;F \leq 2.1 \cdot 10^{+20}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - x \cdot \cos B}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, 1, t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))) (t_1 (- (/ x (tan B)))))
   (if (<= F -2e+68)
     (fma t_0 -1.0 t_1)
     (if (<= F 2.1e+20)
       (/ (- (/ F (sqrt (fma 2.0 x (fma F F 2.0)))) (* x (cos B))) (sin B))
       (fma t_0 1.0 t_1)))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / sin(B);
	double t_1 = -(x / tan(B));
	double tmp;
	if (F <= -2e+68) {
		tmp = fma(t_0, -1.0, t_1);
	} else if (F <= 2.1e+20) {
		tmp = ((F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) - (x * cos(B))) / sin(B);
	} else {
		tmp = fma(t_0, 1.0, t_1);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / sin(B))
	t_1 = Float64(-Float64(x / tan(B)))
	tmp = 0.0
	if (F <= -2e+68)
		tmp = fma(t_0, -1.0, t_1);
	elseif (F <= 2.1e+20)
		tmp = Float64(Float64(Float64(F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) - Float64(x * cos(B))) / sin(B));
	else
		tmp = fma(t_0, 1.0, t_1);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision])}, If[LessEqual[F, -2e+68], N[(t$95$0 * -1.0 + t$95$1), $MachinePrecision], If[LessEqual[F, 2.1e+20], N[(N[(N[(F / N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * 1.0 + t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.99999999999999991e68

   1. Initial program 53.7%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 74.7%

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

   5. Taylor expanded in F around -inf

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

   7. Simplified 99.9%

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

   if -1.99999999999999991e68 < F < 2.1e20

   1. Initial program 98.7%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 99.5%

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

      1. unsub-neg N/A

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

      2. div-inv N/A

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

      3. associate-*l/ N/A

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

      4. tan-quot N/A

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

      5. clear-num N/A

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

      6. associate-/l* N/A

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

      7. sub-div N/A

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

      8. /-lowering-/.f64 N/A

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

   6. Applied egg-rr 99.6%

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

   if 2.1e20 < F

   1. Initial program 53.0%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 70.7%

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

   5. Taylor expanded in F around inf

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

   7. Simplified 99.7%

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

Alternative 2: 99.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := -\frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.45:\\ \;\;\;\;\mathsf{fma}\left(t\_0, -1, t\_1\right)\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, 2\right)}} - x \cdot \cos B}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, 1, t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))) (t_1 (- (/ x (tan B)))))
   (if (<= F -1.45)
     (fma t_0 -1.0 t_1)
     (if (<= F 1.4)
       (/ (- (/ F (sqrt (fma x 2.0 2.0))) (* x (cos B))) (sin B))
       (fma t_0 1.0 t_1)))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / sin(B);
	double t_1 = -(x / tan(B));
	double tmp;
	if (F <= -1.45) {
		tmp = fma(t_0, -1.0, t_1);
	} else if (F <= 1.4) {
		tmp = ((F / sqrt(fma(x, 2.0, 2.0))) - (x * cos(B))) / sin(B);
	} else {
		tmp = fma(t_0, 1.0, t_1);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / sin(B))
	t_1 = Float64(-Float64(x / tan(B)))
	tmp = 0.0
	if (F <= -1.45)
		tmp = fma(t_0, -1.0, t_1);
	elseif (F <= 1.4)
		tmp = Float64(Float64(Float64(F / sqrt(fma(x, 2.0, 2.0))) - Float64(x * cos(B))) / sin(B));
	else
		tmp = fma(t_0, 1.0, t_1);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision])}, If[LessEqual[F, -1.45], N[(t$95$0 * -1.0 + t$95$1), $MachinePrecision], If[LessEqual[F, 1.4], N[(N[(N[(F / N[Sqrt[N[(x * 2.0 + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * 1.0 + t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.44999999999999996

   1. Initial program 57.0%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 77.2%

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

   5. Taylor expanded in F around -inf

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

   7. Simplified 99.9%

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

   if -1.44999999999999996 < F < 1.3999999999999999

   1. Initial program 99.4%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 99.5%

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

      1. unsub-neg N/A

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

      2. div-inv N/A

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

      3. associate-*l/ N/A

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

      4. tan-quot N/A

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

      5. clear-num N/A

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

      6. associate-/l* N/A

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

      7. sub-div N/A

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

      8. /-lowering-/.f64 N/A

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in F around 0

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

      1. sqrt-lowering-sqrt.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-fma.f64 99.0

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

   9. Simplified 99.0%

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

   if 1.3999999999999999 < 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. Step-by-step derivation

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 73.1%

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

   5. Taylor expanded in F around inf

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

   7. Simplified 99.4%

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

Alternative 3: 92.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := -\frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.22 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, -1, t\_1\right)\\ \mathbf{elif}\;F \leq 1.15 \cdot 10^{-137}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \frac{F}{B}\\ \mathbf{elif}\;F \leq 7500000:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, 1, t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))) (t_1 (- (/ x (tan B)))))
   (if (<= F -1.22e-23)
     (fma t_0 -1.0 t_1)
     (if (<= F 1.15e-137)
       (+
        (* x (/ -1.0 (tan B)))
        (* (sqrt (/ 1.0 (fma x 2.0 (fma F F 2.0)))) (/ F B)))
       (if (<= F 7500000.0)
         (/ (- (/ F (sqrt (fma 2.0 x (fma F F 2.0)))) x) (sin B))
         (fma t_0 1.0 t_1))))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / sin(B);
	double t_1 = -(x / tan(B));
	double tmp;
	if (F <= -1.22e-23) {
		tmp = fma(t_0, -1.0, t_1);
	} else if (F <= 1.15e-137) {
		tmp = (x * (-1.0 / tan(B))) + (sqrt((1.0 / fma(x, 2.0, fma(F, F, 2.0)))) * (F / B));
	} else if (F <= 7500000.0) {
		tmp = ((F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) - x) / sin(B);
	} else {
		tmp = fma(t_0, 1.0, t_1);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / sin(B))
	t_1 = Float64(-Float64(x / tan(B)))
	tmp = 0.0
	if (F <= -1.22e-23)
		tmp = fma(t_0, -1.0, t_1);
	elseif (F <= 1.15e-137)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(sqrt(Float64(1.0 / fma(x, 2.0, fma(F, F, 2.0)))) * Float64(F / B)));
	elseif (F <= 7500000.0)
		tmp = Float64(Float64(Float64(F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) - x) / sin(B));
	else
		tmp = fma(t_0, 1.0, t_1);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision])}, If[LessEqual[F, -1.22e-23], N[(t$95$0 * -1.0 + t$95$1), $MachinePrecision], If[LessEqual[F, 1.15e-137], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 / N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7500000.0], N[(N[(N[(F / N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * 1.0 + t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.22000000000000007e-23

   1. Initial program 60.3%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 79.0%

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

   5. Taylor expanded in F around -inf

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

   7. Simplified 96.3%

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

   if -1.22000000000000007e-23 < F < 1.15000000000000004e-137

   1. Initial program 99.5%

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

   3. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. /-lowering-/.f64 88.6

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

   5. Simplified 88.6%

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

   if 1.15000000000000004e-137 < F < 7.5e6

   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. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 99.4%

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

      1. unsub-neg N/A

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

      2. div-inv N/A

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

      3. associate-*l/ N/A

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

      4. tan-quot N/A

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

      5. clear-num N/A

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

      6. associate-/l* N/A

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

      7. sub-div N/A

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

      8. /-lowering-/.f64 N/A

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in B around 0

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

   9. Simplified 92.3%

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

   if 7.5e6 < F

   1. Initial program 56.3%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 72.8%

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

   5. Taylor expanded in F around inf

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

   7. Simplified 99.7%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.22 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sin B}, -1, -\frac{x}{\tan B}\right)\\ \mathbf{elif}\;F \leq 1.15 \cdot 10^{-137}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \frac{F}{B}\\ \mathbf{elif}\;F \leq 7500000:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sin B}, 1, -\frac{x}{\tan B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{\sin B}\\ \mathbf{if}\;F \leq -8.8 \cdot 10^{+133}:\\ \;\;\;\;-\frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.65 \cdot 10^{-103}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 6.8 \cdot 10^{-138}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \frac{F}{B}\\ \mathbf{elif}\;F \leq 820000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- (/ F (sqrt (fma 2.0 x (fma F F 2.0)))) x) (sin B))))
   (if (<= F -8.8e+133)
     (- (/ x (tan B)))
     (if (<= F -1.65e-103)
       t_0
       (if (<= F 6.8e-138)
         (+
          (* x (/ -1.0 (tan B)))
          (* (sqrt (/ 1.0 (fma x 2.0 (fma F F 2.0)))) (/ F B)))
         (if (<= F 820000.0) t_0 (/ (- 1.0 (* x (cos B))) (sin B))))))))

C

double code(double F, double B, double x) {
	double t_0 = ((F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) - x) / sin(B);
	double tmp;
	if (F <= -8.8e+133) {
		tmp = -(x / tan(B));
	} else if (F <= -1.65e-103) {
		tmp = t_0;
	} else if (F <= 6.8e-138) {
		tmp = (x * (-1.0 / tan(B))) + (sqrt((1.0 / fma(x, 2.0, fma(F, F, 2.0)))) * (F / B));
	} else if (F <= 820000.0) {
		tmp = t_0;
	} else {
		tmp = (1.0 - (x * cos(B))) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(Float64(F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) - x) / sin(B))
	tmp = 0.0
	if (F <= -8.8e+133)
		tmp = Float64(-Float64(x / tan(B)));
	elseif (F <= -1.65e-103)
		tmp = t_0;
	elseif (F <= 6.8e-138)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(sqrt(Float64(1.0 / fma(x, 2.0, fma(F, F, 2.0)))) * Float64(F / B)));
	elseif (F <= 820000.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 - Float64(x * cos(B))) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(N[(F / N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -8.8e+133], (-N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, -1.65e-103], t$95$0, If[LessEqual[F, 6.8e-138], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 / N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 820000.0], t$95$0, N[(N[(1.0 - N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -8.8e133

   1. Initial program 37.8%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. cos-lowering-cos.f64 N/A

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

      6. sin-lowering-sin.f64 62.6

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

   5. Simplified 62.6%

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

      1. clear-num N/A

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

      2. distribute-neg-frac2 N/A

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

      3. associate-/l/ N/A

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

      4. tan-quot N/A

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

      5. distribute-neg-frac2 N/A

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

      6. clear-num N/A

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

      7. /-lowering-/.f64 N/A

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

      8. neg-lowering-neg.f64 N/A

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

      9. tan-lowering-tan.f64 62.8

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

   7. Applied egg-rr 62.8%

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

   if -8.8e133 < F < -1.64999999999999995e-103 or 6.8000000000000003e-138 < F < 8.2e5

   1. Initial program 96.9%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 99.6%

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

      1. unsub-neg N/A

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

      2. div-inv N/A

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

      3. associate-*l/ N/A

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

      4. tan-quot N/A

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

      5. clear-num N/A

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

      6. associate-/l* N/A

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

      7. sub-div N/A

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

      8. /-lowering-/.f64 N/A

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in B around 0

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

   9. Simplified 82.7%

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

   if -1.64999999999999995e-103 < F < 6.8000000000000003e-138

   1. Initial program 99.5%

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

   3. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. /-lowering-/.f64 91.0

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

   5. Simplified 91.0%

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

   if 8.2e5 < F

   1. Initial program 56.3%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 72.8%

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

      1. unsub-neg N/A

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

      2. div-inv N/A

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

      3. associate-*l/ N/A

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

      4. tan-quot N/A

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

      5. clear-num N/A

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

      6. associate-/l* N/A

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

      7. sub-div N/A

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

      8. /-lowering-/.f64 N/A

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

   6. Applied egg-rr 72.7%

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

   7. Taylor expanded in F around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. cos-lowering-cos.f64 N/A

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

      5. sin-lowering-sin.f64 99.7

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

   9. Simplified 99.7%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -8.8 \cdot 10^{+133}:\\ \;\;\;\;-\frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.65 \cdot 10^{-103}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{\sin B}\\ \mathbf{elif}\;F \leq 6.8 \cdot 10^{-138}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \frac{F}{B}\\ \mathbf{elif}\;F \leq 820000:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.22 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sin B}, -1, -\frac{x}{\tan B}\right)\\ \mathbf{elif}\;F \leq 3.3 \cdot 10^{-139}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \frac{F}{B}\\ \mathbf{elif}\;F \leq 3100000:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.22e-23)
   (fma (/ 1.0 (sin B)) -1.0 (- (/ x (tan B))))
   (if (<= F 3.3e-139)
     (+
      (* x (/ -1.0 (tan B)))
      (* (sqrt (/ 1.0 (fma x 2.0 (fma F F 2.0)))) (/ F B)))
     (if (<= F 3100000.0)
       (/ (- (/ F (sqrt (fma 2.0 x (fma F F 2.0)))) x) (sin B))
       (/ (- 1.0 (* x (cos B))) (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.22e-23) {
		tmp = fma((1.0 / sin(B)), -1.0, -(x / tan(B)));
	} else if (F <= 3.3e-139) {
		tmp = (x * (-1.0 / tan(B))) + (sqrt((1.0 / fma(x, 2.0, fma(F, F, 2.0)))) * (F / B));
	} else if (F <= 3100000.0) {
		tmp = ((F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) - x) / sin(B);
	} else {
		tmp = (1.0 - (x * cos(B))) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.22e-23)
		tmp = fma(Float64(1.0 / sin(B)), -1.0, Float64(-Float64(x / tan(B))));
	elseif (F <= 3.3e-139)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(sqrt(Float64(1.0 / fma(x, 2.0, fma(F, F, 2.0)))) * Float64(F / B)));
	elseif (F <= 3100000.0)
		tmp = Float64(Float64(Float64(F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) - x) / sin(B));
	else
		tmp = Float64(Float64(1.0 - Float64(x * cos(B))) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.22e-23], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] * -1.0 + (-N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 3.3e-139], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 / N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3100000.0], N[(N[(N[(F / N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.22000000000000007e-23

   1. Initial program 60.3%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 79.0%

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

   5. Taylor expanded in F around -inf

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

   7. Simplified 96.3%

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

   if -1.22000000000000007e-23 < F < 3.3e-139

   1. Initial program 99.5%

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

   3. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. /-lowering-/.f64 88.6

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

   5. Simplified 88.6%

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

   if 3.3e-139 < F < 3.1e6

   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. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 99.4%

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

      1. unsub-neg N/A

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

      2. div-inv N/A

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

      3. associate-*l/ N/A

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

      4. tan-quot N/A

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

      5. clear-num N/A

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

      6. associate-/l* N/A

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

      7. sub-div N/A

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

      8. /-lowering-/.f64 N/A

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in B around 0

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

   9. Simplified 92.3%

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

   if 3.1e6 < F

   1. Initial program 56.3%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 72.8%

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

      1. unsub-neg N/A

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

      2. div-inv N/A

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

      3. associate-*l/ N/A

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

      4. tan-quot N/A

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

      5. clear-num N/A

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

      6. associate-/l* N/A

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

      7. sub-div N/A

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

      8. /-lowering-/.f64 N/A

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

   6. Applied egg-rr 72.7%

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

   7. Taylor expanded in F around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. cos-lowering-cos.f64 N/A

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

      5. sin-lowering-sin.f64 99.7

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

   9. Simplified 99.7%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.22 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sin B}, -1, -\frac{x}{\tan B}\right)\\ \mathbf{elif}\;F \leq 3.3 \cdot 10^{-139}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \frac{F}{B}\\ \mathbf{elif}\;F \leq 3100000:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.3 \cdot 10^{-25}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos B, -x, -1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 4.3 \cdot 10^{-138}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \frac{F}{B}\\ \mathbf{elif}\;F \leq 860000:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.3e-25)
   (/ (fma (cos B) (- x) -1.0) (sin B))
   (if (<= F 4.3e-138)
     (+
      (* x (/ -1.0 (tan B)))
      (* (sqrt (/ 1.0 (fma x 2.0 (fma F F 2.0)))) (/ F B)))
     (if (<= F 860000.0)
       (/ (- (/ F (sqrt (fma 2.0 x (fma F F 2.0)))) x) (sin B))
       (/ (- 1.0 (* x (cos B))) (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.3e-25) {
		tmp = fma(cos(B), -x, -1.0) / sin(B);
	} else if (F <= 4.3e-138) {
		tmp = (x * (-1.0 / tan(B))) + (sqrt((1.0 / fma(x, 2.0, fma(F, F, 2.0)))) * (F / B));
	} else if (F <= 860000.0) {
		tmp = ((F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) - x) / sin(B);
	} else {
		tmp = (1.0 - (x * cos(B))) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.3e-25)
		tmp = Float64(fma(cos(B), Float64(-x), -1.0) / sin(B));
	elseif (F <= 4.3e-138)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(sqrt(Float64(1.0 / fma(x, 2.0, fma(F, F, 2.0)))) * Float64(F / B)));
	elseif (F <= 860000.0)
		tmp = Float64(Float64(Float64(F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) - x) / sin(B));
	else
		tmp = Float64(Float64(1.0 - Float64(x * cos(B))) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.3e-25], N[(N[(N[Cos[B], $MachinePrecision] * (-x) + -1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.3e-138], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 / N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 860000.0], N[(N[(N[(F / N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.2999999999999998e-25

   1. Initial program 60.7%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 79.2%

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

      1. unsub-neg N/A

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

      2. div-inv N/A

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

      3. associate-*l/ N/A

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

      4. tan-quot N/A

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

      5. clear-num N/A

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

      6. associate-/l* N/A

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

      7. sub-div N/A

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

      8. /-lowering-/.f64 N/A

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

   6. Applied egg-rr 79.1%

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

   7. Taylor expanded in F around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. /-lowering-/.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-neg-in N/A

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

      6. *-commutative N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. metadata-eval N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. cos-lowering-cos.f64 N/A

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

      11. neg-lowering-neg.f64 N/A

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

      12. sin-lowering-sin.f64 96.2

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

   9. Simplified 96.2%

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

   if -3.2999999999999998e-25 < F < 4.3e-138

   1. Initial program 99.5%

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

   3. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. /-lowering-/.f64 88.5

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

   5. Simplified 88.5%

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

   if 4.3e-138 < F < 8.6e5

   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. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 99.4%

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

      1. unsub-neg N/A

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

      2. div-inv N/A

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

      3. associate-*l/ N/A

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

      4. tan-quot N/A

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

      5. clear-num N/A

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

      6. associate-/l* N/A

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

      7. sub-div N/A

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

      8. /-lowering-/.f64 N/A

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in B around 0

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

   9. Simplified 92.3%

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

   if 8.6e5 < F

   1. Initial program 56.3%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 72.8%

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

      1. unsub-neg N/A

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

      2. div-inv N/A

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

      3. associate-*l/ N/A

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

      4. tan-quot N/A

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

      5. clear-num N/A

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

      6. associate-/l* N/A

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

      7. sub-div N/A

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

      8. /-lowering-/.f64 N/A

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

   6. Applied egg-rr 72.7%

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

   7. Taylor expanded in F around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. cos-lowering-cos.f64 N/A

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

      5. sin-lowering-sin.f64 99.7

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

   9. Simplified 99.7%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.3 \cdot 10^{-25}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos B, -x, -1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 4.3 \cdot 10^{-138}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \frac{F}{B}\\ \mathbf{elif}\;F \leq 860000:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-43}:\\ \;\;\;\;\cos B \cdot \frac{-x}{\sin B}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-82}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= x -7.5e-43)
   (* (cos B) (/ (- x) (sin B)))
   (if (<= x 2e-82)
     (/ (- (/ F (sqrt (fma 2.0 x (fma F F 2.0)))) x) (sin B))
     (- (/ x (tan B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (x <= -7.5e-43) {
		tmp = cos(B) * (-x / sin(B));
	} else if (x <= 2e-82) {
		tmp = ((F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) - x) / sin(B);
	} else {
		tmp = -(x / tan(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (x <= -7.5e-43)
		tmp = Float64(cos(B) * Float64(Float64(-x) / sin(B)));
	elseif (x <= 2e-82)
		tmp = Float64(Float64(Float64(F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) - x) / sin(B));
	else
		tmp = Float64(-Float64(x / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[x, -7.5e-43], N[(N[Cos[B], $MachinePrecision] * N[((-x) / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2e-82], N[(N[(N[(F / N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], (-N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if x < -7.50000000000000068e-43

   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 x around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. cos-lowering-cos.f64 N/A

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

      6. sin-lowering-sin.f64 79.6

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

   5. Simplified 79.6%

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

      1. clear-num N/A

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

      2. distribute-neg-frac2 N/A

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

      3. associate-/l/ N/A

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

      4. tan-quot N/A

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

      5. distribute-neg-frac2 N/A

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

      6. clear-num N/A

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

      7. tan-quot N/A

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

      8. associate-/r/ N/A

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

      9. *-lowering-*.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. neg-lowering-neg.f64 N/A

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

      12. sin-lowering-sin.f64 N/A

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

      13. cos-lowering-cos.f64 79.7

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

   7. Applied egg-rr 79.7%

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

   if -7.50000000000000068e-43 < x < 2e-82

   1. Initial program 73.4%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 77.1%

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

      1. unsub-neg N/A

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

      2. div-inv N/A

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

      3. associate-*l/ N/A

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

      4. tan-quot N/A

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

      5. clear-num N/A

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

      6. associate-/l* N/A

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

      7. sub-div N/A

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

      8. /-lowering-/.f64 N/A

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

   6. Applied egg-rr 77.2%

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

   7. Taylor expanded in B around 0

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

   9. Simplified 68.8%

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

   if 2e-82 < x

   1. Initial program 82.3%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. cos-lowering-cos.f64 N/A

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

      6. sin-lowering-sin.f64 90.4

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

   5. Simplified 90.4%

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

      1. clear-num N/A

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

      2. distribute-neg-frac2 N/A

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

      3. associate-/l/ N/A

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

      4. tan-quot N/A

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

      5. distribute-neg-frac2 N/A

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

      6. clear-num N/A

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

      7. /-lowering-/.f64 N/A

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

      8. neg-lowering-neg.f64 N/A

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

      9. tan-lowering-tan.f64 90.5

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

   7. Applied egg-rr 90.5%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-43}:\\ \;\;\;\;\cos B \cdot \frac{-x}{\sin B}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-82}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\frac{x}{\tan B}\\ \mathbf{if}\;x \leq -3.5 \cdot 10^{-39}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-82}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ x (tan B)))))
   (if (<= x -3.5e-39)
     t_0
     (if (<= x 1.8e-82)
       (/ (- (/ F (sqrt (fma 2.0 x (fma F F 2.0)))) x) (sin B))
       t_0))))

C

double code(double F, double B, double x) {
	double t_0 = -(x / tan(B));
	double tmp;
	if (x <= -3.5e-39) {
		tmp = t_0;
	} else if (x <= 1.8e-82) {
		tmp = ((F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) - x) / sin(B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(-Float64(x / tan(B)))
	tmp = 0.0
	if (x <= -3.5e-39)
		tmp = t_0;
	elseif (x <= 1.8e-82)
		tmp = Float64(Float64(Float64(F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) - x) / sin(B));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = (-N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision])}, If[LessEqual[x, -3.5e-39], t$95$0, If[LessEqual[x, 1.8e-82], N[(N[(N[(F / N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.5e-39 or 1.79999999999999999e-82 < x

   1. Initial program 77.9%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. cos-lowering-cos.f64 N/A

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

      6. sin-lowering-sin.f64 87.8

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

   5. Simplified 87.8%

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

      1. clear-num N/A

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

      2. distribute-neg-frac2 N/A

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

      3. associate-/l/ N/A

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

      4. tan-quot N/A

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

      5. distribute-neg-frac2 N/A

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

      6. clear-num N/A

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

      7. /-lowering-/.f64 N/A

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

      8. neg-lowering-neg.f64 N/A

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

      9. tan-lowering-tan.f64 87.9

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

   7. Applied egg-rr 87.9%

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

   if -3.5e-39 < x < 1.79999999999999999e-82

   1. Initial program 73.4%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 77.1%

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

      1. unsub-neg N/A

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

      2. div-inv N/A

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

      3. associate-*l/ N/A

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

      4. tan-quot N/A

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

      5. clear-num N/A

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

      6. associate-/l* N/A

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

      7. sub-div N/A

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

      8. /-lowering-/.f64 N/A

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

   6. Applied egg-rr 77.2%

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

   7. Taylor expanded in B around 0

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

   9. Simplified 68.8%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-39}:\\ \;\;\;\;-\frac{x}{\tan B}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-82}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\frac{x}{\tan B}\\ \mathbf{if}\;x \leq -6.2 \cdot 10^{-70}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-84}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ x (tan B)))))
   (if (<= x -6.2e-70)
     t_0
     (if (<= x 6.5e-84) (/ (* F (sqrt (/ 1.0 (fma F F 2.0)))) (sin B)) t_0))))

C

double code(double F, double B, double x) {
	double t_0 = -(x / tan(B));
	double tmp;
	if (x <= -6.2e-70) {
		tmp = t_0;
	} else if (x <= 6.5e-84) {
		tmp = (F * sqrt((1.0 / fma(F, F, 2.0)))) / sin(B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(-Float64(x / tan(B)))
	tmp = 0.0
	if (x <= -6.2e-70)
		tmp = t_0;
	elseif (x <= 6.5e-84)
		tmp = Float64(Float64(F * sqrt(Float64(1.0 / fma(F, F, 2.0)))) / sin(B));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = (-N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision])}, If[LessEqual[x, -6.2e-70], t$95$0, If[LessEqual[x, 6.5e-84], N[(N[(F * N[Sqrt[N[(1.0 / N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.2e-70 or 6.50000000000000022e-84 < x

   1. Initial program 77.2%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. cos-lowering-cos.f64 N/A

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

      6. sin-lowering-sin.f64 85.4

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

   5. Simplified 85.4%

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

      1. clear-num N/A

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

      2. distribute-neg-frac2 N/A

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

      3. associate-/l/ N/A

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

      4. tan-quot N/A

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

      5. distribute-neg-frac2 N/A

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

      6. clear-num N/A

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

      7. /-lowering-/.f64 N/A

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

      8. neg-lowering-neg.f64 N/A

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

      9. tan-lowering-tan.f64 85.5

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

   7. Applied egg-rr 85.5%

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

   if -6.2e-70 < x < 6.50000000000000022e-84

   1. Initial program 74.1%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 78.2%

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

      1. unsub-neg N/A

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

      2. div-inv N/A

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

      3. associate-*l/ N/A

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

      4. tan-quot N/A

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

      5. clear-num N/A

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

      6. associate-/l* N/A

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

      7. sub-div N/A

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

      8. /-lowering-/.f64 N/A

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

   6. Applied egg-rr 78.3%

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

   7. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. accelerator-lowering-fma.f64 59.8

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

   9. Simplified 59.8%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-70}:\\ \;\;\;\;-\frac{x}{\tan B}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-84}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\frac{x}{\tan B}\\ \mathbf{if}\;x \leq -9.5 \cdot 10^{-67}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-86}:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ x (tan B)))))
   (if (<= x -9.5e-67)
     t_0
     (if (<= x 5.2e-86) (* F (/ (sqrt (/ 1.0 (fma F F 2.0))) (sin B))) t_0))))

C

double code(double F, double B, double x) {
	double t_0 = -(x / tan(B));
	double tmp;
	if (x <= -9.5e-67) {
		tmp = t_0;
	} else if (x <= 5.2e-86) {
		tmp = F * (sqrt((1.0 / fma(F, F, 2.0))) / sin(B));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(-Float64(x / tan(B)))
	tmp = 0.0
	if (x <= -9.5e-67)
		tmp = t_0;
	elseif (x <= 5.2e-86)
		tmp = Float64(F * Float64(sqrt(Float64(1.0 / fma(F, F, 2.0))) / sin(B)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = (-N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision])}, If[LessEqual[x, -9.5e-67], t$95$0, If[LessEqual[x, 5.2e-86], N[(F * N[(N[Sqrt[N[(1.0 / N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -9.4999999999999994e-67 or 5.2000000000000002e-86 < x

   1. Initial program 77.2%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. cos-lowering-cos.f64 N/A

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

      6. sin-lowering-sin.f64 85.4

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

   5. Simplified 85.4%

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

      1. clear-num N/A

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

      2. distribute-neg-frac2 N/A

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

      3. associate-/l/ N/A

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

      4. tan-quot N/A

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

      5. distribute-neg-frac2 N/A

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

      6. clear-num N/A

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

      7. /-lowering-/.f64 N/A

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

      8. neg-lowering-neg.f64 N/A

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

      9. tan-lowering-tan.f64 85.5

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

   7. Applied egg-rr 85.5%

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

   if -9.4999999999999994e-67 < x < 5.2000000000000002e-86

   1. Initial program 74.1%

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

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. sin-lowering-sin.f64 59.8

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

   5. Simplified 59.8%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-67}:\\ \;\;\;\;-\frac{x}{\tan B}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-86}:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}\\ \mathbf{if}\;B \leq 0.062:\\ \;\;\;\;\frac{\mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(t\_0, \mathsf{fma}\left(F \cdot \left(B \cdot B\right), 0.019444444444444445, F \cdot 0.16666666666666666\right), x \cdot \mathsf{fma}\left(0.022222222222222223, B \cdot B, 0.3333333333333333\right)\right), \mathsf{fma}\left(F, t\_0, -x\right)\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (fma x 2.0 (fma F F 2.0))))))
   (if (<= B 0.062)
     (/
      (fma
       (* B B)
       (fma
        t_0
        (fma (* F (* B B)) 0.019444444444444445 (* F 0.16666666666666666))
        (* x (fma 0.022222222222222223 (* B B) 0.3333333333333333)))
       (fma F t_0 (- x)))
      B)
     (- (/ x (tan B))))))

C

double code(double F, double B, double x) {
	double t_0 = sqrt((1.0 / fma(x, 2.0, fma(F, F, 2.0))));
	double tmp;
	if (B <= 0.062) {
		tmp = fma((B * B), fma(t_0, fma((F * (B * B)), 0.019444444444444445, (F * 0.16666666666666666)), (x * fma(0.022222222222222223, (B * B), 0.3333333333333333))), fma(F, t_0, -x)) / B;
	} else {
		tmp = -(x / tan(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = sqrt(Float64(1.0 / fma(x, 2.0, fma(F, F, 2.0))))
	tmp = 0.0
	if (B <= 0.062)
		tmp = Float64(fma(Float64(B * B), fma(t_0, fma(Float64(F * Float64(B * B)), 0.019444444444444445, Float64(F * 0.16666666666666666)), Float64(x * fma(0.022222222222222223, Float64(B * B), 0.3333333333333333))), fma(F, t_0, Float64(-x))) / B);
	else
		tmp = Float64(-Float64(x / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[B, 0.062], N[(N[(N[(B * B), $MachinePrecision] * N[(t$95$0 * N[(N[(F * N[(B * B), $MachinePrecision]), $MachinePrecision] * 0.019444444444444445 + N[(F * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + N[(x * N[(0.022222222222222223 * N[(B * B), $MachinePrecision] + 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(F * t$95$0 + (-x)), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], (-N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision])]]

Derivation

1. Split input into 2 regimes

2. if B < 0.062

   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

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

   5. Simplified 56.9%

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

   if 0.062 < B

   1. Initial program 91.1%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. cos-lowering-cos.f64 N/A

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

      6. sin-lowering-sin.f64 65.6

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

   5. Simplified 65.6%

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

      1. clear-num N/A

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

      2. distribute-neg-frac2 N/A

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

      3. associate-/l/ N/A

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

      4. tan-quot N/A

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

      5. distribute-neg-frac2 N/A

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

      6. clear-num N/A

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

      7. /-lowering-/.f64 N/A

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

      8. neg-lowering-neg.f64 N/A

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

      9. tan-lowering-tan.f64 65.7

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

   7. Applied egg-rr 65.7%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 0.062:\\ \;\;\;\;\frac{\mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \mathsf{fma}\left(F \cdot \left(B \cdot B\right), 0.019444444444444445, F \cdot 0.16666666666666666\right), x \cdot \mathsf{fma}\left(0.022222222222222223, B \cdot B, 0.3333333333333333\right)\right), \mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 57.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.036:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \mathsf{fma}\left(0.16666666666666666, F \cdot \left(B \cdot B\right), F\right), 0.3333333333333333 \cdot \left(x \cdot \left(B \cdot B\right)\right) - x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 0.036)
   (/
    (fma
     (sqrt (/ 1.0 (fma x 2.0 (fma F F 2.0))))
     (fma 0.16666666666666666 (* F (* B B)) F)
     (- (* 0.3333333333333333 (* x (* B B))) x))
    B)
   (- (/ x (tan B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 0.036) {
		tmp = fma(sqrt((1.0 / fma(x, 2.0, fma(F, F, 2.0)))), fma(0.16666666666666666, (F * (B * B)), F), ((0.3333333333333333 * (x * (B * B))) - x)) / B;
	} else {
		tmp = -(x / tan(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 0.036)
		tmp = Float64(fma(sqrt(Float64(1.0 / fma(x, 2.0, fma(F, F, 2.0)))), fma(0.16666666666666666, Float64(F * Float64(B * B)), F), Float64(Float64(0.3333333333333333 * Float64(x * Float64(B * B))) - x)) / B);
	else
		tmp = Float64(-Float64(x / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 0.036], N[(N[(N[Sqrt[N[(1.0 / N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.16666666666666666 * N[(F * N[(B * B), $MachinePrecision]), $MachinePrecision] + F), $MachinePrecision] + N[(N[(0.3333333333333333 * N[(x * N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], (-N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if B < 0.0359999999999999973

   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

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

   5. Simplified 56.9%

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

   if 0.0359999999999999973 < B

   1. Initial program 91.1%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. cos-lowering-cos.f64 N/A

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

      6. sin-lowering-sin.f64 65.6

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

   5. Simplified 65.6%

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

      1. clear-num N/A

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

      2. distribute-neg-frac2 N/A

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

      3. associate-/l/ N/A

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

      4. tan-quot N/A

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

      5. distribute-neg-frac2 N/A

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

      6. clear-num N/A

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

      7. /-lowering-/.f64 N/A

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

      8. neg-lowering-neg.f64 N/A

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

      9. tan-lowering-tan.f64 65.7

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

   7. Applied egg-rr 65.7%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 0.036:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \mathsf{fma}\left(0.16666666666666666, F \cdot \left(B \cdot B\right), F\right), 0.3333333333333333 \cdot \left(x \cdot \left(B \cdot B\right)\right) - x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 51.9% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3 \cdot 10^{-21}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, -1 - x\right)}{B}\\ \mathbf{elif}\;F \leq 50000000:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3e-21)
   (/ (fma 0.5 (/ (fma 2.0 x 2.0) (* F F)) (- -1.0 x)) B)
   (if (<= F 50000000.0)
     (/ (- (/ F (sqrt (fma 2.0 x (fma F F 2.0)))) x) B)
     (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3e-21) {
		tmp = fma(0.5, (fma(2.0, x, 2.0) / (F * F)), (-1.0 - x)) / B;
	} else if (F <= 50000000.0) {
		tmp = ((F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) - x) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3e-21)
		tmp = Float64(fma(0.5, Float64(fma(2.0, x, 2.0) / Float64(F * F)), Float64(-1.0 - x)) / B);
	elseif (F <= 50000000.0)
		tmp = Float64(Float64(Float64(F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) - x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3e-21], N[(N[(0.5 * N[(N[(2.0 * x + 2.0), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 50000000.0], N[(N[(N[(F / N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.99999999999999991e-21

   1. Initial program 59.8%

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

   3. Taylor expanded in 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. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. neg-lowering-neg.f64 36.2

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

   5. Simplified 36.2%

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

   6. Taylor expanded in F around -inf

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

      1. sub-neg N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. distribute-neg-in N/A

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

      9. metadata-eval N/A

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

      10. unsub-neg N/A

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

      11. --lowering--.f64 50.0

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

   8. Simplified 50.0%

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

   if -2.99999999999999991e-21 < F < 5e7

   1. Initial program 99.4%

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

   3. Taylor expanded in B around 0

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

      1. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. neg-lowering-neg.f64 49.7

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

   5. Simplified 49.7%

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

      1. /-lowering-/.f64 N/A

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

   7. Applied egg-rr 49.8%

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

   if 5e7 < F

   1. Initial program 56.3%

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

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. neg-lowering-neg.f64 39.5

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

   5. Simplified 39.5%

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

   6. Taylor expanded in F around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 54.0

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

   8. Simplified 54.0%

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

Alternative 14: 51.9% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3 \cdot 10^{-21}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, -1 - x\right)}{B}\\ \mathbf{elif}\;F \leq 50000000:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3e-21)
   (/ (fma 0.5 (/ (fma 2.0 x 2.0) (* F F)) (- -1.0 x)) B)
   (if (<= F 50000000.0)
     (/ (- (/ F (sqrt (fma F F 2.0))) x) B)
     (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3e-21) {
		tmp = fma(0.5, (fma(2.0, x, 2.0) / (F * F)), (-1.0 - x)) / B;
	} else if (F <= 50000000.0) {
		tmp = ((F / sqrt(fma(F, F, 2.0))) - x) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3e-21)
		tmp = Float64(fma(0.5, Float64(fma(2.0, x, 2.0) / Float64(F * F)), Float64(-1.0 - x)) / B);
	elseif (F <= 50000000.0)
		tmp = Float64(Float64(Float64(F / sqrt(fma(F, F, 2.0))) - x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3e-21], N[(N[(0.5 * N[(N[(2.0 * x + 2.0), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 50000000.0], N[(N[(N[(F / N[Sqrt[N[(F * F + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.99999999999999991e-21

   1. Initial program 59.8%

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

   3. Taylor expanded in 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. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. neg-lowering-neg.f64 36.2

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

   5. Simplified 36.2%

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

   6. Taylor expanded in F around -inf

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

      1. sub-neg N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. distribute-neg-in N/A

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

      9. metadata-eval N/A

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

      10. unsub-neg N/A

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

      11. --lowering--.f64 50.0

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

   8. Simplified 50.0%

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

   if -2.99999999999999991e-21 < F < 5e7

   1. Initial program 99.4%

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

   3. Taylor expanded in B around 0

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

      1. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. neg-lowering-neg.f64 49.7

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

   5. Simplified 49.7%

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

   6. Taylor expanded in x around 0

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

      1. sqrt-lowering-sqrt.f64 N/A

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

      2. /-lowering-/.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. accelerator-lowering-fma.f64 49.2

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

   8. Simplified 49.2%

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

      1. /-lowering-/.f64 N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. un-div-inv N/A

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

      7. /-lowering-/.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. accelerator-lowering-fma.f64 49.2

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

   10. Applied egg-rr 49.2%

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

   if 5e7 < F

   1. Initial program 56.3%

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

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. neg-lowering-neg.f64 39.5

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

   5. Simplified 39.5%

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

   6. Taylor expanded in F around inf

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 54.0

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

   8. Simplified 54.0%

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

Alternative 15: 52.2% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3 \cdot 10^{-21}:\\ \;\;\;\;\frac{\left(-1 - x\right) + \frac{1}{F \cdot F}}{B}\\ \mathbf{elif}\;F \leq 2 \cdot 10^{+97}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3e-21)
   (/ (+ (- -1.0 x) (/ 1.0 (* F F))) B)
   (if (<= F 2e+97) (/ (- (/ F (sqrt (fma F F 2.0))) x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3e-21) {
		tmp = ((-1.0 - x) + (1.0 / (F * F))) / B;
	} else if (F <= 2e+97) {
		tmp = ((F / sqrt(fma(F, F, 2.0))) - x) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3e-21)
		tmp = Float64(Float64(Float64(-1.0 - x) + Float64(1.0 / Float64(F * F))) / B);
	elseif (F <= 2e+97)
		tmp = Float64(Float64(Float64(F / sqrt(fma(F, F, 2.0))) - x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3e-21], N[(N[(N[(-1.0 - x), $MachinePrecision] + N[(1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 2e+97], N[(N[(N[(F / N[Sqrt[N[(F * F + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.99999999999999991e-21

   1. Initial program 59.8%

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

   3. Taylor expanded in 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. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. neg-lowering-neg.f64 36.2

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

   5. Simplified 36.2%

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

   6. Taylor expanded in x around 0

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

      1. sqrt-lowering-sqrt.f64 N/A

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

      2. /-lowering-/.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. accelerator-lowering-fma.f64 36.2

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

   8. Simplified 36.2%

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

   9. Taylor expanded in F around -inf

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

       1. sub-neg N/A

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

       2. +-lowering-+.f64 N/A

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

       3. /-lowering-/.f64 N/A

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

       4. unpow2 N/A

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

       5. *-lowering-*.f64 N/A

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

       6. distribute-neg-in N/A

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

       7. metadata-eval N/A

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

       8. unsub-neg N/A

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

       9. --lowering--.f64 49.9

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

   11. Simplified 49.9%

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

   if -2.99999999999999991e-21 < F < 2.0000000000000001e97

   1. Initial program 97.1%

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

   3. Taylor expanded in B around 0

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

      1. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. neg-lowering-neg.f64 50.8

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

   5. Simplified 50.8%

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

   6. Taylor expanded in x around 0

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

      1. sqrt-lowering-sqrt.f64 N/A

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

      2. /-lowering-/.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. accelerator-lowering-fma.f64 50.4

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

   8. Simplified 50.4%

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

      1. /-lowering-/.f64 N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. un-div-inv N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{F}{\sqrt{F \cdot F + 2}}} - x}{B} \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\frac{F}{\color{blue}{\sqrt{F \cdot F + 2}}} - x}{B} \]

      9. accelerator-lowering-fma.f64 50.4

         \[\leadsto \frac{\frac{F}{\sqrt{\color{blue}{\mathsf{fma}\left(F, F, 2\right)}}} - x}{B} \]

   10. Applied egg-rr 50.4%

       \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}} - x}{B}} \]

   if 2.0000000000000001e97 < F

   1. Initial program 44.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, \mathsf{neg}\left(x\right)\right)}}{B} \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      14. neg-lowering-neg.f64 32.5

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{-x}\right)}{B} \]

   5. Simplified 32.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

      2. --lowering--.f64 52.8

         \[\leadsto \frac{\color{blue}{1 - x}}{B} \]

   8. Simplified 52.8%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3 \cdot 10^{-21}:\\ \;\;\;\;\frac{\left(-1 - x\right) + \frac{1}{F \cdot F}}{B}\\ \mathbf{elif}\;F \leq 2 \cdot 10^{+97}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 52.1% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.75 \cdot 10^{-20}:\\ \;\;\;\;\frac{\left(-1 - x\right) + \frac{1}{F \cdot F}}{B}\\ \mathbf{elif}\;F \leq 1.3:\\ \;\;\;\;\frac{\mathsf{fma}\left(F, \sqrt{\mathsf{fma}\left(F \cdot F, -0.25, 0.5\right)}, -x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - x\right) + \frac{-1}{F \cdot F}}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.75e-20)
   (/ (+ (- -1.0 x) (/ 1.0 (* F F))) B)
   (if (<= F 1.3)
     (/ (fma F (sqrt (fma (* F F) -0.25 0.5)) (- x)) B)
     (/ (+ (- 1.0 x) (/ -1.0 (* F F))) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.75e-20) {
		tmp = ((-1.0 - x) + (1.0 / (F * F))) / B;
	} else if (F <= 1.3) {
		tmp = fma(F, sqrt(fma((F * F), -0.25, 0.5)), -x) / B;
	} else {
		tmp = ((1.0 - x) + (-1.0 / (F * F))) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.75e-20)
		tmp = Float64(Float64(Float64(-1.0 - x) + Float64(1.0 / Float64(F * F))) / B);
	elseif (F <= 1.3)
		tmp = Float64(fma(F, sqrt(fma(Float64(F * F), -0.25, 0.5)), Float64(-x)) / B);
	else
		tmp = Float64(Float64(Float64(1.0 - x) + Float64(-1.0 / Float64(F * F))) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.75e-20], N[(N[(N[(-1.0 - x), $MachinePrecision] + N[(1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.3], N[(N[(F * N[Sqrt[N[(N[(F * F), $MachinePrecision] * -0.25 + 0.5), $MachinePrecision]], $MachinePrecision] + (-x)), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(1.0 - x), $MachinePrecision] + N[(-1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.75000000000000002e-20

   1. Initial program 59.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, \mathsf{neg}\left(x\right)\right)}}{B} \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      14. neg-lowering-neg.f64 36.7

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{-x}\right)}{B} \]

   5. Simplified 36.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{B}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}}, \mathsf{neg}\left(x\right)\right)}{B} \]
   7. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + {F}^{2}}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      3. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{{F}^{2} + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. accelerator-lowering-fma.f64 36.7

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2\right)}}}, -x\right)}{B} \]

   8. Simplified 36.7%

      \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}, -x\right)}{B} \]

   9. Taylor expanded in F around -inf

      \[\leadsto \frac{\color{blue}{\frac{1}{{F}^{2}} - \left(1 + x\right)}}{B} \]
   10. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \frac{\color{blue}{\frac{1}{{F}^{2}} + \left(\mathsf{neg}\left(\left(1 + x\right)\right)\right)}}{B} \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{1}{{F}^{2}} + \left(\mathsf{neg}\left(\left(1 + x\right)\right)\right)}}{B} \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{1}{{F}^{2}}} + \left(\mathsf{neg}\left(\left(1 + x\right)\right)\right)}{B} \]

       4. unpow2 N/A

          \[\leadsto \frac{\frac{1}{\color{blue}{F \cdot F}} + \left(\mathsf{neg}\left(\left(1 + x\right)\right)\right)}{B} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{1}{\color{blue}{F \cdot F}} + \left(\mathsf{neg}\left(\left(1 + x\right)\right)\right)}{B} \]

       6. distribute-neg-in N/A

          \[\leadsto \frac{\frac{1}{F \cdot F} + \color{blue}{\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(x\right)\right)\right)}}{B} \]

       7. metadata-eval N/A

          \[\leadsto \frac{\frac{1}{F \cdot F} + \left(\color{blue}{-1} + \left(\mathsf{neg}\left(x\right)\right)\right)}{B} \]

       8. unsub-neg N/A

          \[\leadsto \frac{\frac{1}{F \cdot F} + \color{blue}{\left(-1 - x\right)}}{B} \]

       9. --lowering--.f64 50.6

          \[\leadsto \frac{\frac{1}{F \cdot F} + \color{blue}{\left(-1 - x\right)}}{B} \]

   11. Simplified 50.6%

       \[\leadsto \frac{\color{blue}{\frac{1}{F \cdot F} + \left(-1 - x\right)}}{B} \]

   if -1.75000000000000002e-20 < F < 1.30000000000000004

   1. Initial program 99.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, \mathsf{neg}\left(x\right)\right)}}{B} \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      14. neg-lowering-neg.f64 48.8

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{-x}\right)}{B} \]

   5. Simplified 48.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{B}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}}, \mathsf{neg}\left(x\right)\right)}{B} \]
   7. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + {F}^{2}}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      3. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{{F}^{2} + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. accelerator-lowering-fma.f64 48.3

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2\right)}}}, -x\right)}{B} \]

   8. Simplified 48.3%

      \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}, -x\right)}{B} \]

   9. Taylor expanded in F around 0

      \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2} + \frac{-1}{4} \cdot {F}^{2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{-1}{4} \cdot {F}^{2} + \frac{1}{2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

       2. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{{F}^{2} \cdot \frac{-1}{4}} + \frac{1}{2}}, \mathsf{neg}\left(x\right)\right)}{B} \]

       3. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\mathsf{fma}\left({F}^{2}, \frac{-1}{4}, \frac{1}{2}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

       4. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\mathsf{fma}\left(\color{blue}{F \cdot F}, \frac{-1}{4}, \frac{1}{2}\right)}, \mathsf{neg}\left(x\right)\right)}{B} \]

       5. *-lowering-*.f64 48.3

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\mathsf{fma}\left(\color{blue}{F \cdot F}, -0.25, 0.5\right)}, -x\right)}{B} \]

   11. Simplified 48.3%

       \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\mathsf{fma}\left(F \cdot F, -0.25, 0.5\right)}}, -x\right)}{B} \]

   if 1.30000000000000004 < 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. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, \mathsf{neg}\left(x\right)\right)}}{B} \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      14. neg-lowering-neg.f64 40.4

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{-x}\right)}{B} \]

   5. Simplified 40.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{B}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}}, \mathsf{neg}\left(x\right)\right)}{B} \]
   7. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + {F}^{2}}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      3. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{{F}^{2} + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. accelerator-lowering-fma.f64 40.4

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2\right)}}}, -x\right)}{B} \]

   8. Simplified 40.4%

      \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}, -x\right)}{B} \]

   9. Taylor expanded in F around inf

      \[\leadsto \frac{\color{blue}{1 - \left(x + \frac{1}{{F}^{2}}\right)}}{B} \]
   10. Step-by-step derivation

       1. associate--r+ N/A

          \[\leadsto \frac{\color{blue}{\left(1 - x\right) - \frac{1}{{F}^{2}}}}{B} \]

       2. sub-neg N/A

          \[\leadsto \frac{\color{blue}{\left(1 - x\right) + \left(\mathsf{neg}\left(\frac{1}{{F}^{2}}\right)\right)}}{B} \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(1 - x\right) + \left(\mathsf{neg}\left(\frac{1}{{F}^{2}}\right)\right)}}{B} \]

       4. --lowering--.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(1 - x\right)} + \left(\mathsf{neg}\left(\frac{1}{{F}^{2}}\right)\right)}{B} \]

       5. distribute-neg-frac N/A

          \[\leadsto \frac{\left(1 - x\right) + \color{blue}{\frac{\mathsf{neg}\left(1\right)}{{F}^{2}}}}{B} \]

       6. metadata-eval N/A

          \[\leadsto \frac{\left(1 - x\right) + \frac{\color{blue}{-1}}{{F}^{2}}}{B} \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \frac{\left(1 - x\right) + \color{blue}{\frac{-1}{{F}^{2}}}}{B} \]

       8. unpow2 N/A

          \[\leadsto \frac{\left(1 - x\right) + \frac{-1}{\color{blue}{F \cdot F}}}{B} \]

       9. *-lowering-*.f64 54.6

          \[\leadsto \frac{\left(1 - x\right) + \frac{-1}{\color{blue}{F \cdot F}}}{B} \]

   11. Simplified 54.6%

       \[\leadsto \frac{\color{blue}{\left(1 - x\right) + \frac{-1}{F \cdot F}}}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.75 \cdot 10^{-20}:\\ \;\;\;\;\frac{\left(-1 - x\right) + \frac{1}{F \cdot F}}{B}\\ \mathbf{elif}\;F \leq 1.3:\\ \;\;\;\;\frac{\mathsf{fma}\left(F, \sqrt{\mathsf{fma}\left(F \cdot F, -0.25, 0.5\right)}, -x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - x\right) + \frac{-1}{F \cdot F}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 52.1% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3 \cdot 10^{-21}:\\ \;\;\;\;\frac{\left(-1 - x\right) + \frac{1}{F \cdot F}}{B}\\ \mathbf{elif}\;F \leq 1.45:\\ \;\;\;\;\frac{\mathsf{fma}\left(F, \sqrt{0.5}, -x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - x\right) + \frac{-1}{F \cdot F}}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3e-21)
   (/ (+ (- -1.0 x) (/ 1.0 (* F F))) B)
   (if (<= F 1.45)
     (/ (fma F (sqrt 0.5) (- x)) B)
     (/ (+ (- 1.0 x) (/ -1.0 (* F F))) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3e-21) {
		tmp = ((-1.0 - x) + (1.0 / (F * F))) / B;
	} else if (F <= 1.45) {
		tmp = fma(F, sqrt(0.5), -x) / B;
	} else {
		tmp = ((1.0 - x) + (-1.0 / (F * F))) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3e-21)
		tmp = Float64(Float64(Float64(-1.0 - x) + Float64(1.0 / Float64(F * F))) / B);
	elseif (F <= 1.45)
		tmp = Float64(fma(F, sqrt(0.5), Float64(-x)) / B);
	else
		tmp = Float64(Float64(Float64(1.0 - x) + Float64(-1.0 / Float64(F * F))) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3e-21], N[(N[(N[(-1.0 - x), $MachinePrecision] + N[(1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.45], N[(N[(F * N[Sqrt[0.5], $MachinePrecision] + (-x)), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(1.0 - x), $MachinePrecision] + N[(-1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.99999999999999991e-21

   1. Initial program 59.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in 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. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, \mathsf{neg}\left(x\right)\right)}}{B} \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      14. neg-lowering-neg.f64 36.2

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{-x}\right)}{B} \]

   5. Simplified 36.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{B}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}}, \mathsf{neg}\left(x\right)\right)}{B} \]
   7. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + {F}^{2}}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      3. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{{F}^{2} + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. accelerator-lowering-fma.f64 36.2

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2\right)}}}, -x\right)}{B} \]

   8. Simplified 36.2%

      \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}, -x\right)}{B} \]

   9. Taylor expanded in F around -inf

      \[\leadsto \frac{\color{blue}{\frac{1}{{F}^{2}} - \left(1 + x\right)}}{B} \]
   10. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \frac{\color{blue}{\frac{1}{{F}^{2}} + \left(\mathsf{neg}\left(\left(1 + x\right)\right)\right)}}{B} \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{1}{{F}^{2}} + \left(\mathsf{neg}\left(\left(1 + x\right)\right)\right)}}{B} \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{1}{{F}^{2}}} + \left(\mathsf{neg}\left(\left(1 + x\right)\right)\right)}{B} \]

       4. unpow2 N/A

          \[\leadsto \frac{\frac{1}{\color{blue}{F \cdot F}} + \left(\mathsf{neg}\left(\left(1 + x\right)\right)\right)}{B} \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{1}{\color{blue}{F \cdot F}} + \left(\mathsf{neg}\left(\left(1 + x\right)\right)\right)}{B} \]

       6. distribute-neg-in N/A

          \[\leadsto \frac{\frac{1}{F \cdot F} + \color{blue}{\left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(x\right)\right)\right)}}{B} \]

       7. metadata-eval N/A

          \[\leadsto \frac{\frac{1}{F \cdot F} + \left(\color{blue}{-1} + \left(\mathsf{neg}\left(x\right)\right)\right)}{B} \]

       8. unsub-neg N/A

          \[\leadsto \frac{\frac{1}{F \cdot F} + \color{blue}{\left(-1 - x\right)}}{B} \]

       9. --lowering--.f64 49.9

          \[\leadsto \frac{\frac{1}{F \cdot F} + \color{blue}{\left(-1 - x\right)}}{B} \]

   11. Simplified 49.9%

       \[\leadsto \frac{\color{blue}{\frac{1}{F \cdot F} + \left(-1 - x\right)}}{B} \]

   if -2.99999999999999991e-21 < F < 1.44999999999999996

   1. Initial program 99.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, \mathsf{neg}\left(x\right)\right)}}{B} \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      14. neg-lowering-neg.f64 49.2

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{-x}\right)}{B} \]

   5. Simplified 49.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{B}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}}, \mathsf{neg}\left(x\right)\right)}{B} \]
   7. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + {F}^{2}}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      3. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{{F}^{2} + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. accelerator-lowering-fma.f64 48.7

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2\right)}}}, -x\right)}{B} \]

   8. Simplified 48.7%

      \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}, -x\right)}{B} \]

   9. Taylor expanded in F around 0

      \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]
   10. Step-by-step derivation

       1. sqrt-lowering-sqrt.f64 48.7

          \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{0.5}}, -x\right)}{B} \]

   11. Simplified 48.7%

       \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{0.5}}, -x\right)}{B} \]

   if 1.44999999999999996 < 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. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, \mathsf{neg}\left(x\right)\right)}}{B} \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      14. neg-lowering-neg.f64 40.4

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{-x}\right)}{B} \]

   5. Simplified 40.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{B}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}}, \mathsf{neg}\left(x\right)\right)}{B} \]
   7. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + {F}^{2}}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      3. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{{F}^{2} + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. accelerator-lowering-fma.f64 40.4

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2\right)}}}, -x\right)}{B} \]

   8. Simplified 40.4%

      \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}, -x\right)}{B} \]

   9. Taylor expanded in F around inf

      \[\leadsto \frac{\color{blue}{1 - \left(x + \frac{1}{{F}^{2}}\right)}}{B} \]
   10. Step-by-step derivation

       1. associate--r+ N/A

          \[\leadsto \frac{\color{blue}{\left(1 - x\right) - \frac{1}{{F}^{2}}}}{B} \]

       2. sub-neg N/A

          \[\leadsto \frac{\color{blue}{\left(1 - x\right) + \left(\mathsf{neg}\left(\frac{1}{{F}^{2}}\right)\right)}}{B} \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(1 - x\right) + \left(\mathsf{neg}\left(\frac{1}{{F}^{2}}\right)\right)}}{B} \]

       4. --lowering--.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(1 - x\right)} + \left(\mathsf{neg}\left(\frac{1}{{F}^{2}}\right)\right)}{B} \]

       5. distribute-neg-frac N/A

          \[\leadsto \frac{\left(1 - x\right) + \color{blue}{\frac{\mathsf{neg}\left(1\right)}{{F}^{2}}}}{B} \]

       6. metadata-eval N/A

          \[\leadsto \frac{\left(1 - x\right) + \frac{\color{blue}{-1}}{{F}^{2}}}{B} \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \frac{\left(1 - x\right) + \color{blue}{\frac{-1}{{F}^{2}}}}{B} \]

       8. unpow2 N/A

          \[\leadsto \frac{\left(1 - x\right) + \frac{-1}{\color{blue}{F \cdot F}}}{B} \]

       9. *-lowering-*.f64 54.6

          \[\leadsto \frac{\left(1 - x\right) + \frac{-1}{\color{blue}{F \cdot F}}}{B} \]

   11. Simplified 54.6%

       \[\leadsto \frac{\color{blue}{\left(1 - x\right) + \frac{-1}{F \cdot F}}}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3 \cdot 10^{-21}:\\ \;\;\;\;\frac{\left(-1 - x\right) + \frac{1}{F \cdot F}}{B}\\ \mathbf{elif}\;F \leq 1.45:\\ \;\;\;\;\frac{\mathsf{fma}\left(F, \sqrt{0.5}, -x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - x\right) + \frac{-1}{F \cdot F}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 52.1% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.22 \cdot 10^{-23}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.46:\\ \;\;\;\;\frac{\mathsf{fma}\left(F, \sqrt{0.5}, -x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - x\right) + \frac{-1}{F \cdot F}}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.22e-23)
   (/ (- -1.0 x) B)
   (if (<= F 1.46)
     (/ (fma F (sqrt 0.5) (- x)) B)
     (/ (+ (- 1.0 x) (/ -1.0 (* F F))) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.22e-23) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.46) {
		tmp = fma(F, sqrt(0.5), -x) / B;
	} else {
		tmp = ((1.0 - x) + (-1.0 / (F * F))) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.22e-23)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 1.46)
		tmp = Float64(fma(F, sqrt(0.5), Float64(-x)) / B);
	else
		tmp = Float64(Float64(Float64(1.0 - x) + Float64(-1.0 / Float64(F * F))) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.22e-23], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.46], N[(N[(F * N[Sqrt[0.5], $MachinePrecision] + (-x)), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(1.0 - x), $MachinePrecision] + N[(-1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.22000000000000007e-23

   1. Initial program 60.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, \mathsf{neg}\left(x\right)\right)}}{B} \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      14. neg-lowering-neg.f64 35.8

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{-x}\right)}{B} \]

   5. Simplified 35.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\left(1 + x\right)\right)}}{B} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(1 + x\right)\right)}{B}} \]

      4. distribute-neg-in N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-1} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      6. unsub-neg N/A

         \[\leadsto \frac{\color{blue}{-1 - x}}{B} \]

      7. --lowering--.f64 49.3

         \[\leadsto \frac{\color{blue}{-1 - x}}{B} \]

   8. Simplified 49.3%

      \[\leadsto \color{blue}{\frac{-1 - x}{B}} \]

   if -1.22000000000000007e-23 < F < 1.46

   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. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, \mathsf{neg}\left(x\right)\right)}}{B} \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      14. neg-lowering-neg.f64 49.6

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{-x}\right)}{B} \]

   5. Simplified 49.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{B}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}}, \mathsf{neg}\left(x\right)\right)}{B} \]
   7. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + {F}^{2}}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      3. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{{F}^{2} + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. accelerator-lowering-fma.f64 49.1

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2\right)}}}, -x\right)}{B} \]

   8. Simplified 49.1%

      \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}, -x\right)}{B} \]

   9. Taylor expanded in F around 0

      \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]
   10. Step-by-step derivation

       1. sqrt-lowering-sqrt.f64 49.1

          \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{0.5}}, -x\right)}{B} \]

   11. Simplified 49.1%

       \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{0.5}}, -x\right)}{B} \]

   if 1.46 < 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. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, \mathsf{neg}\left(x\right)\right)}}{B} \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      14. neg-lowering-neg.f64 40.4

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{-x}\right)}{B} \]

   5. Simplified 40.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{B}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}}, \mathsf{neg}\left(x\right)\right)}{B} \]
   7. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + {F}^{2}}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      3. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{{F}^{2} + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. accelerator-lowering-fma.f64 40.4

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2\right)}}}, -x\right)}{B} \]

   8. Simplified 40.4%

      \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}, -x\right)}{B} \]

   9. Taylor expanded in F around inf

      \[\leadsto \frac{\color{blue}{1 - \left(x + \frac{1}{{F}^{2}}\right)}}{B} \]
   10. Step-by-step derivation

       1. associate--r+ N/A

          \[\leadsto \frac{\color{blue}{\left(1 - x\right) - \frac{1}{{F}^{2}}}}{B} \]

       2. sub-neg N/A

          \[\leadsto \frac{\color{blue}{\left(1 - x\right) + \left(\mathsf{neg}\left(\frac{1}{{F}^{2}}\right)\right)}}{B} \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(1 - x\right) + \left(\mathsf{neg}\left(\frac{1}{{F}^{2}}\right)\right)}}{B} \]

       4. --lowering--.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(1 - x\right)} + \left(\mathsf{neg}\left(\frac{1}{{F}^{2}}\right)\right)}{B} \]

       5. distribute-neg-frac N/A

          \[\leadsto \frac{\left(1 - x\right) + \color{blue}{\frac{\mathsf{neg}\left(1\right)}{{F}^{2}}}}{B} \]

       6. metadata-eval N/A

          \[\leadsto \frac{\left(1 - x\right) + \frac{\color{blue}{-1}}{{F}^{2}}}{B} \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \frac{\left(1 - x\right) + \color{blue}{\frac{-1}{{F}^{2}}}}{B} \]

       8. unpow2 N/A

          \[\leadsto \frac{\left(1 - x\right) + \frac{-1}{\color{blue}{F \cdot F}}}{B} \]

       9. *-lowering-*.f64 54.6

          \[\leadsto \frac{\left(1 - x\right) + \frac{-1}{\color{blue}{F \cdot F}}}{B} \]

   11. Simplified 54.6%

       \[\leadsto \frac{\color{blue}{\left(1 - x\right) + \frac{-1}{F \cdot F}}}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 52.1% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.22 \cdot 10^{-23}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.45:\\ \;\;\;\;\frac{\mathsf{fma}\left(F, \sqrt{0.5}, -x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.22e-23)
   (/ (- -1.0 x) B)
   (if (<= F 1.45) (/ (fma F (sqrt 0.5) (- x)) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.22e-23) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.45) {
		tmp = fma(F, sqrt(0.5), -x) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.22e-23)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 1.45)
		tmp = Float64(fma(F, sqrt(0.5), Float64(-x)) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.22e-23], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.45], N[(N[(F * N[Sqrt[0.5], $MachinePrecision] + (-x)), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.22000000000000007e-23

   1. Initial program 60.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, \mathsf{neg}\left(x\right)\right)}}{B} \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      14. neg-lowering-neg.f64 35.8

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{-x}\right)}{B} \]

   5. Simplified 35.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\left(1 + x\right)\right)}}{B} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(1 + x\right)\right)}{B}} \]

      4. distribute-neg-in N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-1} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      6. unsub-neg N/A

         \[\leadsto \frac{\color{blue}{-1 - x}}{B} \]

      7. --lowering--.f64 49.3

         \[\leadsto \frac{\color{blue}{-1 - x}}{B} \]

   8. Simplified 49.3%

      \[\leadsto \color{blue}{\frac{-1 - x}{B}} \]

   if -1.22000000000000007e-23 < F < 1.44999999999999996

   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. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, \mathsf{neg}\left(x\right)\right)}}{B} \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      14. neg-lowering-neg.f64 49.6

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{-x}\right)}{B} \]

   5. Simplified 49.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{B}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}}, \mathsf{neg}\left(x\right)\right)}{B} \]
   7. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + {F}^{2}}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      3. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{{F}^{2} + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. accelerator-lowering-fma.f64 49.1

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2\right)}}}, -x\right)}{B} \]

   8. Simplified 49.1%

      \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}, -x\right)}{B} \]

   9. Taylor expanded in F around 0

      \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]
   10. Step-by-step derivation

       1. sqrt-lowering-sqrt.f64 49.1

          \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{0.5}}, -x\right)}{B} \]

   11. Simplified 49.1%

       \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{0.5}}, -x\right)}{B} \]

   if 1.44999999999999996 < 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. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, \mathsf{neg}\left(x\right)\right)}}{B} \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      14. neg-lowering-neg.f64 40.4

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{-x}\right)}{B} \]

   5. Simplified 40.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

      2. --lowering--.f64 54.3

         \[\leadsto \frac{\color{blue}{1 - x}}{B} \]

   8. Simplified 54.3%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 45.1% accurate, 9.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.2 \cdot 10^{-46}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 3.1 \cdot 10^{-83}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(B \cdot B, 0.3333333333333333, -1\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.2e-46)
   (/ (- -1.0 x) B)
   (if (<= F 3.1e-83)
     (/ (* x (fma (* B B) 0.3333333333333333 -1.0)) B)
     (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.2e-46) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 3.1e-83) {
		tmp = (x * fma((B * B), 0.3333333333333333, -1.0)) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.2e-46)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 3.1e-83)
		tmp = Float64(Float64(x * fma(Float64(B * B), 0.3333333333333333, -1.0)) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.2e-46], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 3.1e-83], N[(N[(x * N[(N[(B * B), $MachinePrecision] * 0.3333333333333333 + -1.0), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.20000000000000007e-46

   1. Initial program 62.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, \mathsf{neg}\left(x\right)\right)}}{B} \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      14. neg-lowering-neg.f64 38.6

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{-x}\right)}{B} \]

   5. Simplified 38.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\left(1 + x\right)\right)}}{B} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(1 + x\right)\right)}{B}} \]

      4. distribute-neg-in N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-1} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      6. unsub-neg N/A

         \[\leadsto \frac{\color{blue}{-1 - x}}{B} \]

      7. --lowering--.f64 49.0

         \[\leadsto \frac{\color{blue}{-1 - x}}{B} \]

   8. Simplified 49.0%

      \[\leadsto \color{blue}{\frac{-1 - x}{B}} \]

   if -1.20000000000000007e-46 < F < 3.09999999999999992e-83

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right)} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right)} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{x \cdot \cos B}{\sin B}}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{x \cdot \cos B}}{\sin B}\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{x \cdot \color{blue}{\cos B}}{\sin B}\right) \]

      6. sin-lowering-sin.f64 77.9

         \[\leadsto -\frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

   5. Simplified 77.9%

      \[\leadsto \color{blue}{-\frac{x \cdot \cos B}{\sin B}} \]
   6. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{1}{\frac{\sin B}{x \cdot \cos B}}}\right) \]

      2. distribute-neg-frac N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\frac{\sin B}{x \cdot \cos B}}} \]

      3. associate-/l/ N/A

         \[\leadsto \frac{\mathsf{neg}\left(1\right)}{\color{blue}{\frac{\frac{\sin B}{\cos B}}{x}}} \]

      4. tan-quot N/A

         \[\leadsto \frac{\mathsf{neg}\left(1\right)}{\frac{\color{blue}{\tan B}}{x}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-1}}{\frac{\tan B}{x}} \]

      6. div-inv N/A

         \[\leadsto \frac{-1}{\color{blue}{\tan B \cdot \frac{1}{x}}} \]

      7. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{\tan B}}{\frac{1}{x}}} \]

      8. frac-2neg N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\tan B\right)}}}{\frac{1}{x}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\frac{\color{blue}{1}}{\mathsf{neg}\left(\tan B\right)}}{\frac{1}{x}} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{\mathsf{neg}\left(\tan B\right)}}{\frac{1}{x}}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\mathsf{neg}\left(\tan B\right)}}{\frac{1}{x}} \]

      12. frac-2neg N/A

          \[\leadsto \frac{\color{blue}{\frac{-1}{\tan B}}}{\frac{1}{x}} \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{-1}{\tan B}}}{\frac{1}{x}} \]

      14. tan-lowering-tan.f64 N/A

          \[\leadsto \frac{\frac{-1}{\color{blue}{\tan B}}}{\frac{1}{x}} \]

      15. /-lowering-/.f64 77.7

          \[\leadsto \frac{\frac{-1}{\tan B}}{\color{blue}{\frac{1}{x}}} \]

   7. Applied egg-rr 77.7%

      \[\leadsto \color{blue}{\frac{\frac{-1}{\tan B}}{\frac{1}{x}}} \]

   8. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{-1 \cdot x + \frac{1}{3} \cdot \left({B}^{2} \cdot x\right)}{B}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot x + \frac{1}{3} \cdot \left({B}^{2} \cdot x\right)}{B}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(x\right)\right)} + \frac{1}{3} \cdot \left({B}^{2} \cdot x\right)}{B} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{3} \cdot {B}^{2}\right) \cdot x} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      5. mul-1-neg N/A

         \[\leadsto \frac{\left(\frac{1}{3} \cdot {B}^{2}\right) \cdot x + \color{blue}{-1 \cdot x}}{B} \]

      6. distribute-rgt-out N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left(\frac{1}{3} \cdot {B}^{2} + -1\right)}}{B} \]

      7. metadata-eval N/A

         \[\leadsto \frac{x \cdot \left(\frac{1}{3} \cdot {B}^{2} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)}{B} \]

      8. sub-neg N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\frac{1}{3} \cdot {B}^{2} - 1\right)}}{B} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left(\frac{1}{3} \cdot {B}^{2} - 1\right)}}{B} \]

      10. sub-neg N/A

          \[\leadsto \frac{x \cdot \color{blue}{\left(\frac{1}{3} \cdot {B}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right)}}{B} \]

      11. *-commutative N/A

          \[\leadsto \frac{x \cdot \left(\color{blue}{{B}^{2} \cdot \frac{1}{3}} + \left(\mathsf{neg}\left(1\right)\right)\right)}{B} \]

      12. metadata-eval N/A

          \[\leadsto \frac{x \cdot \left({B}^{2} \cdot \frac{1}{3} + \color{blue}{-1}\right)}{B} \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left({B}^{2}, \frac{1}{3}, -1\right)}}{B} \]

      14. unpow2 N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(\color{blue}{B \cdot B}, \frac{1}{3}, -1\right)}{B} \]

      15. *-lowering-*.f64 37.4

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(\color{blue}{B \cdot B}, 0.3333333333333333, -1\right)}{B} \]

   10. Simplified 37.4%

       \[\leadsto \color{blue}{\frac{x \cdot \mathsf{fma}\left(B \cdot B, 0.3333333333333333, -1\right)}{B}} \]

   if 3.09999999999999992e-83 < F

   1. Initial program 64.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, \mathsf{neg}\left(x\right)\right)}}{B} \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      14. neg-lowering-neg.f64 43.8

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{-x}\right)}{B} \]

   5. Simplified 43.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

      2. --lowering--.f64 49.7

         \[\leadsto \frac{\color{blue}{1 - x}}{B} \]

   8. Simplified 49.7%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 21: 45.0% accurate, 9.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.15 \cdot 10^{-47}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{-83}:\\ \;\;\;\;x \cdot \frac{\mathsf{fma}\left(B \cdot B, 0.3333333333333333, -1\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.15e-47)
   (/ (- -1.0 x) B)
   (if (<= F 4.5e-83)
     (* x (/ (fma (* B B) 0.3333333333333333 -1.0) B))
     (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.15e-47) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 4.5e-83) {
		tmp = x * (fma((B * B), 0.3333333333333333, -1.0) / B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.15e-47)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 4.5e-83)
		tmp = Float64(x * Float64(fma(Float64(B * B), 0.3333333333333333, -1.0) / B));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.15e-47], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 4.5e-83], N[(x * N[(N[(N[(B * B), $MachinePrecision] * 0.3333333333333333 + -1.0), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.1499999999999999e-47

   1. Initial program 62.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, \mathsf{neg}\left(x\right)\right)}}{B} \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      14. neg-lowering-neg.f64 38.6

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{-x}\right)}{B} \]

   5. Simplified 38.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\left(1 + x\right)\right)}}{B} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(1 + x\right)\right)}{B}} \]

      4. distribute-neg-in N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-1} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      6. unsub-neg N/A

         \[\leadsto \frac{\color{blue}{-1 - x}}{B} \]

      7. --lowering--.f64 49.0

         \[\leadsto \frac{\color{blue}{-1 - x}}{B} \]

   8. Simplified 49.0%

      \[\leadsto \color{blue}{\frac{-1 - x}{B}} \]

   if -2.1499999999999999e-47 < F < 4.49999999999999997e-83

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right)} \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right)} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{x \cdot \cos B}{\sin B}}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{x \cdot \cos B}}{\sin B}\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{x \cdot \color{blue}{\cos B}}{\sin B}\right) \]

      6. sin-lowering-sin.f64 77.9

         \[\leadsto -\frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

   5. Simplified 77.9%

      \[\leadsto \color{blue}{-\frac{x \cdot \cos B}{\sin B}} \]
   6. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{1}{\frac{\sin B}{x \cdot \cos B}}}\right) \]

      2. distribute-neg-frac N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\frac{\sin B}{x \cdot \cos B}}} \]

      3. associate-/l/ N/A

         \[\leadsto \frac{\mathsf{neg}\left(1\right)}{\color{blue}{\frac{\frac{\sin B}{\cos B}}{x}}} \]

      4. tan-quot N/A

         \[\leadsto \frac{\mathsf{neg}\left(1\right)}{\frac{\color{blue}{\tan B}}{x}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-1}}{\frac{\tan B}{x}} \]

      6. div-inv N/A

         \[\leadsto \frac{-1}{\color{blue}{\tan B \cdot \frac{1}{x}}} \]

      7. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{\tan B}}{\frac{1}{x}}} \]

      8. frac-2neg N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\tan B\right)}}}{\frac{1}{x}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\frac{\color{blue}{1}}{\mathsf{neg}\left(\tan B\right)}}{\frac{1}{x}} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{\mathsf{neg}\left(\tan B\right)}}{\frac{1}{x}}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\mathsf{neg}\left(\tan B\right)}}{\frac{1}{x}} \]

      12. frac-2neg N/A

          \[\leadsto \frac{\color{blue}{\frac{-1}{\tan B}}}{\frac{1}{x}} \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{-1}{\tan B}}}{\frac{1}{x}} \]

      14. tan-lowering-tan.f64 N/A

          \[\leadsto \frac{\frac{-1}{\color{blue}{\tan B}}}{\frac{1}{x}} \]

      15. /-lowering-/.f64 77.7

          \[\leadsto \frac{\frac{-1}{\tan B}}{\color{blue}{\frac{1}{x}}} \]

   7. Applied egg-rr 77.7%

      \[\leadsto \color{blue}{\frac{\frac{-1}{\tan B}}{\frac{1}{x}}} \]

   8. Taylor expanded in B around 0

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{3} \cdot {B}^{2} - 1}{B}}}{\frac{1}{x}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{1}{3} \cdot {B}^{2} - 1}{B}}}{\frac{1}{x}} \]

      2. sub-neg N/A

         \[\leadsto \frac{\frac{\color{blue}{\frac{1}{3} \cdot {B}^{2} + \left(\mathsf{neg}\left(1\right)\right)}}{B}}{\frac{1}{x}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{\color{blue}{{B}^{2} \cdot \frac{1}{3}} + \left(\mathsf{neg}\left(1\right)\right)}{B}}{\frac{1}{x}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\frac{{B}^{2} \cdot \frac{1}{3} + \color{blue}{-1}}{B}}{\frac{1}{x}} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left({B}^{2}, \frac{1}{3}, -1\right)}}{B}}{\frac{1}{x}} \]

      6. unpow2 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{B \cdot B}, \frac{1}{3}, -1\right)}{B}}{\frac{1}{x}} \]

      7. *-lowering-*.f64 37.2

         \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{B \cdot B}, 0.3333333333333333, -1\right)}{B}}{\frac{1}{x}} \]

   10. Simplified 37.2%

       \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(B \cdot B, 0.3333333333333333, -1\right)}{B}}}{\frac{1}{x}} \]
   11. Step-by-step derivation

       1. associate-/r/ N/A

          \[\leadsto \color{blue}{\frac{\frac{\left(B \cdot B\right) \cdot \frac{1}{3} + -1}{B}}{1} \cdot x} \]

       2. /-rgt-identity N/A

          \[\leadsto \color{blue}{\frac{\left(B \cdot B\right) \cdot \frac{1}{3} + -1}{B}} \cdot x \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\left(B \cdot B\right) \cdot \frac{1}{3} + -1}{B} \cdot x} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left(B \cdot B\right) \cdot \frac{1}{3} + -1}{B}} \cdot x \]

       5. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(B \cdot B, \frac{1}{3}, -1\right)}}{B} \cdot x \]

       6. *-lowering-*.f64 37.3

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{B \cdot B}, 0.3333333333333333, -1\right)}{B} \cdot x \]

   12. Applied egg-rr 37.3%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(B \cdot B, 0.3333333333333333, -1\right)}{B} \cdot x} \]

   if 4.49999999999999997e-83 < F

   1. Initial program 64.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, \mathsf{neg}\left(x\right)\right)}}{B} \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      14. neg-lowering-neg.f64 43.8

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{-x}\right)}{B} \]

   5. Simplified 43.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

      2. --lowering--.f64 49.7

         \[\leadsto \frac{\color{blue}{1 - x}}{B} \]

   8. Simplified 49.7%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.15 \cdot 10^{-47}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{-83}:\\ \;\;\;\;x \cdot \frac{\mathsf{fma}\left(B \cdot B, 0.3333333333333333, -1\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 44.7% accurate, 13.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.55 \cdot 10^{-46}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{-136}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.55e-46)
   (/ (- -1.0 x) B)
   (if (<= F 6.2e-136) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.55e-46) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 6.2e-136) {
		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 <= (-1.55d-46)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 6.2d-136) 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 <= -1.55e-46) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 6.2e-136) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.55e-46:
		tmp = (-1.0 - x) / B
	elif F <= 6.2e-136:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.55e-46)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 6.2e-136)
		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 <= -1.55e-46)
		tmp = (-1.0 - x) / B;
	elseif (F <= 6.2e-136)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.55e-46], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 6.2e-136], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.55e-46

   1. Initial program 62.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, \mathsf{neg}\left(x\right)\right)}}{B} \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      14. neg-lowering-neg.f64 38.6

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{-x}\right)}{B} \]

   5. Simplified 38.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\left(1 + x\right)\right)}}{B} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(1 + x\right)\right)}{B}} \]

      4. distribute-neg-in N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-1} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      6. unsub-neg N/A

         \[\leadsto \frac{\color{blue}{-1 - x}}{B} \]

      7. --lowering--.f64 49.0

         \[\leadsto \frac{\color{blue}{-1 - x}}{B} \]

   8. Simplified 49.0%

      \[\leadsto \color{blue}{\frac{-1 - x}{B}} \]

   if -1.55e-46 < F < 6.2e-136

   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. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, \mathsf{neg}\left(x\right)\right)}}{B} \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      14. neg-lowering-neg.f64 46.3

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{-x}\right)}{B} \]

   5. Simplified 46.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot x}{B}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(x\right)}}{B} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(x\right)}{B}} \]

      4. neg-lowering-neg.f64 37.1

         \[\leadsto \frac{\color{blue}{-x}}{B} \]

   8. Simplified 37.1%

      \[\leadsto \color{blue}{\frac{-x}{B}} \]

   if 6.2e-136 < F

   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

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, \mathsf{neg}\left(x\right)\right)}}{B} \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      14. neg-lowering-neg.f64 43.7

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{-x}\right)}{B} \]

   5. Simplified 43.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

      2. --lowering--.f64 49.3

         \[\leadsto \frac{\color{blue}{1 - x}}{B} \]

   8. Simplified 49.3%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 23: 37.5% accurate, 17.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.15 \cdot 10^{-46}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.15e-46) (/ (- -1.0 x) B) (/ (- x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.15e-46) {
		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 <= (-1.15d-46)) 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 <= -1.15e-46) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.15e-46:
		tmp = (-1.0 - x) / B
	else:
		tmp = -x / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.15e-46)
		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 <= -1.15e-46)
		tmp = (-1.0 - x) / B;
	else
		tmp = -x / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.15e-46], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[((-x) / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -1.15e-46

   1. Initial program 62.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, \mathsf{neg}\left(x\right)\right)}}{B} \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      14. neg-lowering-neg.f64 38.6

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{-x}\right)}{B} \]

   5. Simplified 38.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\left(1 + x\right)\right)}}{B} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(1 + x\right)\right)}{B}} \]

      4. distribute-neg-in N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-1} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

      6. unsub-neg N/A

         \[\leadsto \frac{\color{blue}{-1 - x}}{B} \]

      7. --lowering--.f64 49.0

         \[\leadsto \frac{\color{blue}{-1 - x}}{B} \]

   8. Simplified 49.0%

      \[\leadsto \color{blue}{\frac{-1 - x}{B}} \]

   if -1.15e-46 < F

   1. Initial program 82.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, \mathsf{neg}\left(x\right)\right)}}{B} \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      7. associate-+l+ N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      9. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      12. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

      14. neg-lowering-neg.f64 45.0

          \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{-x}\right)}{B} \]

   5. Simplified 45.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot x}{B}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(x\right)}}{B} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(x\right)}{B}} \]

      4. neg-lowering-neg.f64 34.2

         \[\leadsto \frac{\color{blue}{-x}}{B} \]

   8. Simplified 34.2%

      \[\leadsto \color{blue}{\frac{-x}{B}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 30.0% accurate, 26.3× speedup

Math

\[\begin{array}{l} \\ \frac{-x}{B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (/ (- x) B))

C

double code(double F, double B, double x) {
	return -x / B;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = -x / b
end function

Java

public static double code(double F, double B, double x) {
	return -x / B;
}

Python

def code(F, B, x):
	return -x / B

Julia

function code(F, B, x)
	return Float64(Float64(-x) / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = -x / B;
end

Wolfram

code[F_, B_, x_] := N[((-x) / B), $MachinePrecision]

Derivation

1. Initial program 76.0%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
2. Add Preprocessing

3. Taylor expanded in B around 0

   \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
4. Step-by-step derivation

   1. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, \mathsf{neg}\left(x\right)\right)}}{B} \]

   4. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

   6. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

   7. associate-+l+ N/A

      \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

   8. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

   9. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

   10. accelerator-lowering-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

   11. +-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

   12. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

   13. accelerator-lowering-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

   14. neg-lowering-neg.f64 43.0

       \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{-x}\right)}{B} \]

5. Simplified 43.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{B}} \]

6. Taylor expanded in F around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
7. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{-1 \cdot x}{B}} \]

   2. mul-1-neg N/A

      \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(x\right)}}{B} \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(x\right)}{B}} \]

   4. neg-lowering-neg.f64 32.5

      \[\leadsto \frac{\color{blue}{-x}}{B} \]

8. Simplified 32.5%

   \[\leadsto \color{blue}{\frac{-x}{B}} \]
9. Add Preprocessing

Alternative 25: 10.8% accurate, 30.7× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (/ -1.0 B))

C

double code(double F, double B, double x) {
	return -1.0 / B;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = (-1.0d0) / b
end function

Java

public static double code(double F, double B, double x) {
	return -1.0 / B;
}

Python

def code(F, B, x):
	return -1.0 / B

Julia

function code(F, B, x)
	return Float64(-1.0 / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = -1.0 / B;
end

Wolfram

code[F_, B_, x_] := N[(-1.0 / B), $MachinePrecision]

Derivation

1. Initial program 76.0%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
2. Add Preprocessing

3. Taylor expanded in B around 0

   \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
4. Step-by-step derivation

   1. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(F, \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}, \mathsf{neg}\left(x\right)\right)}}{B} \]

   4. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(F, \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

   6. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

   7. associate-+l+ N/A

      \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left({F}^{2} + 2\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

   8. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{x \cdot 2} + \left({F}^{2} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

   9. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{x \cdot 2 + \color{blue}{\left(2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

   10. accelerator-lowering-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(x, 2, 2 + {F}^{2}\right)}}}, \mathsf{neg}\left(x\right)\right)}{B} \]

   11. +-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{{F}^{2} + 2}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

   12. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F} + 2\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

   13. accelerator-lowering-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, \mathsf{neg}\left(x\right)\right)}{B} \]

   14. neg-lowering-neg.f64 43.0

       \[\leadsto \frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, \color{blue}{-x}\right)}{B} \]

5. Simplified 43.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \sqrt{\frac{1}{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, -x\right)}{B}} \]

6. Taylor expanded in F around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
7. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

   2. mul-1-neg N/A

      \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\left(1 + x\right)\right)}}{B} \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(1 + x\right)\right)}{B}} \]

   4. distribute-neg-in N/A

      \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(x\right)\right)}}{B} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{-1} + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

   6. unsub-neg N/A

      \[\leadsto \frac{\color{blue}{-1 - x}}{B} \]

   7. --lowering--.f64 32.5

      \[\leadsto \frac{\color{blue}{-1 - x}}{B} \]

8. Simplified 32.5%

   \[\leadsto \color{blue}{\frac{-1 - x}{B}} \]

9. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{-1}{B}} \]
10. Step-by-step derivation

    1. /-lowering-/.f64 9.9

       \[\leadsto \color{blue}{\frac{-1}{B}} \]

11. Simplified 9.9%

    \[\leadsto \color{blue}{\frac{-1}{B}} \]
12. Add Preprocessing

Alternative 26: 2.8% accurate, 33.5× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(B \cdot 0.3333333333333333\right) \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (* x (* B 0.3333333333333333)))

C

double code(double F, double B, double x) {
	return x * (B * 0.3333333333333333);
}

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 * 0.3333333333333333d0)
end function

Java

public static double code(double F, double B, double x) {
	return x * (B * 0.3333333333333333);
}

Python

def code(F, B, x):
	return x * (B * 0.3333333333333333)

Julia

function code(F, B, x)
	return Float64(x * Float64(B * 0.3333333333333333))
end

MATLAB

function tmp = code(F, B, x)
	tmp = x * (B * 0.3333333333333333);
end

Wolfram

code[F_, B_, x_] := N[(x * N[(B * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.0%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right)} \]

   2. neg-lowering-neg.f64 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right)} \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{x \cdot \cos B}{\sin B}}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{neg}\left(\frac{\color{blue}{x \cdot \cos B}}{\sin B}\right) \]

   5. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{neg}\left(\frac{x \cdot \color{blue}{\cos B}}{\sin B}\right) \]

   6. sin-lowering-sin.f64 61.2

      \[\leadsto -\frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

5. Simplified 61.2%

   \[\leadsto \color{blue}{-\frac{x \cdot \cos B}{\sin B}} \]
6. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{1}{\frac{\sin B}{x \cdot \cos B}}}\right) \]

   2. distribute-neg-frac N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\frac{\sin B}{x \cdot \cos B}}} \]

   3. associate-/l/ N/A

      \[\leadsto \frac{\mathsf{neg}\left(1\right)}{\color{blue}{\frac{\frac{\sin B}{\cos B}}{x}}} \]

   4. tan-quot N/A

      \[\leadsto \frac{\mathsf{neg}\left(1\right)}{\frac{\color{blue}{\tan B}}{x}} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{-1}}{\frac{\tan B}{x}} \]

   6. div-inv N/A

      \[\leadsto \frac{-1}{\color{blue}{\tan B \cdot \frac{1}{x}}} \]

   7. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{-1}{\tan B}}{\frac{1}{x}}} \]

   8. frac-2neg N/A

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\tan B\right)}}}{\frac{1}{x}} \]

   9. metadata-eval N/A

      \[\leadsto \frac{\frac{\color{blue}{1}}{\mathsf{neg}\left(\tan B\right)}}{\frac{1}{x}} \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\frac{1}{\mathsf{neg}\left(\tan B\right)}}{\frac{1}{x}}} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\mathsf{neg}\left(\tan B\right)}}{\frac{1}{x}} \]

   12. frac-2neg N/A

       \[\leadsto \frac{\color{blue}{\frac{-1}{\tan B}}}{\frac{1}{x}} \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \frac{\color{blue}{\frac{-1}{\tan B}}}{\frac{1}{x}} \]

   14. tan-lowering-tan.f64 N/A

       \[\leadsto \frac{\frac{-1}{\color{blue}{\tan B}}}{\frac{1}{x}} \]

   15. /-lowering-/.f64 61.1

       \[\leadsto \frac{\frac{-1}{\tan B}}{\color{blue}{\frac{1}{x}}} \]

7. Applied egg-rr 61.1%

   \[\leadsto \color{blue}{\frac{\frac{-1}{\tan B}}{\frac{1}{x}}} \]

8. Taylor expanded in B around 0

   \[\leadsto \frac{\color{blue}{\frac{\frac{1}{3} \cdot {B}^{2} - 1}{B}}}{\frac{1}{x}} \]
9. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{3} \cdot {B}^{2} - 1}{B}}}{\frac{1}{x}} \]

   2. sub-neg N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{3} \cdot {B}^{2} + \left(\mathsf{neg}\left(1\right)\right)}}{B}}{\frac{1}{x}} \]

   3. *-commutative N/A

      \[\leadsto \frac{\frac{\color{blue}{{B}^{2} \cdot \frac{1}{3}} + \left(\mathsf{neg}\left(1\right)\right)}{B}}{\frac{1}{x}} \]

   4. metadata-eval N/A

      \[\leadsto \frac{\frac{{B}^{2} \cdot \frac{1}{3} + \color{blue}{-1}}{B}}{\frac{1}{x}} \]

   5. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left({B}^{2}, \frac{1}{3}, -1\right)}}{B}}{\frac{1}{x}} \]

   6. unpow2 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{B \cdot B}, \frac{1}{3}, -1\right)}{B}}{\frac{1}{x}} \]

   7. *-lowering-*.f64 32.5

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{B \cdot B}, 0.3333333333333333, -1\right)}{B}}{\frac{1}{x}} \]

10. Simplified 32.5%

    \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(B \cdot B, 0.3333333333333333, -1\right)}{B}}}{\frac{1}{x}} \]

11. Taylor expanded in B around inf

    \[\leadsto \color{blue}{\frac{1}{3} \cdot \left(B \cdot x\right)} \]
12. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \color{blue}{\left(B \cdot x\right) \cdot \frac{1}{3}} \]

    2. *-commutative N/A

       \[\leadsto \color{blue}{\left(x \cdot B\right)} \cdot \frac{1}{3} \]

    3. associate-*l* N/A

       \[\leadsto \color{blue}{x \cdot \left(B \cdot \frac{1}{3}\right)} \]

    4. *-commutative N/A

       \[\leadsto x \cdot \color{blue}{\left(\frac{1}{3} \cdot B\right)} \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \color{blue}{x \cdot \left(\frac{1}{3} \cdot B\right)} \]

    6. *-commutative N/A

       \[\leadsto x \cdot \color{blue}{\left(B \cdot \frac{1}{3}\right)} \]

    7. *-lowering-*.f64 2.8

       \[\leadsto x \cdot \color{blue}{\left(B \cdot 0.3333333333333333\right)} \]

13. Simplified 2.8%

    \[\leadsto \color{blue}{x \cdot \left(B \cdot 0.3333333333333333\right)} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024198 
(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))))))