VandenBroeck and Keller, Equation (23)

Percentage Accurate: 63.8% → 84.1%

Time: 30.2s

Precision: binary64

Cost: 79240

• 24.9% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ t_1 := \frac{F}{\sin B}\\ \mathbf{if}\;F \leq -0.00019:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -1.1 \cdot 10^{-296}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{F \cdot -0.25}{\sin B \cdot \sqrt{0.5}} - \frac{\cos B}{\sin B}, \mathsf{fma}\left(0.5, t_1 \cdot \frac{0.5 - {\left(\frac{-0.25}{\sqrt{0.5}}\right)}^{2}}{\frac{\sqrt{0.5}}{x \cdot x}}, \frac{F \cdot \sqrt{0.5}}{\sin B}\right)\right)\\ \mathbf{elif}\;F \leq 1850000:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{1}{t_1}} - t_0\\ \mathbf{else}:\\ \;\;\;\;F \cdot \left(\frac{\frac{1}{\sin B}}{F} + \frac{-1 - x}{\sin B \cdot {F}^{3}}\right) - t_0\\ \end{array} \]

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))))))

↓

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))) (t_1 (/ F (sin B))))
   (if (<= F -0.00019)
     (+ (* x (/ -1.0 (tan B))) (/ -1.0 (sin B)))
     (if (<= F -1.1e-296)
       (fma
        x
        (- (/ (* F -0.25) (* (sin B) (sqrt 0.5))) (/ (cos B) (sin B)))
        (fma
         0.5
         (*
          t_1
          (/ (- 0.5 (pow (/ -0.25 (sqrt 0.5)) 2.0)) (/ (sqrt 0.5) (* x x))))
         (/ (* F (sqrt 0.5)) (sin B))))
       (if (<= F 1850000.0)
         (- (/ (sqrt 0.5) (/ 1.0 t_1)) t_0)
         (-
          (*
           F
           (+ (/ (/ 1.0 (sin B)) F) (/ (- -1.0 x) (* (sin B) (pow F 3.0)))))
          t_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)));
}

↓

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double t_1 = F / sin(B);
	double tmp;
	if (F <= -0.00019) {
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / sin(B));
	} else if (F <= -1.1e-296) {
		tmp = fma(x, (((F * -0.25) / (sin(B) * sqrt(0.5))) - (cos(B) / sin(B))), fma(0.5, (t_1 * ((0.5 - pow((-0.25 / sqrt(0.5)), 2.0)) / (sqrt(0.5) / (x * x)))), ((F * sqrt(0.5)) / sin(B))));
	} else if (F <= 1850000.0) {
		tmp = (sqrt(0.5) / (1.0 / t_1)) - t_0;
	} else {
		tmp = (F * (((1.0 / sin(B)) / F) + ((-1.0 - x) / (sin(B) * pow(F, 3.0))))) - t_0;
	}
	return tmp;
}

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

↓

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	t_1 = Float64(F / sin(B))
	tmp = 0.0
	if (F <= -0.00019)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(-1.0 / sin(B)));
	elseif (F <= -1.1e-296)
		tmp = fma(x, Float64(Float64(Float64(F * -0.25) / Float64(sin(B) * sqrt(0.5))) - Float64(cos(B) / sin(B))), fma(0.5, Float64(t_1 * Float64(Float64(0.5 - (Float64(-0.25 / sqrt(0.5)) ^ 2.0)) / Float64(sqrt(0.5) / Float64(x * x)))), Float64(Float64(F * sqrt(0.5)) / sin(B))));
	elseif (F <= 1850000.0)
		tmp = Float64(Float64(sqrt(0.5) / Float64(1.0 / t_1)) - t_0);
	else
		tmp = Float64(Float64(F * Float64(Float64(Float64(1.0 / sin(B)) / F) + Float64(Float64(-1.0 - x) / Float64(sin(B) * (F ^ 3.0))))) - t_0);
	end
	return tmp
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]

↓

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.00019], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.1e-296], N[(x * N[(N[(N[(F * -0.25), $MachinePrecision] / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Cos[B], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(t$95$1 * N[(N[(0.5 - N[Power[N[(-0.25 / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[0.5], $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1850000.0], N[(N[(N[Sqrt[0.5], $MachinePrecision] / N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(F * N[(N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision] + N[(N[(-1.0 - x), $MachinePrecision] / N[(N[Sin[B], $MachinePrecision] * N[Power[F, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.9000000000000001e-4

   1. Initial program 58.9%

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

   2. Taylor expanded in F around -inf 96.2%

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

   if -1.9000000000000001e-4 < F < -1.10000000000000006e-296

   1. Initial program 65.2%

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

   2. Simplified 65.3%

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

      [Start] 65.2%	\[ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
      +-commutative [=>] 65.2%	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]
      unsub-neg [=>] 65.2%	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]
      associate-*l/ [=>] 65.2%	\[ \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]
      associate-*r/ [<=] 65.2%	\[ \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]
      *-commutative [<=] 65.2%	\[ \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \cdot F} - x \cdot \frac{1}{\tan B} \]

   3. Taylor expanded in F around 0 65.1%

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

   4. Taylor expanded in x around 0 67.6%

      \[\leadsto \color{blue}{x \cdot \left(-0.25 \cdot \frac{F}{\sin B \cdot \sqrt{0.5}} - \frac{\cos B}{\sin B}\right) + \left(0.5 \cdot \frac{F \cdot \left(\left(0.5 - {\left(\frac{-0.25}{\sqrt{0.5}}\right)}^{2}\right) \cdot {x}^{2}\right)}{\sqrt{0.5} \cdot \sin B} + \frac{\sqrt{0.5} \cdot F}{\sin B}\right)} \]

   5. Simplified 75.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{-0.25 \cdot F}{\sin B \cdot \sqrt{0.5}} - \frac{\cos B}{\sin B}, \mathsf{fma}\left(0.5, \frac{F}{\sin B} \cdot \frac{0.5 - {\left(\frac{-0.25}{\sqrt{0.5}}\right)}^{2}}{\frac{\sqrt{0.5}}{x \cdot x}}, \frac{F \cdot \sqrt{0.5}}{\sin B}\right)\right)} \]
      Step-by-step derivation

      [Start] 67.6%	\[ x \cdot \left(-0.25 \cdot \frac{F}{\sin B \cdot \sqrt{0.5}} - \frac{\cos B}{\sin B}\right) + \left(0.5 \cdot \frac{F \cdot \left(\left(0.5 - {\left(\frac{-0.25}{\sqrt{0.5}}\right)}^{2}\right) \cdot {x}^{2}\right)}{\sqrt{0.5} \cdot \sin B} + \frac{\sqrt{0.5} \cdot F}{\sin B}\right) \]
      fma-def [=>] 75.8%	\[ \color{blue}{\mathsf{fma}\left(x, -0.25 \cdot \frac{F}{\sin B \cdot \sqrt{0.5}} - \frac{\cos B}{\sin B}, 0.5 \cdot \frac{F \cdot \left(\left(0.5 - {\left(\frac{-0.25}{\sqrt{0.5}}\right)}^{2}\right) \cdot {x}^{2}\right)}{\sqrt{0.5} \cdot \sin B} + \frac{\sqrt{0.5} \cdot F}{\sin B}\right)} \]
      associate-*r/ [=>] 75.8%	\[ \mathsf{fma}\left(x, \color{blue}{\frac{-0.25 \cdot F}{\sin B \cdot \sqrt{0.5}}} - \frac{\cos B}{\sin B}, 0.5 \cdot \frac{F \cdot \left(\left(0.5 - {\left(\frac{-0.25}{\sqrt{0.5}}\right)}^{2}\right) \cdot {x}^{2}\right)}{\sqrt{0.5} \cdot \sin B} + \frac{\sqrt{0.5} \cdot F}{\sin B}\right) \]
      fma-def [=>] 75.8%	\[ \mathsf{fma}\left(x, \frac{-0.25 \cdot F}{\sin B \cdot \sqrt{0.5}} - \frac{\cos B}{\sin B}, \color{blue}{\mathsf{fma}\left(0.5, \frac{F \cdot \left(\left(0.5 - {\left(\frac{-0.25}{\sqrt{0.5}}\right)}^{2}\right) \cdot {x}^{2}\right)}{\sqrt{0.5} \cdot \sin B}, \frac{\sqrt{0.5} \cdot F}{\sin B}\right)}\right) \]

   if -1.10000000000000006e-296 < F < 1.85e6

   1. Initial program 75.6%

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

   2. Simplified 75.6%

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

      [Start] 75.6%	\[ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
      +-commutative [=>] 75.6%	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]
      unsub-neg [=>] 75.6%	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]
      associate-*l/ [=>] 75.6%	\[ \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]
      associate-*r/ [<=] 75.6%	\[ \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]
      *-commutative [<=] 75.6%	\[ \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \cdot F} - x \cdot \frac{1}{\tan B} \]

   3. Taylor expanded in F around 0 75.6%

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

   4. Taylor expanded in x around 0 79.2%

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

   5. Simplified 79.1%

      \[\leadsto \color{blue}{\frac{\sqrt{0.5}}{\frac{\sin B}{F}}} - \frac{x}{\tan B} \]
      Step-by-step derivation

      [Start] 79.2%	\[ \frac{\sqrt{0.5} \cdot F}{\sin B} - \frac{x}{\tan B} \]
      associate-/l* [=>] 79.1%	\[ \color{blue}{\frac{\sqrt{0.5}}{\frac{\sin B}{F}}} - \frac{x}{\tan B} \]

   6. Applied egg-rr 79.2%

      \[\leadsto \frac{\sqrt{0.5}}{\color{blue}{{\left(\frac{F}{\sin B}\right)}^{-1}}} - \frac{x}{\tan B} \]
      Step-by-step derivation

      [Start] 79.1%	\[ \frac{\sqrt{0.5}}{\frac{\sin B}{F}} - \frac{x}{\tan B} \]
      clear-num [=>] 79.2%	\[ \frac{\sqrt{0.5}}{\color{blue}{\frac{1}{\frac{F}{\sin B}}}} - \frac{x}{\tan B} \]
      inv-pow [=>] 79.2%	\[ \frac{\sqrt{0.5}}{\color{blue}{{\left(\frac{F}{\sin B}\right)}^{-1}}} - \frac{x}{\tan B} \]

   7. Simplified 79.2%

      \[\leadsto \frac{\sqrt{0.5}}{\color{blue}{\frac{1}{\frac{F}{\sin B}}}} - \frac{x}{\tan B} \]
      Step-by-step derivation

      [Start] 79.2%	\[ \frac{\sqrt{0.5}}{{\left(\frac{F}{\sin B}\right)}^{-1}} - \frac{x}{\tan B} \]
      unpow-1 [=>] 79.2%	\[ \frac{\sqrt{0.5}}{\color{blue}{\frac{1}{\frac{F}{\sin B}}}} - \frac{x}{\tan B} \]

   if 1.85e6 < F

   1. Initial program 54.8%

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

   2. Simplified 69.2%

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

      [Start] 54.8%	\[ \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
      +-commutative [=>] 54.8%	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]
      unsub-neg [=>] 54.8%	\[ \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]
      associate-*l/ [=>] 69.2%	\[ \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]
      associate-*r/ [<=] 69.1%	\[ \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} - x \cdot \frac{1}{\tan B} \]
      *-commutative [<=] 69.1%	\[ \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B} \cdot F} - x \cdot \frac{1}{\tan B} \]

   3. Taylor expanded in F around inf 92.3%

      \[\leadsto F \cdot \color{blue}{\left(-0.5 \cdot \frac{2 + 2 \cdot x}{\sin B \cdot {F}^{3}} + \frac{1}{\sin B \cdot F}\right)} - \frac{x}{\tan B} \]

   4. Simplified 94.0%

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

      [Start] 92.3%	\[ F \cdot \left(-0.5 \cdot \frac{2 + 2 \cdot x}{\sin B \cdot {F}^{3}} + \frac{1}{\sin B \cdot F}\right) - \frac{x}{\tan B} \]
      +-commutative [=>] 92.3%	\[ F \cdot \color{blue}{\left(\frac{1}{\sin B \cdot F} + -0.5 \cdot \frac{2 + 2 \cdot x}{\sin B \cdot {F}^{3}}\right)} - \frac{x}{\tan B} \]
      associate-/r* [=>] 92.6%	\[ F \cdot \left(\color{blue}{\frac{\frac{1}{\sin B}}{F}} + -0.5 \cdot \frac{2 + 2 \cdot x}{\sin B \cdot {F}^{3}}\right) - \frac{x}{\tan B} \]
      associate-*r/ [=>] 92.6%	\[ F \cdot \left(\frac{\frac{1}{\sin B}}{F} + \color{blue}{\frac{-0.5 \cdot \left(2 + 2 \cdot x\right)}{\sin B \cdot {F}^{3}}}\right) - \frac{x}{\tan B} \]
      distribute-lft-in [=>] 92.6%	\[ F \cdot \left(\frac{\frac{1}{\sin B}}{F} + \frac{\color{blue}{-0.5 \cdot 2 + -0.5 \cdot \left(2 \cdot x\right)}}{\sin B \cdot {F}^{3}}\right) - \frac{x}{\tan B} \]
      metadata-eval [=>] 92.6%	\[ F \cdot \left(\frac{\frac{1}{\sin B}}{F} + \frac{\color{blue}{-1} + -0.5 \cdot \left(2 \cdot x\right)}{\sin B \cdot {F}^{3}}\right) - \frac{x}{\tan B} \]
      associate-*r* [=>] 94.0%	\[ F \cdot \left(\frac{\frac{1}{\sin B}}{F} + \frac{-1 + \color{blue}{\left(-0.5 \cdot 2\right) \cdot x}}{\sin B \cdot {F}^{3}}\right) - \frac{x}{\tan B} \]
      metadata-eval [=>] 94.0%	\[ F \cdot \left(\frac{\frac{1}{\sin B}}{F} + \frac{-1 + \color{blue}{-1} \cdot x}{\sin B \cdot {F}^{3}}\right) - \frac{x}{\tan B} \]
      neg-mul-1 [<=] 94.0%	\[ F \cdot \left(\frac{\frac{1}{\sin B}}{F} + \frac{-1 + \color{blue}{\left(-x\right)}}{\sin B \cdot {F}^{3}}\right) - \frac{x}{\tan B} \]
      *-commutative [=>] 94.0%	\[ F \cdot \left(\frac{\frac{1}{\sin B}}{F} + \frac{-1 + \left(-x\right)}{\color{blue}{{F}^{3} \cdot \sin B}}\right) - \frac{x}{\tan B} \]

3. Recombined 4 regimes into one program.

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.00019:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -1.1 \cdot 10^{-296}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{F \cdot -0.25}{\sin B \cdot \sqrt{0.5}} - \frac{\cos B}{\sin B}, \mathsf{fma}\left(0.5, \frac{F}{\sin B} \cdot \frac{0.5 - {\left(\frac{-0.25}{\sqrt{0.5}}\right)}^{2}}{\frac{\sqrt{0.5}}{x \cdot x}}, \frac{F \cdot \sqrt{0.5}}{\sin B}\right)\right)\\ \mathbf{elif}\;F \leq 1850000:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{1}{\frac{F}{\sin B}}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \left(\frac{\frac{1}{\sin B}}{F} + \frac{-1 - x}{\sin B \cdot {F}^{3}}\right) - \frac{x}{\tan B}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	84.1%
Cost	79240

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ t_1 := \frac{F}{\sin B}\\ \mathbf{if}\;F \leq -0.00019:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -1.1 \cdot 10^{-296}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{F \cdot -0.25}{\sin B \cdot \sqrt{0.5}} - \frac{\cos B}{\sin B}, \mathsf{fma}\left(0.5, t_1 \cdot \frac{0.5 - {\left(\frac{-0.25}{\sqrt{0.5}}\right)}^{2}}{\frac{\sqrt{0.5}}{x \cdot x}}, \frac{F \cdot \sqrt{0.5}}{\sin B}\right)\right)\\ \mathbf{elif}\;F \leq 1850000:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{1}{t_1}} - t_0\\ \mathbf{else}:\\ \;\;\;\;F \cdot \left(\frac{\frac{1}{\sin B}}{F} + \frac{-1 - x}{\sin B \cdot {F}^{3}}\right) - t_0\\ \end{array} \]

Alternative 2
Accuracy	85.5%
Cost	27144

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -122000:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1850000:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{1}{\frac{F}{\sin B}}} - t_0\\ \mathbf{else}:\\ \;\;\;\;F \cdot \left(\frac{\frac{1}{\sin B}}{F} + \frac{-1 - x}{\sin B \cdot {F}^{3}}\right) - t_0\\ \end{array} \]

Alternative 3
Accuracy	85.5%
Cost	20168

\[\begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -122000:\\ \;\;\;\;t_0 + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.26 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{1}{\frac{F}{\sin B}}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{1}{\sin B}\\ \end{array} \]

Alternative 4
Accuracy	85.5%
Cost	20040

\[\begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -122000:\\ \;\;\;\;t_0 + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.26 \cdot 10^{-8}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{1}{\sin B}\\ \end{array} \]

Alternative 5
Accuracy	85.5%
Cost	20040

\[\begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -122000:\\ \;\;\;\;t_0 + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.26 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\sin B}{F}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{1}{\sin B}\\ \end{array} \]

Alternative 6
Accuracy	78.2%
Cost	14284

\[\begin{array}{l} t_0 := \frac{\sqrt{0.5}}{\frac{\sin B}{F}}\\ t_1 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -0.22:\\ \;\;\;\;t_1 + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -7.2 \cdot 10^{-64}:\\ \;\;\;\;t_0 - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(x \cdot B\right)\right)\\ \mathbf{elif}\;F \leq 9 \cdot 10^{-112}:\\ \;\;\;\;F \cdot \left(\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{1}{B}\right) - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{-14}:\\ \;\;\;\;t_0 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t_1 + \frac{1}{\sin B}\\ \end{array} \]

Alternative 7
Accuracy	76.8%
Cost	14024

\[\begin{array}{l} t_0 := \frac{\sqrt{0.5}}{\frac{\sin B}{F}}\\ t_1 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -9.2 \cdot 10^{-5}:\\ \;\;\;\;t_1 + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -8.5 \cdot 10^{-174}:\\ \;\;\;\;t_0 - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(x \cdot B\right)\right)\\ \mathbf{elif}\;F \leq 9.8 \cdot 10^{-172}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 8.6 \cdot 10^{-15}:\\ \;\;\;\;t_0 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t_1 + \frac{1}{\sin B}\\ \end{array} \]

Alternative 8
Accuracy	67.0%
Cost	13904

\[\begin{array}{l} t_0 := \frac{\sqrt{0.5}}{\frac{\sin B}{F}} - \frac{x}{B}\\ \mathbf{if}\;F \leq -1.42:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -8.2 \cdot 10^{-174}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{-176}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 1.26 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 9
Accuracy	72.2%
Cost	13904

\[\begin{array}{l} t_0 := \frac{\sqrt{0.5}}{\frac{\sin B}{F}} - \frac{x}{B}\\ \mathbf{if}\;F \leq -0.44:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -8.5 \cdot 10^{-174}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 1.28 \cdot 10^{-169}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 1.26 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 10
Accuracy	77.6%
Cost	13904

\[\begin{array}{l} t_0 := \frac{\sqrt{0.5}}{\frac{\sin B}{F}} - \frac{x}{B}\\ t_1 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -0.082:\\ \;\;\;\;t_1 + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -8.5 \cdot 10^{-174}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 2.6 \cdot 10^{-177}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{-14}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1 + \frac{1}{\sin B}\\ \end{array} \]

Alternative 11
Accuracy	63.7%
Cost	13780

\[\begin{array}{l} t_0 := F \cdot \frac{\sqrt{0.5}}{\sin B}\\ \mathbf{if}\;F \leq -122000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -8.8 \cdot 10^{-41}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -1.6 \cdot 10^{-56}:\\ \;\;\;\;\left|B \cdot \left(x \cdot 0.3333333333333333\right)\right| - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.04 \cdot 10^{-109}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 1.38 \cdot 10^{-11}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 12
Accuracy	63.6%
Cost	13780

\[\begin{array}{l} \mathbf{if}\;F \leq -122000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.35 \cdot 10^{-41}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq -1.9 \cdot 10^{-61}:\\ \;\;\;\;\left|B \cdot \left(x \cdot 0.3333333333333333\right)\right| - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4 \cdot 10^{-109}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 4.6 \cdot 10^{-13}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 13
Accuracy	56.2%
Cost	7376

\[\begin{array}{l} t_0 := \frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{if}\;F \leq -92000000000:\\ \;\;\;\;B \cdot -0.16666666666666666 + \left(\frac{-1}{B} - \frac{x}{B}\right)\\ \mathbf{elif}\;F \leq -8.5 \cdot 10^{-174}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 3.1 \cdot 10^{-161}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 1.55 \cdot 10^{-37}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 14
Accuracy	62.6%
Cost	7244

\[\begin{array}{l} \mathbf{if}\;F \leq -122000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 9.2 \cdot 10^{-166}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 1.55 \cdot 10^{-37}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 15
Accuracy	55.2%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;F \leq 5.4 \cdot 10^{+41}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 16
Accuracy	49.7%
Cost	6656

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

Alternative 17
Accuracy	40.8%
Cost	968

\[\begin{array}{l} \mathbf{if}\;F \leq -92000000000:\\ \;\;\;\;B \cdot -0.16666666666666666 + \left(\frac{-1}{B} - \frac{x}{B}\right)\\ \mathbf{elif}\;F \leq 3.1 \cdot 10^{-88}:\\ \;\;\;\;\left(x \cdot B\right) \cdot 0.3333333333333333 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{B} + B \cdot 0.16666666666666666\right) - \frac{x}{B}\\ \end{array} \]

Alternative 18
Accuracy	35.5%
Cost	836

\[\begin{array}{l} \mathbf{if}\;F \leq -92000000000:\\ \;\;\;\;B \cdot -0.16666666666666666 + \left(\frac{-1}{B} - \frac{x}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot B\right) \cdot 0.3333333333333333 - \frac{x}{B}\\ \end{array} \]

Alternative 19
Accuracy	29.8%
Cost	576

\[x \cdot \left(B \cdot 0.3333333333333333 + \frac{-1}{B}\right) \]

Alternative 20
Accuracy	29.8%
Cost	576

\[\left(x \cdot B\right) \cdot 0.3333333333333333 - \frac{x}{B} \]

Alternative 21
Accuracy	27.5%
Cost	256

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

Reproduce

herbie shell --seed 2023255 
(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))))))