VandenBroeck and Keller, Equation (23)

Percentage Accurate: 75.7% → 99.7%

Time: 18.0s

Alternatives: 21

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 75.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3.5 \cdot 10^{+18}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 80000000:\\ \;\;\;\;F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -3.5e+18)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 80000000.0)
       (- (* F (/ (pow (fma x 2.0 (fma F F 2.0)) -0.5) (sin B))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -3.5e+18) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 80000000.0) {
		tmp = (F * (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) / sin(B))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -3.5e+18)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 80000000.0)
		tmp = Float64(Float64(F * Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) / sin(B))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -3.5e+18], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 80000000.0], N[(N[(F * N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.5e18

   1. Initial program 44.4%

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

   3. Simplified 63.4%

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

      1. clear-num 63.4%

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

      2. inv-pow 63.4%

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

      3. fma-define 63.4%

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

      4. fma-undefine 63.4%

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

      5. *-commutative 63.4%

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

      6. fma-define 63.4%

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

      7. fma-define 63.4%

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

   6. Applied egg-rr 63.4%

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

      1. unpow-1 63.4%

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

   8. Simplified 63.4%

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

   9. Taylor expanded in F around -inf 99.9%

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

   if -3.5e18 < F < 8e7

   1. Initial program 99.4%

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

   3. Simplified 99.6%

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

   if 8e7 < F

   1. Initial program 54.5%

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

   3. Simplified 68.8%

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

      1. clear-num 68.7%

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

      2. inv-pow 68.7%

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

      3. fma-define 68.7%

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

      4. fma-undefine 68.7%

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

      5. *-commutative 68.7%

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

      6. fma-define 68.7%

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

      7. fma-define 68.7%

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

   6. Applied egg-rr 68.7%

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

      1. unpow-1 68.7%

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

   8. Simplified 68.7%

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

   9. Taylor expanded in F around inf 99.9%

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

Alternative 2: 99.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.8 \cdot 10^{+51}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 80000000:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.8e+51)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 80000000.0)
       (- (/ (/ F (sin B)) (sqrt (fma 2.0 x (fma F F 2.0)))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.8e+51) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 80000000.0) {
		tmp = ((F / sin(B)) / sqrt(fma(2.0, x, fma(F, F, 2.0)))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.8e+51)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 80000000.0)
		tmp = Float64(Float64(Float64(F / sin(B)) / sqrt(fma(2.0, x, fma(F, F, 2.0)))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.8e+51], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 80000000.0], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.80000000000000005e51

   1. Initial program 41.4%

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

   3. Simplified 61.4%

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

      1. clear-num 61.5%

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

      2. inv-pow 61.5%

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

      3. fma-define 61.5%

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

      4. fma-undefine 61.5%

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

      5. *-commutative 61.5%

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

      6. fma-define 61.5%

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

      7. fma-define 61.5%

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

   6. Applied egg-rr 61.5%

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

      1. unpow-1 61.5%

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

   8. Simplified 61.5%

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

   9. Taylor expanded in F around -inf 99.9%

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

   if -1.80000000000000005e51 < F < 8e7

   1. Initial program 99.4%

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

   3. Simplified 99.6%

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

      1. clear-num 99.5%

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

      2. inv-pow 99.5%

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

      3. fma-define 99.5%

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

      4. fma-undefine 99.5%

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

      5. *-commutative 99.5%

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

      6. fma-define 99.5%

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

      7. fma-define 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. unpow-1 99.5%

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

   8. Simplified 99.5%

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

      1. un-div-inv 99.6%

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

      2. div-inv 99.6%

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

      3. pow-flip 99.6%

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

      4. metadata-eval 99.6%

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

      5. pow1/2 99.6%

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

   10. Applied egg-rr 99.6%

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

       1. associate-/r* 99.5%

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

   12. Simplified 99.5%

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

   if 8e7 < F

   1. Initial program 54.5%

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

   3. Simplified 68.8%

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

      1. clear-num 68.7%

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

      2. inv-pow 68.7%

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

      3. fma-define 68.7%

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

      4. fma-undefine 68.7%

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

      5. *-commutative 68.7%

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

      6. fma-define 68.7%

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

      7. fma-define 68.7%

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

   6. Applied egg-rr 68.7%

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

      1. unpow-1 68.7%

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

   8. Simplified 68.7%

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

   9. Taylor expanded in F around inf 99.9%

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

Alternative 3: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -9.2 \cdot 10^{-18}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 42000000:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -9.2e-18)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 42000000.0)
       (+
        (/ -1.0 (/ (tan B) x))
        (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)))
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -9.2e-18) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 42000000.0) {
		tmp = (-1.0 / (tan(B) / x)) + ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5));
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-9.2d-18)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 42000000.0d0) then
        tmp = ((-1.0d0) / (tan(b) / x)) + ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)))
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -9.2e-18) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 42000000.0) {
		tmp = (-1.0 / (Math.tan(B) / x)) + ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5));
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -9.2e-18:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 42000000.0:
		tmp = (-1.0 / (math.tan(B) / x)) + ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5))
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -9.2e-18)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 42000000.0)
		tmp = Float64(Float64(-1.0 / Float64(tan(B) / x)) + Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -9.2e-18)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 42000000.0)
		tmp = (-1.0 / (tan(B) / x)) + ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5));
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -9.2e-18], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 42000000.0], N[(N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -9.2000000000000004e-18

   1. Initial program 47.0%

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

   3. Simplified 65.1%

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

      1. clear-num 65.1%

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

      2. inv-pow 65.1%

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

      3. fma-define 65.1%

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

      4. fma-undefine 65.1%

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

      5. *-commutative 65.1%

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

      6. fma-define 65.1%

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

      7. fma-define 65.1%

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

   6. Applied egg-rr 65.1%

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

      1. unpow-1 65.1%

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

   8. Simplified 65.1%

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

   9. Taylor expanded in F around -inf 99.9%

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

   if -9.2000000000000004e-18 < F < 4.2e7

   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. div-inv 99.6%

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

      2. clear-num 99.4%

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

   4. Applied egg-rr 99.4%

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

      1. metadata-eval 99.4%

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

      2. metadata-eval 99.4%

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

   6. Applied egg-rr 99.4%

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

   if 4.2e7 < F

   1. Initial program 54.5%

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

   3. Simplified 68.8%

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

      1. clear-num 68.7%

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

      2. inv-pow 68.7%

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

      3. fma-define 68.7%

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

      4. fma-undefine 68.7%

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

      5. *-commutative 68.7%

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

      6. fma-define 68.7%

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

      7. fma-define 68.7%

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

   6. Applied egg-rr 68.7%

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

      1. unpow-1 68.7%

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

   8. Simplified 68.7%

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

   9. Taylor expanded in F around inf 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -9.2 \cdot 10^{-18}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 42000000:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -9.2 \cdot 10^{-18}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 20000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} + x \cdot \frac{-1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -9.2e-18)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 20000000.0)
       (+
        (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5))
        (* x (/ -1.0 (tan B))))
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -9.2e-18) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 20000000.0) {
		tmp = ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) + (x * (-1.0 / tan(B)));
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-9.2d-18)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 20000000.0d0) then
        tmp = ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0))) + (x * ((-1.0d0) / tan(b)))
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -9.2e-18) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 20000000.0) {
		tmp = ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) + (x * (-1.0 / Math.tan(B)));
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -9.2e-18:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 20000000.0:
		tmp = ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) + (x * (-1.0 / math.tan(B)))
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -9.2e-18)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 20000000.0)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)) + Float64(x * Float64(-1.0 / tan(B))));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -9.2e-18)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 20000000.0)
		tmp = ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5)) + (x * (-1.0 / tan(B)));
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -9.2e-18], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 20000000.0], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] + N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -9.2000000000000004e-18

   1. Initial program 47.0%

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

   3. Simplified 65.1%

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

      1. clear-num 65.1%

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

      2. inv-pow 65.1%

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

      3. fma-define 65.1%

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

      4. fma-undefine 65.1%

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

      5. *-commutative 65.1%

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

      6. fma-define 65.1%

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

      7. fma-define 65.1%

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

   6. Applied egg-rr 65.1%

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

      1. unpow-1 65.1%

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

   8. Simplified 65.1%

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

   9. Taylor expanded in F around -inf 99.9%

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

   if -9.2000000000000004e-18 < F < 2e7

   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. metadata-eval 99.4%

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

      2. metadata-eval 99.4%

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

   4. Applied egg-rr 99.4%

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

   if 2e7 < F

   1. Initial program 54.5%

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

   3. Simplified 68.8%

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

      1. clear-num 68.7%

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

      2. inv-pow 68.7%

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

      3. fma-define 68.7%

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

      4. fma-undefine 68.7%

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

      5. *-commutative 68.7%

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

      6. fma-define 68.7%

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

      7. fma-define 68.7%

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

   6. Applied egg-rr 68.7%

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

      1. unpow-1 68.7%

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

   8. Simplified 68.7%

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

   9. Taylor expanded in F around inf 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -9.2 \cdot 10^{-18}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 20000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} + x \cdot \frac{-1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 0.92:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{\sin B \cdot \left(F \cdot \left(1 + \frac{1}{{F}^{2}}\right)\right)} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.4)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 0.92)
       (- (* F (/ (sqrt 0.5) (sin B))) t_0)
       (- (/ F (* (sin B) (* F (+ 1.0 (/ 1.0 (pow F 2.0)))))) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.4) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 0.92) {
		tmp = (F * (sqrt(0.5) / sin(B))) - t_0;
	} else {
		tmp = (F / (sin(B) * (F * (1.0 + (1.0 / pow(F, 2.0)))))) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.4d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 0.92d0) then
        tmp = (f * (sqrt(0.5d0) / sin(b))) - t_0
    else
        tmp = (f / (sin(b) * (f * (1.0d0 + (1.0d0 / (f ** 2.0d0)))))) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.4) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 0.92) {
		tmp = (F * (Math.sqrt(0.5) / Math.sin(B))) - t_0;
	} else {
		tmp = (F / (Math.sin(B) * (F * (1.0 + (1.0 / Math.pow(F, 2.0)))))) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.4:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 0.92:
		tmp = (F * (math.sqrt(0.5) / math.sin(B))) - t_0
	else:
		tmp = (F / (math.sin(B) * (F * (1.0 + (1.0 / math.pow(F, 2.0)))))) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.4)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 0.92)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / sin(B))) - t_0);
	else
		tmp = Float64(Float64(F / Float64(sin(B) * Float64(F * Float64(1.0 + Float64(1.0 / (F ^ 2.0)))))) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.4)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 0.92)
		tmp = (F * (sqrt(0.5) / sin(B))) - t_0;
	else
		tmp = (F / (sin(B) * (F * (1.0 + (1.0 / (F ^ 2.0)))))) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.4], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.92], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * N[(F * N[(1.0 + N[(1.0 / N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.3999999999999999

   1. Initial program 45.3%

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

   3. Simplified 64.0%

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

      1. clear-num 64.0%

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

      2. inv-pow 64.0%

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

      3. fma-define 64.0%

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

      4. fma-undefine 64.0%

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

      5. *-commutative 64.0%

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

      6. fma-define 64.0%

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

      7. fma-define 64.0%

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

   6. Applied egg-rr 64.0%

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

      1. unpow-1 64.0%

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

   8. Simplified 64.0%

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

   9. Taylor expanded in F around -inf 99.9%

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

   if -1.3999999999999999 < F < 0.92000000000000004

   1. Initial program 99.4%

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

   3. Simplified 99.6%

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

      1. clear-num 99.5%

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

      2. inv-pow 99.5%

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

      3. fma-define 99.5%

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

      4. fma-undefine 99.5%

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

      5. *-commutative 99.5%

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

      6. fma-define 99.5%

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

      7. fma-define 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. unpow-1 99.5%

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

   8. Simplified 99.5%

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

   9. Taylor expanded in x around 0 99.6%

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

       1. +-commutative 99.6%

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

       2. unpow2 99.6%

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

       3. fma-undefine 99.6%

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

   11. Simplified 99.6%

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

   12. Taylor expanded in F around 0 98.8%

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

   if 0.92000000000000004 < F

   1. Initial program 55.6%

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

   3. Simplified 69.5%

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

      1. clear-num 69.5%

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

      2. inv-pow 69.5%

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

      3. fma-define 69.5%

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

      4. fma-undefine 69.5%

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

      5. *-commutative 69.5%

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

      6. fma-define 69.5%

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

      7. fma-define 69.5%

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

   6. Applied egg-rr 69.5%

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

      1. unpow-1 69.5%

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

   8. Simplified 69.5%

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

   9. Taylor expanded in x around 0 69.2%

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

       1. +-commutative 69.2%

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

       2. unpow2 69.2%

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

       3. fma-undefine 69.2%

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

   11. Simplified 69.2%

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

       1. un-div-inv 69.2%

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

       2. *-commutative 69.2%

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

   13. Applied egg-rr 69.2%

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

   14. Taylor expanded in F around inf 99.3%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.92:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{\sin B \cdot \left(F \cdot \left(1 + \frac{1}{{F}^{2}}\right)\right)} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if F < -1.3999999999999999

   1. Initial program 45.3%

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

   3. Simplified 64.0%

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

      1. clear-num 64.0%

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

      2. inv-pow 64.0%

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

      3. fma-define 64.0%

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

      4. fma-undefine 64.0%

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

      5. *-commutative 64.0%

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

      6. fma-define 64.0%

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

      7. fma-define 64.0%

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

   6. Applied egg-rr 64.0%

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

      1. unpow-1 64.0%

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

   8. Simplified 64.0%

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

   9. Taylor expanded in F around -inf 99.9%

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

   if -1.3999999999999999 < F < 1.3999999999999999

   1. Initial program 99.4%

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

   3. Simplified 99.6%

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

      1. clear-num 99.5%

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

      2. inv-pow 99.5%

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

      3. fma-define 99.5%

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

      4. fma-undefine 99.5%

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

      5. *-commutative 99.5%

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

      6. fma-define 99.5%

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

      7. fma-define 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. unpow-1 99.5%

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

   8. Simplified 99.5%

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

   9. Taylor expanded in x around 0 99.6%

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

       1. +-commutative 99.6%

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

       2. unpow2 99.6%

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

       3. fma-undefine 99.6%

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

   11. Simplified 99.6%

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

   12. Taylor expanded in F around 0 98.8%

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

   if 1.3999999999999999 < F

   1. Initial program 55.6%

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

   3. Simplified 69.5%

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

      1. clear-num 69.5%

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

      2. inv-pow 69.5%

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

      3. fma-define 69.5%

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

      4. fma-undefine 69.5%

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

      5. *-commutative 69.5%

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

      6. fma-define 69.5%

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

      7. fma-define 69.5%

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

   6. Applied egg-rr 69.5%

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

      1. unpow-1 69.5%

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

   8. Simplified 69.5%

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

   9. Taylor expanded in F around inf 99.1%

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

Alternative 7: 91.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ t_1 := \frac{F}{\sin B} \cdot t\_0 - \frac{x}{B}\\ t_2 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3.1 \cdot 10^{-16}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_2\\ \mathbf{elif}\;F \leq -7.5 \cdot 10^{-149}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;F \leq 10^{-148}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + t\_0 \cdot \frac{F}{B}\\ \mathbf{elif}\;F \leq 9500:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5))
        (t_1 (- (* (/ F (sin B)) t_0) (/ x B)))
        (t_2 (/ x (tan B))))
   (if (<= F -3.1e-16)
     (- (/ -1.0 (sin B)) t_2)
     (if (<= F -7.5e-149)
       t_1
       (if (<= F 1e-148)
         (+ (/ -1.0 (/ (tan B) x)) (* t_0 (/ F B)))
         (if (<= F 9500.0) t_1 (- (/ 1.0 (sin B)) t_2)))))))

C

double code(double F, double B, double x) {
	double t_0 = pow(((2.0 + (F * F)) + (x * 2.0)), -0.5);
	double t_1 = ((F / sin(B)) * t_0) - (x / B);
	double t_2 = x / tan(B);
	double tmp;
	if (F <= -3.1e-16) {
		tmp = (-1.0 / sin(B)) - t_2;
	} else if (F <= -7.5e-149) {
		tmp = t_1;
	} else if (F <= 1e-148) {
		tmp = (-1.0 / (tan(B) / x)) + (t_0 * (F / B));
	} else if (F <= 9500.0) {
		tmp = t_1;
	} else {
		tmp = (1.0 / sin(B)) - t_2;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = ((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)
    t_1 = ((f / sin(b)) * t_0) - (x / b)
    t_2 = x / tan(b)
    if (f <= (-3.1d-16)) then
        tmp = ((-1.0d0) / sin(b)) - t_2
    else if (f <= (-7.5d-149)) then
        tmp = t_1
    else if (f <= 1d-148) then
        tmp = ((-1.0d0) / (tan(b) / x)) + (t_0 * (f / b))
    else if (f <= 9500.0d0) then
        tmp = t_1
    else
        tmp = (1.0d0 / sin(b)) - t_2
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5);
	double t_1 = ((F / Math.sin(B)) * t_0) - (x / B);
	double t_2 = x / Math.tan(B);
	double tmp;
	if (F <= -3.1e-16) {
		tmp = (-1.0 / Math.sin(B)) - t_2;
	} else if (F <= -7.5e-149) {
		tmp = t_1;
	} else if (F <= 1e-148) {
		tmp = (-1.0 / (Math.tan(B) / x)) + (t_0 * (F / B));
	} else if (F <= 9500.0) {
		tmp = t_1;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_2;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)
	t_1 = ((F / math.sin(B)) * t_0) - (x / B)
	t_2 = x / math.tan(B)
	tmp = 0
	if F <= -3.1e-16:
		tmp = (-1.0 / math.sin(B)) - t_2
	elif F <= -7.5e-149:
		tmp = t_1
	elif F <= 1e-148:
		tmp = (-1.0 / (math.tan(B) / x)) + (t_0 * (F / B))
	elif F <= 9500.0:
		tmp = t_1
	else:
		tmp = (1.0 / math.sin(B)) - t_2
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5
	t_1 = Float64(Float64(Float64(F / sin(B)) * t_0) - Float64(x / B))
	t_2 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -3.1e-16)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_2);
	elseif (F <= -7.5e-149)
		tmp = t_1;
	elseif (F <= 1e-148)
		tmp = Float64(Float64(-1.0 / Float64(tan(B) / x)) + Float64(t_0 * Float64(F / B)));
	elseif (F <= 9500.0)
		tmp = t_1;
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = ((2.0 + (F * F)) + (x * 2.0)) ^ -0.5;
	t_1 = ((F / sin(B)) * t_0) - (x / B);
	t_2 = x / tan(B);
	tmp = 0.0;
	if (F <= -3.1e-16)
		tmp = (-1.0 / sin(B)) - t_2;
	elseif (F <= -7.5e-149)
		tmp = t_1;
	elseif (F <= 1e-148)
		tmp = (-1.0 / (tan(B) / x)) + (t_0 * (F / B));
	elseif (F <= 9500.0)
		tmp = t_1;
	else
		tmp = (1.0 / sin(B)) - t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -3.1e-16], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision], If[LessEqual[F, -7.5e-149], t$95$1, If[LessEqual[F, 1e-148], N[(N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(F / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 9500.0], t$95$1, N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.1000000000000001e-16

   1. Initial program 46.2%

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

   3. Simplified 64.5%

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

      1. clear-num 64.6%

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

      2. inv-pow 64.6%

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

      3. fma-define 64.6%

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

      4. fma-undefine 64.6%

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

      5. *-commutative 64.6%

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

      6. fma-define 64.6%

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

      7. fma-define 64.6%

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

   6. Applied egg-rr 64.6%

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

      1. unpow-1 64.6%

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

   8. Simplified 64.6%

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

   9. Taylor expanded in F around -inf 99.9%

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

   if -3.1000000000000001e-16 < F < -7.49999999999999995e-149 or 9.99999999999999936e-149 < F < 9500

   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. Taylor expanded in B around 0 86.0%

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

      1. associate-*r/ 86.0%

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

      2. neg-mul-1 86.0%

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

   5. Simplified 86.0%

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

   if -7.49999999999999995e-149 < F < 9.99999999999999936e-149

   1. Initial program 99.5%

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

      1. div-inv 99.7%

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

      2. clear-num 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in B around 0 89.3%

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

   if 9500 < F

   1. Initial program 54.5%

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

   3. Simplified 68.8%

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

      1. clear-num 68.7%

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

      2. inv-pow 68.7%

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

      3. fma-define 68.7%

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

      4. fma-undefine 68.7%

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

      5. *-commutative 68.7%

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

      6. fma-define 68.7%

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

      7. fma-define 68.7%

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

   6. Applied egg-rr 68.7%

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

      1. unpow-1 68.7%

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

   8. Simplified 68.7%

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

   9. Taylor expanded in F around inf 99.9%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.1 \cdot 10^{-16}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -7.5 \cdot 10^{-149}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 10^{-148}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B}\\ \mathbf{elif}\;F \leq 9500:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3.1 \cdot 10^{-16}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_1\\ \mathbf{elif}\;F \leq -1.1 \cdot 10^{-150}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{-150}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{elif}\;F \leq 4100000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0
         (-
          (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5))
          (/ x B)))
        (t_1 (/ x (tan B))))
   (if (<= F -3.1e-16)
     (- (/ -1.0 (sin B)) t_1)
     (if (<= F -1.1e-150)
       t_0
       (if (<= F 4.5e-150)
         (/ (* x (cos B)) (- (sin B)))
         (if (<= F 4100000.0) t_0 (- (/ 1.0 (sin B)) t_1)))))))

C

double code(double F, double B, double x) {
	double t_0 = ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B);
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -3.1e-16) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= -1.1e-150) {
		tmp = t_0;
	} else if (F <= 4.5e-150) {
		tmp = (x * cos(B)) / -sin(B);
	} else if (F <= 4100000.0) {
		tmp = t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_1;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0))) - (x / b)
    t_1 = x / tan(b)
    if (f <= (-3.1d-16)) then
        tmp = ((-1.0d0) / sin(b)) - t_1
    else if (f <= (-1.1d-150)) then
        tmp = t_0
    else if (f <= 4.5d-150) then
        tmp = (x * cos(b)) / -sin(b)
    else if (f <= 4100000.0d0) then
        tmp = t_0
    else
        tmp = (1.0d0 / sin(b)) - t_1
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B);
	double t_1 = x / Math.tan(B);
	double tmp;
	if (F <= -3.1e-16) {
		tmp = (-1.0 / Math.sin(B)) - t_1;
	} else if (F <= -1.1e-150) {
		tmp = t_0;
	} else if (F <= 4.5e-150) {
		tmp = (x * Math.cos(B)) / -Math.sin(B);
	} else if (F <= 4100000.0) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B)
	t_1 = x / math.tan(B)
	tmp = 0
	if F <= -3.1e-16:
		tmp = (-1.0 / math.sin(B)) - t_1
	elif F <= -1.1e-150:
		tmp = t_0
	elif F <= 4.5e-150:
		tmp = (x * math.cos(B)) / -math.sin(B)
	elif F <= 4100000.0:
		tmp = t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)) - Float64(x / B))
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -3.1e-16)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= -1.1e-150)
		tmp = t_0;
	elseif (F <= 4.5e-150)
		tmp = Float64(Float64(x * cos(B)) / Float64(-sin(B)));
	elseif (F <= 4100000.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5)) - (x / B);
	t_1 = x / tan(B);
	tmp = 0.0;
	if (F <= -3.1e-16)
		tmp = (-1.0 / sin(B)) - t_1;
	elseif (F <= -1.1e-150)
		tmp = t_0;
	elseif (F <= 4.5e-150)
		tmp = (x * cos(B)) / -sin(B);
	elseif (F <= 4100000.0)
		tmp = t_0;
	else
		tmp = (1.0 / sin(B)) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -3.1e-16], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, -1.1e-150], t$95$0, If[LessEqual[F, 4.5e-150], N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 4100000.0], t$95$0, N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.1000000000000001e-16

   1. Initial program 46.2%

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

   3. Simplified 64.5%

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

      1. clear-num 64.6%

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

      2. inv-pow 64.6%

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

      3. fma-define 64.6%

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

      4. fma-undefine 64.6%

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

      5. *-commutative 64.6%

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

      6. fma-define 64.6%

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

      7. fma-define 64.6%

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

   6. Applied egg-rr 64.6%

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

      1. unpow-1 64.6%

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

   8. Simplified 64.6%

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

   9. Taylor expanded in F around -inf 99.9%

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

   if -3.1000000000000001e-16 < F < -1.1e-150 or 4.5000000000000002e-150 < F < 4.1e6

   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. Taylor expanded in B around 0 86.0%

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

      1. associate-*r/ 86.0%

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

      2. neg-mul-1 86.0%

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

   5. Simplified 86.0%

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

   if -1.1e-150 < F < 4.5000000000000002e-150

   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 F around -inf 33.6%

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

   4. Taylor expanded in x around inf 81.3%

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

   if 4.1e6 < F

   1. Initial program 54.5%

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

   3. Simplified 68.8%

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

      1. clear-num 68.7%

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

      2. inv-pow 68.7%

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

      3. fma-define 68.7%

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

      4. fma-undefine 68.7%

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

      5. *-commutative 68.7%

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

      6. fma-define 68.7%

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

      7. fma-define 68.7%

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

   6. Applied egg-rr 68.7%

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

      1. unpow-1 68.7%

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

   8. Simplified 68.7%

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

   9. Taylor expanded in F around inf 99.9%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.1 \cdot 10^{-16}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.1 \cdot 10^{-150}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{-150}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{elif}\;F \leq 4100000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3.1 \cdot 10^{-16}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq -5.2 \cdot 10^{-150}:\\ \;\;\;\;\frac{F}{B} \cdot \frac{1}{\sqrt{\mathsf{fma}\left(2, x, 2\right)}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.6 \cdot 10^{-25}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -3.1e-16)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F -5.2e-150)
       (- (* (/ F B) (/ 1.0 (sqrt (fma 2.0 x 2.0)))) (/ x B))
       (if (<= F 2.6e-25)
         (/ (* x (cos B)) (- (sin B)))
         (- (/ 1.0 (sin B)) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -3.1e-16) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= -5.2e-150) {
		tmp = ((F / B) * (1.0 / sqrt(fma(2.0, x, 2.0)))) - (x / B);
	} else if (F <= 2.6e-25) {
		tmp = (x * cos(B)) / -sin(B);
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -3.1e-16)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= -5.2e-150)
		tmp = Float64(Float64(Float64(F / B) * Float64(1.0 / sqrt(fma(2.0, x, 2.0)))) - Float64(x / B));
	elseif (F <= 2.6e-25)
		tmp = Float64(Float64(x * cos(B)) / Float64(-sin(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -3.1e-16], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -5.2e-150], N[(N[(N[(F / B), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(2.0 * x + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.6e-25], N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.1000000000000001e-16

   1. Initial program 46.2%

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

   3. Simplified 64.5%

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

      1. clear-num 64.6%

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

      2. inv-pow 64.6%

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

      3. fma-define 64.6%

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

      4. fma-undefine 64.6%

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

      5. *-commutative 64.6%

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

      6. fma-define 64.6%

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

      7. fma-define 64.6%

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

   6. Applied egg-rr 64.6%

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

      1. unpow-1 64.6%

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

   8. Simplified 64.6%

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

   9. Taylor expanded in F around -inf 99.9%

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

   if -3.1000000000000001e-16 < F < -5.1999999999999995e-150

   1. Initial program 99.3%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in B around 0 69.7%

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

   6. Taylor expanded in F around 0 69.6%

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

      1. mul-1-neg 69.6%

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

      2. +-commutative 69.6%

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

      3. unsub-neg 69.6%

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

      4. *-commutative 69.6%

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

   8. Simplified 69.6%

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

      1. sqrt-div 69.9%

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

      2. metadata-eval 69.9%

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

      3. +-commutative 69.9%

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

      4. fma-define 69.9%

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

   10. Applied egg-rr 69.9%

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

   if -5.1999999999999995e-150 < F < 2.6e-25

   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 F around -inf 33.5%

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

   4. Taylor expanded in x around inf 74.5%

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

   if 2.6e-25 < F

   1. Initial program 58.7%

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

   3. Simplified 71.6%

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

      1. clear-num 71.6%

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

      2. inv-pow 71.6%

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

      3. fma-define 71.6%

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

      4. fma-undefine 71.6%

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

      5. *-commutative 71.6%

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

      6. fma-define 71.6%

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

      7. fma-define 71.6%

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

   6. Applied egg-rr 71.6%

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

      1. unpow-1 71.6%

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

   8. Simplified 71.6%

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

   9. Taylor expanded in F around inf 95.1%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.1 \cdot 10^{-16}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -5.2 \cdot 10^{-150}:\\ \;\;\;\;\frac{F}{B} \cdot \frac{1}{\sqrt{\mathsf{fma}\left(2, x, 2\right)}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.6 \cdot 10^{-25}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 84.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3.1 \cdot 10^{-16}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq -9.2 \cdot 10^{-150}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 9.8 \cdot 10^{-30}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -3.1e-16)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F -9.2e-150)
       (- (/ (* F (sqrt 0.5)) B) (/ x B))
       (if (<= F 9.8e-30)
         (/ (* x (cos B)) (- (sin B)))
         (- (/ 1.0 (sin B)) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -3.1e-16) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= -9.2e-150) {
		tmp = ((F * sqrt(0.5)) / B) - (x / B);
	} else if (F <= 9.8e-30) {
		tmp = (x * cos(B)) / -sin(B);
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-3.1d-16)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= (-9.2d-150)) then
        tmp = ((f * sqrt(0.5d0)) / b) - (x / b)
    else if (f <= 9.8d-30) then
        tmp = (x * cos(b)) / -sin(b)
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -3.1e-16) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= -9.2e-150) {
		tmp = ((F * Math.sqrt(0.5)) / B) - (x / B);
	} else if (F <= 9.8e-30) {
		tmp = (x * Math.cos(B)) / -Math.sin(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -3.1e-16:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= -9.2e-150:
		tmp = ((F * math.sqrt(0.5)) / B) - (x / B)
	elif F <= 9.8e-30:
		tmp = (x * math.cos(B)) / -math.sin(B)
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -3.1e-16)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= -9.2e-150)
		tmp = Float64(Float64(Float64(F * sqrt(0.5)) / B) - Float64(x / B));
	elseif (F <= 9.8e-30)
		tmp = Float64(Float64(x * cos(B)) / Float64(-sin(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -3.1e-16)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= -9.2e-150)
		tmp = ((F * sqrt(0.5)) / B) - (x / B);
	elseif (F <= 9.8e-30)
		tmp = (x * cos(B)) / -sin(B);
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -3.1e-16], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -9.2e-150], N[(N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 9.8e-30], N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.1000000000000001e-16

   1. Initial program 46.2%

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

   3. Simplified 64.5%

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

      1. clear-num 64.6%

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

      2. inv-pow 64.6%

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

      3. fma-define 64.6%

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

      4. fma-undefine 64.6%

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

      5. *-commutative 64.6%

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

      6. fma-define 64.6%

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

      7. fma-define 64.6%

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

   6. Applied egg-rr 64.6%

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

      1. unpow-1 64.6%

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

   8. Simplified 64.6%

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

   9. Taylor expanded in F around -inf 99.9%

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

   if -3.1000000000000001e-16 < F < -9.20000000000000011e-150

   1. Initial program 99.3%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in B around 0 69.7%

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

   6. Taylor expanded in F around 0 69.6%

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

      1. mul-1-neg 69.6%

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

      2. +-commutative 69.6%

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

      3. unsub-neg 69.6%

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

      4. *-commutative 69.6%

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

   8. Simplified 69.6%

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

   9. Taylor expanded in x around 0 69.7%

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

   if -9.20000000000000011e-150 < F < 9.79999999999999942e-30

   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 F around -inf 33.5%

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

   4. Taylor expanded in x around inf 74.5%

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

   if 9.79999999999999942e-30 < F

   1. Initial program 58.7%

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

   3. Simplified 71.6%

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

      1. clear-num 71.6%

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

      2. inv-pow 71.6%

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

      3. fma-define 71.6%

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

      4. fma-undefine 71.6%

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

      5. *-commutative 71.6%

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

      6. fma-define 71.6%

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

      7. fma-define 71.6%

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

   6. Applied egg-rr 71.6%

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

      1. unpow-1 71.6%

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

   8. Simplified 71.6%

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

   9. Taylor expanded in F around inf 95.1%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.1 \cdot 10^{-16}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -9.2 \cdot 10^{-150}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 9.8 \cdot 10^{-30}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 77.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3.1 \cdot 10^{-16}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq -6 \cdot 10^{-150}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.1 \cdot 10^{-31}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -3.1e-16)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F -6e-150)
       (- (/ (* F (sqrt 0.5)) B) (/ x B))
       (if (<= F 1.1e-31) (/ (* x (cos B)) (- (sin B))) (- (/ 1.0 B) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -3.1e-16) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= -6e-150) {
		tmp = ((F * sqrt(0.5)) / B) - (x / B);
	} else if (F <= 1.1e-31) {
		tmp = (x * cos(B)) / -sin(B);
	} else {
		tmp = (1.0 / B) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-3.1d-16)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= (-6d-150)) then
        tmp = ((f * sqrt(0.5d0)) / b) - (x / b)
    else if (f <= 1.1d-31) then
        tmp = (x * cos(b)) / -sin(b)
    else
        tmp = (1.0d0 / b) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -3.1e-16) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= -6e-150) {
		tmp = ((F * Math.sqrt(0.5)) / B) - (x / B);
	} else if (F <= 1.1e-31) {
		tmp = (x * Math.cos(B)) / -Math.sin(B);
	} else {
		tmp = (1.0 / B) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -3.1e-16:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= -6e-150:
		tmp = ((F * math.sqrt(0.5)) / B) - (x / B)
	elif F <= 1.1e-31:
		tmp = (x * math.cos(B)) / -math.sin(B)
	else:
		tmp = (1.0 / B) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -3.1e-16)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= -6e-150)
		tmp = Float64(Float64(Float64(F * sqrt(0.5)) / B) - Float64(x / B));
	elseif (F <= 1.1e-31)
		tmp = Float64(Float64(x * cos(B)) / Float64(-sin(B)));
	else
		tmp = Float64(Float64(1.0 / B) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -3.1e-16)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= -6e-150)
		tmp = ((F * sqrt(0.5)) / B) - (x / B);
	elseif (F <= 1.1e-31)
		tmp = (x * cos(B)) / -sin(B);
	else
		tmp = (1.0 / B) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -3.1e-16], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -6e-150], N[(N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.1e-31], N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.1000000000000001e-16

   1. Initial program 46.2%

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

   3. Simplified 64.5%

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

      1. clear-num 64.6%

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

      2. inv-pow 64.6%

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

      3. fma-define 64.6%

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

      4. fma-undefine 64.6%

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

      5. *-commutative 64.6%

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

      6. fma-define 64.6%

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

      7. fma-define 64.6%

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

   6. Applied egg-rr 64.6%

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

      1. unpow-1 64.6%

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

   8. Simplified 64.6%

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

   9. Taylor expanded in F around -inf 99.9%

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

   if -3.1000000000000001e-16 < F < -6.0000000000000003e-150

   1. Initial program 99.3%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in B around 0 69.7%

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

   6. Taylor expanded in F around 0 69.6%

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

      1. mul-1-neg 69.6%

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

      2. +-commutative 69.6%

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

      3. unsub-neg 69.6%

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

      4. *-commutative 69.6%

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

   8. Simplified 69.6%

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

   9. Taylor expanded in x around 0 69.7%

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

   if -6.0000000000000003e-150 < F < 1.10000000000000005e-31

   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 F around -inf 33.5%

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

   4. Taylor expanded in x around inf 74.5%

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

   if 1.10000000000000005e-31 < F

   1. Initial program 58.7%

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

   3. Simplified 71.6%

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

      1. clear-num 71.6%

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

      2. inv-pow 71.6%

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

      3. fma-define 71.6%

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

      4. fma-undefine 71.6%

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

      5. *-commutative 71.6%

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

      6. fma-define 71.6%

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

      7. fma-define 71.6%

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

   6. Applied egg-rr 71.6%

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

      1. unpow-1 71.6%

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

   8. Simplified 71.6%

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

   9. Taylor expanded in F around inf 95.1%

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

   10. Taylor expanded in B around 0 71.6%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.1 \cdot 10^{-16}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -6 \cdot 10^{-150}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.1 \cdot 10^{-31}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 70.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.1 \cdot 10^{-16}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -6.2 \cdot 10^{-148}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 6.3 \cdot 10^{-28}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.1e-16)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F -6.2e-148)
     (- (/ (* F (sqrt 0.5)) B) (/ x B))
     (if (<= F 6.3e-28)
       (/ (* x (cos B)) (- (sin B)))
       (- (/ 1.0 B) (/ x (tan B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.1e-16) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -6.2e-148) {
		tmp = ((F * sqrt(0.5)) / B) - (x / B);
	} else if (F <= 6.3e-28) {
		tmp = (x * cos(B)) / -sin(B);
	} else {
		tmp = (1.0 / B) - (x / tan(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 <= (-3.1d-16)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-6.2d-148)) then
        tmp = ((f * sqrt(0.5d0)) / b) - (x / b)
    else if (f <= 6.3d-28) then
        tmp = (x * cos(b)) / -sin(b)
    else
        tmp = (1.0d0 / b) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.1e-16) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -6.2e-148) {
		tmp = ((F * Math.sqrt(0.5)) / B) - (x / B);
	} else if (F <= 6.3e-28) {
		tmp = (x * Math.cos(B)) / -Math.sin(B);
	} else {
		tmp = (1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.1e-16:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -6.2e-148:
		tmp = ((F * math.sqrt(0.5)) / B) - (x / B)
	elif F <= 6.3e-28:
		tmp = (x * math.cos(B)) / -math.sin(B)
	else:
		tmp = (1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.1e-16)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -6.2e-148)
		tmp = Float64(Float64(Float64(F * sqrt(0.5)) / B) - Float64(x / B));
	elseif (F <= 6.3e-28)
		tmp = Float64(Float64(x * cos(B)) / Float64(-sin(B)));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -3.1e-16)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -6.2e-148)
		tmp = ((F * sqrt(0.5)) / B) - (x / B);
	elseif (F <= 6.3e-28)
		tmp = (x * cos(B)) / -sin(B);
	else
		tmp = (1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.1e-16], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -6.2e-148], N[(N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 6.3e-28], N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.1000000000000001e-16

   1. Initial program 46.2%

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

   3. Taylor expanded in F around -inf 99.8%

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

   4. Taylor expanded in B around 0 80.9%

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

      1. associate-*r/ 27.6%

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

      2. neg-mul-1 27.6%

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

   6. Simplified 80.9%

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

   if -3.1000000000000001e-16 < F < -6.2000000000000003e-148

   1. Initial program 99.3%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in B around 0 69.7%

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

   6. Taylor expanded in F around 0 69.6%

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

      1. mul-1-neg 69.6%

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

      2. +-commutative 69.6%

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

      3. unsub-neg 69.6%

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

      4. *-commutative 69.6%

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

   8. Simplified 69.6%

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

   9. Taylor expanded in x around 0 69.7%

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

   if -6.2000000000000003e-148 < F < 6.2999999999999997e-28

   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 F around -inf 33.5%

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

   4. Taylor expanded in x around inf 74.5%

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

   if 6.2999999999999997e-28 < F

   1. Initial program 58.7%

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

   3. Simplified 71.6%

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

      1. clear-num 71.6%

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

      2. inv-pow 71.6%

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

      3. fma-define 71.6%

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

      4. fma-undefine 71.6%

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

      5. *-commutative 71.6%

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

      6. fma-define 71.6%

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

      7. fma-define 71.6%

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

   6. Applied egg-rr 71.6%

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

      1. unpow-1 71.6%

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

   8. Simplified 71.6%

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

   9. Taylor expanded in F around inf 95.1%

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

   10. Taylor expanded in B around 0 71.6%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.1 \cdot 10^{-16}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -6.2 \cdot 10^{-148}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 6.3 \cdot 10^{-28}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 69.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.1 \cdot 10^{-16}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -7 \cdot 10^{-192}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 6.5 \cdot 10^{-35}:\\ \;\;\;\;x \cdot \frac{\cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.1e-16)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F -7e-192)
     (- (/ (* F (sqrt 0.5)) B) (/ x B))
     (if (<= F 6.5e-35)
       (* x (/ (cos B) (- (sin B))))
       (- (/ 1.0 B) (/ x (tan B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.1e-16) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -7e-192) {
		tmp = ((F * sqrt(0.5)) / B) - (x / B);
	} else if (F <= 6.5e-35) {
		tmp = x * (cos(B) / -sin(B));
	} else {
		tmp = (1.0 / B) - (x / tan(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 <= (-3.1d-16)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-7d-192)) then
        tmp = ((f * sqrt(0.5d0)) / b) - (x / b)
    else if (f <= 6.5d-35) then
        tmp = x * (cos(b) / -sin(b))
    else
        tmp = (1.0d0 / b) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.1e-16) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -7e-192) {
		tmp = ((F * Math.sqrt(0.5)) / B) - (x / B);
	} else if (F <= 6.5e-35) {
		tmp = x * (Math.cos(B) / -Math.sin(B));
	} else {
		tmp = (1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.1e-16:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -7e-192:
		tmp = ((F * math.sqrt(0.5)) / B) - (x / B)
	elif F <= 6.5e-35:
		tmp = x * (math.cos(B) / -math.sin(B))
	else:
		tmp = (1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.1e-16)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -7e-192)
		tmp = Float64(Float64(Float64(F * sqrt(0.5)) / B) - Float64(x / B));
	elseif (F <= 6.5e-35)
		tmp = Float64(x * Float64(cos(B) / Float64(-sin(B))));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -3.1e-16)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -7e-192)
		tmp = ((F * sqrt(0.5)) / B) - (x / B);
	elseif (F <= 6.5e-35)
		tmp = x * (cos(B) / -sin(B));
	else
		tmp = (1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.1e-16], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -7e-192], N[(N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 6.5e-35], N[(x * N[(N[Cos[B], $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.1000000000000001e-16

   1. Initial program 46.2%

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

   3. Taylor expanded in F around -inf 99.8%

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

   4. Taylor expanded in B around 0 80.9%

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

      1. associate-*r/ 27.6%

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

      2. neg-mul-1 27.6%

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

   6. Simplified 80.9%

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

   if -3.1000000000000001e-16 < F < -7.00000000000000029e-192

   1. Initial program 99.4%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in B around 0 71.9%

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

   6. Taylor expanded in F around 0 71.8%

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

      1. mul-1-neg 71.8%

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

      2. +-commutative 71.8%

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

      3. unsub-neg 71.8%

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

      4. *-commutative 71.8%

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

   8. Simplified 71.8%

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

   9. Taylor expanded in x around 0 71.9%

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

   if -7.00000000000000029e-192 < F < 6.4999999999999999e-35

   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 F around -inf 29.3%

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

   4. Taylor expanded in x around inf 73.7%

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

      1. mul-1-neg 73.7%

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

      2. associate-/l* 73.6%

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

      3. distribute-lft-neg-in 73.6%

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

   6. Simplified 73.6%

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

   if 6.4999999999999999e-35 < F

   1. Initial program 58.7%

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

   3. Simplified 71.6%

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

      1. clear-num 71.6%

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

      2. inv-pow 71.6%

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

      3. fma-define 71.6%

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

      4. fma-undefine 71.6%

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

      5. *-commutative 71.6%

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

      6. fma-define 71.6%

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

      7. fma-define 71.6%

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

   6. Applied egg-rr 71.6%

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

      1. unpow-1 71.6%

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

   8. Simplified 71.6%

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

   9. Taylor expanded in F around inf 95.1%

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

   10. Taylor expanded in B around 0 71.6%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.1 \cdot 10^{-16}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -7 \cdot 10^{-192}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 6.5 \cdot 10^{-35}:\\ \;\;\;\;x \cdot \frac{\cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 63.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.1 \cdot 10^{-16}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.05 \cdot 10^{-217}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.1e-16)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 1.05e-217)
     (- (/ (* F (sqrt 0.5)) B) (/ x B))
     (- (/ 1.0 B) (/ x (tan B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.1e-16) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 1.05e-217) {
		tmp = ((F * sqrt(0.5)) / B) - (x / B);
	} else {
		tmp = (1.0 / B) - (x / tan(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 <= (-3.1d-16)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 1.05d-217) then
        tmp = ((f * sqrt(0.5d0)) / b) - (x / b)
    else
        tmp = (1.0d0 / b) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.1e-16) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 1.05e-217) {
		tmp = ((F * Math.sqrt(0.5)) / B) - (x / B);
	} else {
		tmp = (1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.1e-16:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 1.05e-217:
		tmp = ((F * math.sqrt(0.5)) / B) - (x / B)
	else:
		tmp = (1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.1e-16)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 1.05e-217)
		tmp = Float64(Float64(Float64(F * sqrt(0.5)) / B) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -3.1e-16)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 1.05e-217)
		tmp = ((F * sqrt(0.5)) / B) - (x / B);
	else
		tmp = (1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.1e-16], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.05e-217], N[(N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.1000000000000001e-16

   1. Initial program 46.2%

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

   3. Taylor expanded in F around -inf 99.8%

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

   4. Taylor expanded in B around 0 80.9%

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

      1. associate-*r/ 27.6%

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

      2. neg-mul-1 27.6%

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

   6. Simplified 80.9%

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

   if -3.1000000000000001e-16 < F < 1.05e-217

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in B around 0 63.1%

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

   6. Taylor expanded in F around 0 63.1%

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

      1. mul-1-neg 63.1%

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

      2. +-commutative 63.1%

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

      3. unsub-neg 63.1%

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

      4. *-commutative 63.1%

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

   8. Simplified 63.1%

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

   9. Taylor expanded in x around 0 63.2%

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

   if 1.05e-217 < F

   1. Initial program 70.1%

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

   3. Simplified 79.5%

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

      1. clear-num 79.5%

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

      2. inv-pow 79.5%

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

      3. fma-define 79.5%

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

      4. fma-undefine 79.5%

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

      5. *-commutative 79.5%

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

      6. fma-define 79.5%

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

      7. fma-define 79.5%

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

   6. Applied egg-rr 79.5%

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

      1. unpow-1 79.5%

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

   8. Simplified 79.5%

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

   9. Taylor expanded in F around inf 75.0%

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

   10. Taylor expanded in B around 0 64.5%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.1 \cdot 10^{-16}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.05 \cdot 10^{-217}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 63.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.1 \cdot 10^{-16}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{-218}:\\ \;\;\;\;\sqrt{0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.1e-16)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 4.5e-218)
     (- (* (sqrt 0.5) (/ F B)) (/ x B))
     (- (/ 1.0 B) (/ x (tan B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.1e-16) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 4.5e-218) {
		tmp = (sqrt(0.5) * (F / B)) - (x / B);
	} else {
		tmp = (1.0 / B) - (x / tan(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 <= (-3.1d-16)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 4.5d-218) then
        tmp = (sqrt(0.5d0) * (f / b)) - (x / b)
    else
        tmp = (1.0d0 / b) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.1e-16) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 4.5e-218) {
		tmp = (Math.sqrt(0.5) * (F / B)) - (x / B);
	} else {
		tmp = (1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.1e-16:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 4.5e-218:
		tmp = (math.sqrt(0.5) * (F / B)) - (x / B)
	else:
		tmp = (1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.1e-16)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 4.5e-218)
		tmp = Float64(Float64(sqrt(0.5) * Float64(F / B)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -3.1e-16)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 4.5e-218)
		tmp = (sqrt(0.5) * (F / B)) - (x / B);
	else
		tmp = (1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.1e-16], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.5e-218], N[(N[(N[Sqrt[0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.1000000000000001e-16

   1. Initial program 46.2%

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

   3. Taylor expanded in F around -inf 99.8%

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

   4. Taylor expanded in B around 0 80.9%

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

      1. associate-*r/ 27.6%

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

      2. neg-mul-1 27.6%

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

   6. Simplified 80.9%

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

   if -3.1000000000000001e-16 < F < 4.49999999999999977e-218

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in B around 0 63.1%

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

   6. Taylor expanded in F around 0 63.1%

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

      1. mul-1-neg 63.1%

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

      2. +-commutative 63.1%

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

      3. unsub-neg 63.1%

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

      4. *-commutative 63.1%

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

   8. Simplified 63.1%

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

   9. Taylor expanded in x around 0 63.1%

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

   if 4.49999999999999977e-218 < F

   1. Initial program 70.1%

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

   3. Simplified 79.5%

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

      1. clear-num 79.5%

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

      2. inv-pow 79.5%

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

      3. fma-define 79.5%

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

      4. fma-undefine 79.5%

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

      5. *-commutative 79.5%

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

      6. fma-define 79.5%

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

      7. fma-define 79.5%

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

   6. Applied egg-rr 79.5%

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

      1. unpow-1 79.5%

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

   8. Simplified 79.5%

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

   9. Taylor expanded in F around inf 75.0%

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

   10. Taylor expanded in B around 0 64.5%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.1 \cdot 10^{-16}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{-218}:\\ \;\;\;\;\sqrt{0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 61.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.029:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.029) (- (/ -1.0 (sin B)) (/ x B)) (- (/ 1.0 B) (/ x (tan B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.029) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else {
		tmp = (1.0 / B) - (x / tan(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 <= (-0.029d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else
        tmp = (1.0d0 / b) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.029) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else {
		tmp = (1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -0.029:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	else:
		tmp = (1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.029)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -0.029)
		tmp = (-1.0 / sin(B)) - (x / B);
	else
		tmp = (1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.029], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -0.0290000000000000015

   1. Initial program 45.3%

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

   3. Taylor expanded in F around -inf 99.8%

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

   4. Taylor expanded in B around 0 82.2%

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

      1. associate-*r/ 27.9%

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

      2. neg-mul-1 27.9%

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

   6. Simplified 82.2%

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

   if -0.0290000000000000015 < F

   1. Initial program 81.5%

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

   3. Simplified 87.3%

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

      1. clear-num 87.3%

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

      2. inv-pow 87.3%

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

      3. fma-define 87.3%

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

      4. fma-undefine 87.3%

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

      5. *-commutative 87.3%

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

      6. fma-define 87.3%

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

      7. fma-define 87.3%

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

   6. Applied egg-rr 87.3%

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

      1. unpow-1 87.3%

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

   8. Simplified 87.3%

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

   9. Taylor expanded in F around inf 59.1%

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

   10. Taylor expanded in B around 0 55.8%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.029:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 55.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.2:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.2) (/ (- -1.0 x) B) (- (/ 1.0 B) (/ x (tan B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.2) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = (1.0 / B) - (x / tan(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 <= (-1.2d0)) then
        tmp = ((-1.0d0) - x) / b
    else
        tmp = (1.0d0 / b) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.2) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = (1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.2:
		tmp = (-1.0 - x) / B
	else:
		tmp = (1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.2)
		tmp = Float64(Float64(-1.0 - x) / B);
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.2)
		tmp = (-1.0 - x) / B;
	else
		tmp = (1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.2], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -1.19999999999999996

   1. Initial program 45.3%

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

   3. Taylor expanded in F around -inf 99.8%

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

   4. Taylor expanded in B around 0 60.8%

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

      1. associate-*r/ 60.8%

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

      2. distribute-lft-in 60.8%

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

      3. metadata-eval 60.8%

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

      4. neg-mul-1 60.8%

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

      5. unsub-neg 60.8%

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

   6. Simplified 60.8%

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

   if -1.19999999999999996 < F

   1. Initial program 81.5%

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

   3. Simplified 87.3%

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

      1. clear-num 87.3%

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

      2. inv-pow 87.3%

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

      3. fma-define 87.3%

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

      4. fma-undefine 87.3%

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

      5. *-commutative 87.3%

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

      6. fma-define 87.3%

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

      7. fma-define 87.3%

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

   6. Applied egg-rr 87.3%

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

      1. unpow-1 87.3%

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

   8. Simplified 87.3%

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

   9. Taylor expanded in F around inf 59.1%

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

   10. Taylor expanded in B around 0 55.8%

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

Alternative 18: 43.8% accurate, 21.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.1 \cdot 10^{-45}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.1 \cdot 10^{-29}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.1e-45)
   (/ (- -1.0 x) B)
   (if (<= F 1.1e-29) (/ x (- B)) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.1e-45) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.1e-29) {
		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 <= (-1.1d-45)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 1.1d-29) 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 <= -1.1e-45) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.1e-29) {
		tmp = x / -B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.1e-45:
		tmp = (-1.0 - x) / B
	elif F <= 1.1e-29:
		tmp = x / -B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.1e-45)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 1.1e-29)
		tmp = Float64(x / Float64(-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 <= -1.1e-45)
		tmp = (-1.0 - x) / B;
	elseif (F <= 1.1e-29)
		tmp = x / -B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.1e-45], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.1e-29], N[(x / (-B)), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.09999999999999997e-45

   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. Taylor expanded in F around -inf 89.4%

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

   4. Taylor expanded in B around 0 55.2%

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

      1. associate-*r/ 55.2%

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

      2. distribute-lft-in 55.2%

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

      3. metadata-eval 55.2%

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

      4. neg-mul-1 55.2%

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

      5. unsub-neg 55.2%

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

   6. Simplified 55.2%

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

   if -1.09999999999999997e-45 < F < 1.09999999999999995e-29

   1. Initial program 99.4%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in B around 0 54.7%

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

   6. Taylor expanded in F around 0 39.3%

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

      1. associate-*r/ 39.3%

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

      2. neg-mul-1 39.3%

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

   8. Simplified 39.3%

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

   if 1.09999999999999995e-29 < F

   1. Initial program 58.7%

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

   3. Simplified 71.6%

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

   5. Taylor expanded in B around 0 42.1%

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

   6. Taylor expanded in F around inf 56.7%

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

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.1 \cdot 10^{-45}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.1 \cdot 10^{-29}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 36.3% accurate, 32.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -4.1 \cdot 10^{-44}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4.1e-44) (/ (- -1.0 x) B) (/ x (- B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.1e-44) {
		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 <= (-4.1d-44)) 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 <= -4.1e-44) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = x / -B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -4.1e-44:
		tmp = (-1.0 - x) / B
	else:
		tmp = x / -B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -4.1e-44)
		tmp = Float64(Float64(-1.0 - x) / B);
	else
		tmp = Float64(x / Float64(-B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -4.1e-44)
		tmp = (-1.0 - x) / B;
	else
		tmp = x / -B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -4.1e-44], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[(x / (-B)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -4.09999999999999992e-44

   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. Taylor expanded in F around -inf 89.4%

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

   4. Taylor expanded in B around 0 55.2%

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

      1. associate-*r/ 55.2%

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

      2. distribute-lft-in 55.2%

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

      3. metadata-eval 55.2%

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

      4. neg-mul-1 55.2%

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

      5. unsub-neg 55.2%

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

   6. Simplified 55.2%

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

   if -4.09999999999999992e-44 < F

   1. Initial program 80.5%

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

   3. Simplified 86.6%

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

   5. Taylor expanded in B around 0 48.8%

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

   6. Taylor expanded in F around 0 34.5%

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

      1. associate-*r/ 34.5%

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

      2. neg-mul-1 34.5%

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

   8. Simplified 34.5%

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

4. Final simplification 40.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.1 \cdot 10^{-44}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 28.9% accurate, 81.0× 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(x / Float64(-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 73.0%

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

3. Simplified 81.9%

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

5. Taylor expanded in B around 0 46.4%

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

6. Taylor expanded in F around 0 31.9%

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

   1. associate-*r/ 31.9%

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

   2. neg-mul-1 31.9%

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

8. Simplified 31.9%

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

9. Final simplification 31.9%

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

Alternative 21: 2.8% accurate, 108.0× 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(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 73.0%

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

3. Simplified 81.9%

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

5. Taylor expanded in B around 0 46.4%

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

6. Taylor expanded in F around 0 31.9%

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

   1. associate-*r/ 31.9%

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

   2. neg-mul-1 31.9%

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

8. Simplified 31.9%

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

   1. add-sqr-sqrt 9.5%

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

   2. sqrt-unprod 8.8%

      \[\leadsto \frac{\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}}{B} \]

   3. sqr-neg 8.8%

      \[\leadsto \frac{\sqrt{\color{blue}{x \cdot x}}}{B} \]

   4. sqrt-unprod 1.4%

      \[\leadsto \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{B} \]

   5. add-sqr-sqrt 2.9%

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

   6. *-un-lft-identity 2.9%

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

10. Applied egg-rr 2.9%

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

    1. *-lft-identity 2.9%

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

12. Simplified 2.9%

    \[\leadsto \frac{\color{blue}{x}}{B} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024135 
(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))))))