VandenBroeck and Keller, Equation (23)

Percentage Accurate: 77.3% → 99.7%

Time: 19.0s

Alternatives: 27

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: 77.3% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -5 \cdot 10^{+47}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 120000000:\\ \;\;\;\;\frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -5e+47)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 120000000.0)
       (- (/ F (/ (sin B) (pow (fma 2.0 x (fma F F 2.0)) -0.5))) 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 <= -5e+47) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 120000000.0) {
		tmp = (F / (sin(B) / pow(fma(2.0, x, fma(F, F, 2.0)), -0.5))) - 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 <= -5e+47)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 120000000.0)
		tmp = Float64(Float64(F / Float64(sin(B) / (fma(2.0, x, fma(F, F, 2.0)) ^ -0.5))) - 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, -5e+47], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 120000000.0], N[(N[(F / N[(N[Sin[B], $MachinePrecision] / N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $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 < -5.00000000000000022e47

   1. Initial program 58.9%

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

   3. Simplified 73.0%

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

   5. Taylor expanded in F around -inf 99.9%

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

   if -5.00000000000000022e47 < F < 1.2e8

   1. Initial program 98.7%

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

   3. Simplified 99.7%

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

      1. clear-num 99.7%

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

      2. un-div-inv 99.7%

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

      3. fma-define 99.7%

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

      4. fma-undefine 99.7%

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

      5. *-commutative 99.7%

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

      6. fma-define 99.7%

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

      7. fma-define 99.7%

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

   6. Applied egg-rr 99.7%

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

   if 1.2e8 < F

   1. Initial program 68.2%

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

   3. Simplified 82.6%

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

   5. Taylor expanded in F around inf 99.8%

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

Alternative 2: 99.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2.55 \cdot 10^{+47}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 120000000:\\ \;\;\;\;F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -2.55e+47)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 120000000.0)
       (- (* 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 <= -2.55e+47) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 120000000.0) {
		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 <= -2.55e+47)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 120000000.0)
		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, -2.55e+47], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 120000000.0], 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 < -2.5500000000000001e47

   1. Initial program 58.9%

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

   3. Simplified 73.0%

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

   5. Taylor expanded in F around -inf 99.9%

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

   if -2.5500000000000001e47 < F < 1.2e8

   1. Initial program 98.7%

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

   3. Simplified 99.7%

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

   if 1.2e8 < F

   1. Initial program 68.2%

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

   3. Simplified 82.6%

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

   5. Taylor expanded in F around inf 99.8%

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

Alternative 3: 99.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1 \cdot 10^{+44}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 60000000:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1e+44)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 60000000.0)
       (- (/ (/ 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 <= -1e+44) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 60000000.0) {
		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 <= -1e+44)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 60000000.0)
		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, -1e+44], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 60000000.0], 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 < -1.0000000000000001e44

   1. Initial program 57.9%

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

   3. Simplified 73.5%

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

   5. Taylor expanded in F around -inf 99.9%

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

   if -1.0000000000000001e44 < F < 6e7

   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 99.7%

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

      1. clear-num 99.7%

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

      2. un-div-inv 99.7%

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

      3. fma-define 99.7%

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

      4. fma-undefine 99.7%

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

      5. *-commutative 99.7%

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

      6. fma-define 99.7%

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

      7. fma-define 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. div-inv 99.7%

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

      2. div-inv 99.6%

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

      3. fma-define 99.6%

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

      4. fma-undefine 99.6%

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

      5. +-commutative 99.6%

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

      6. pow-flip 99.7%

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

      7. metadata-eval 99.7%

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

      8. pow1/2 99.7%

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

      9. +-commutative 99.7%

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

      10. fma-undefine 99.7%

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

      11. fma-define 99.7%

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

   8. Applied egg-rr 99.7%

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

      1. associate-*r/ 99.7%

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

      2. *-rgt-identity 99.7%

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

      3. associate-/r* 99.6%

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

   10. Simplified 99.6%

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

   if 6e7 < F

   1. Initial program 68.2%

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

   3. Simplified 82.6%

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

   5. Taylor expanded in F around inf 99.8%

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

Alternative 4: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -7 \cdot 10^{+43}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 125000000:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -7e+43)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 125000000.0)
       (+
        (/ -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 <= -7e+43) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 125000000.0) {
		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 <= (-7d+43)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 125000000.0d0) 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 <= -7e+43) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 125000000.0) {
		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 <= -7e+43:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 125000000.0:
		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 <= -7e+43)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 125000000.0)
		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 <= -7e+43)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 125000000.0)
		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, -7e+43], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 125000000.0], 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 < -7.0000000000000002e43

   1. Initial program 57.9%

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

   3. Simplified 73.5%

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

   5. Taylor expanded in F around -inf 99.9%

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

   if -7.0000000000000002e43 < F < 1.25e8

   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. neg-mul-1 99.7%

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

      3. clear-num 99.6%

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

      4. un-div-inv 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. metadata-eval 99.6%

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

      2. metadata-eval 99.6%

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

   6. Applied egg-rr 99.6%

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

   if 1.25e8 < F

   1. Initial program 68.2%

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

   3. Simplified 82.6%

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

   5. Taylor expanded in F around inf 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7 \cdot 10^{+43}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 125000000:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.35 \cdot 10^{+37}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.5 \cdot 10^{-104}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B}\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \cos B \cdot \frac{x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.35e+37)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 1.5e-104)
     (+
      (/ -1.0 (/ (tan B) x))
      (* (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5) (/ F B)))
     (if (<= F 6.2e-6)
       (- (* (/ F (sin B)) (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))) (/ x B))
       (- (/ 1.0 (sin B)) (* (cos B) (/ x (sin B))))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.35e+37) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 1.5e-104) {
		tmp = (-1.0 / (tan(B) / x)) + (pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B));
	} else if (F <= 6.2e-6) {
		tmp = ((F / sin(B)) * sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B);
	} else {
		tmp = (1.0 / sin(B)) - (cos(B) * (x / sin(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 <= (-3.35d+37)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 1.5d-104) then
        tmp = ((-1.0d0) / (tan(b) / x)) + ((((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)) * (f / b))
    else if (f <= 6.2d-6) then
        tmp = ((f / sin(b)) * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))) - (x / b)
    else
        tmp = (1.0d0 / sin(b)) - (cos(b) * (x / sin(b)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.35e+37) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 1.5e-104) {
		tmp = (-1.0 / (Math.tan(B) / x)) + (Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B));
	} else if (F <= 6.2e-6) {
		tmp = ((F / Math.sin(B)) * Math.sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B);
	} else {
		tmp = (1.0 / Math.sin(B)) - (Math.cos(B) * (x / Math.sin(B)));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.35e+37:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 1.5e-104:
		tmp = (-1.0 / (math.tan(B) / x)) + (math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B))
	elif F <= 6.2e-6:
		tmp = ((F / math.sin(B)) * math.sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B)
	else:
		tmp = (1.0 / math.sin(B)) - (math.cos(B) * (x / math.sin(B)))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.35e+37)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 1.5e-104)
		tmp = Float64(Float64(-1.0 / Float64(tan(B) / x)) + Float64((Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5) * Float64(F / B)));
	elseif (F <= 6.2e-6)
		tmp = Float64(Float64(Float64(F / sin(B)) * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(cos(B) * Float64(x / sin(B))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -3.35e+37)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 1.5e-104)
		tmp = (-1.0 / (tan(B) / x)) + ((((2.0 + (F * F)) + (x * 2.0)) ^ -0.5) * (F / B));
	elseif (F <= 6.2e-6)
		tmp = ((F / sin(B)) * sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B);
	else
		tmp = (1.0 / sin(B)) - (cos(B) * (x / sin(B)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.35e+37], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.5e-104], N[(N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 6.2e-6], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[B], $MachinePrecision] * N[(x / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.34999999999999984e37

   1. Initial program 58.6%

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

   3. Simplified 73.9%

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

   5. Taylor expanded in F around -inf 99.9%

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

   if -3.34999999999999984e37 < F < 1.5000000000000001e-104

   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. neg-mul-1 99.7%

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

      3. clear-num 99.6%

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

      4. un-div-inv 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. metadata-eval 99.6%

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

      2. metadata-eval 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in B around 0 89.9%

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

   if 1.5000000000000001e-104 < F < 6.1999999999999999e-6

   1. Initial program 99.7%

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

   3. Taylor expanded in B around 0 98.6%

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

      1. associate-*r/ 98.6%

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

      2. neg-mul-1 98.6%

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

   5. Simplified 98.6%

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

   6. Taylor expanded in F around 0 98.6%

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

      1. +-commutative 98.6%

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

      2. neg-mul-1 98.6%

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

      3. unsub-neg 98.6%

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

      4. *-commutative 98.6%

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

      5. *-commutative 98.6%

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

   8. Simplified 98.6%

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

   if 6.1999999999999999e-6 < 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 83.7%

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

   5. Taylor expanded in F around inf 98.0%

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

      1. associate-*l/ 98.1%

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

      2. *-commutative 98.1%

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

   7. Simplified 98.1%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.35 \cdot 10^{+37}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.5 \cdot 10^{-104}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B}\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \cos B \cdot \frac{x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1.45:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{2}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

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 1.45)
       (- (/ F (* (sin B) (sqrt 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 <= -1.4) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.45) {
		tmp = (F / (sin(B) * sqrt(2.0))) - 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 <= 1.45d0) then
        tmp = (f / (sin(b) * sqrt(2.0d0))) - 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 <= 1.45) {
		tmp = (F / (Math.sin(B) * Math.sqrt(2.0))) - 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 <= 1.45:
		tmp = (F / (math.sin(B) * math.sqrt(2.0))) - 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 <= 1.45)
		tmp = Float64(Float64(F / Float64(sin(B) * sqrt(2.0))) - 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 <= 1.45)
		tmp = (F / (sin(B) * sqrt(2.0))) - 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, 1.45], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[2.0], $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 61.6%

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

   3. Simplified 75.8%

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

   5. Taylor expanded in F around -inf 99.6%

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

   if -1.3999999999999999 < F < 1.44999999999999996

   1. Initial program 99.5%

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

   3. Simplified 99.7%

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

      1. clear-num 99.7%

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

      2. un-div-inv 99.7%

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

      3. fma-define 99.7%

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

      4. fma-undefine 99.7%

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

      5. *-commutative 99.7%

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

      6. fma-define 99.7%

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

      7. fma-define 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in F around 0 99.2%

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

   8. Taylor expanded in x around 0 99.2%

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

   if 1.44999999999999996 < F

   1. Initial program 69.0%

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

   3. Simplified 83.1%

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

   5. 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: 91.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3.35 \cdot 10^{+37}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1.58 \cdot 10^{-108}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B}\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -3.35e+37)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.58e-108)
       (+
        (/ -1.0 (/ (tan B) x))
        (* (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5) (/ F B)))
       (if (<= F 6.2e-6)
         (- (* (/ F (sin B)) (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))) (/ x 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 <= -3.35e+37) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.58e-108) {
		tmp = (-1.0 / (tan(B) / x)) + (pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B));
	} else if (F <= 6.2e-6) {
		tmp = ((F / sin(B)) * sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / 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 <= (-3.35d+37)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.58d-108) then
        tmp = ((-1.0d0) / (tan(b) / x)) + ((((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)) * (f / b))
    else if (f <= 6.2d-6) then
        tmp = ((f / sin(b)) * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))) - (x / 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 <= -3.35e+37) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.58e-108) {
		tmp = (-1.0 / (Math.tan(B) / x)) + (Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B));
	} else if (F <= 6.2e-6) {
		tmp = ((F / Math.sin(B)) * Math.sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / 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 <= -3.35e+37:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.58e-108:
		tmp = (-1.0 / (math.tan(B) / x)) + (math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B))
	elif F <= 6.2e-6:
		tmp = ((F / math.sin(B)) * math.sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / 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 <= -3.35e+37)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.58e-108)
		tmp = Float64(Float64(-1.0 / Float64(tan(B) / x)) + Float64((Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5) * Float64(F / B)));
	elseif (F <= 6.2e-6)
		tmp = Float64(Float64(Float64(F / sin(B)) * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))) - Float64(x / 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 <= -3.35e+37)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.58e-108)
		tmp = (-1.0 / (tan(B) / x)) + ((((2.0 + (F * F)) + (x * 2.0)) ^ -0.5) * (F / B));
	elseif (F <= 6.2e-6)
		tmp = ((F / sin(B)) * sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / 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, -3.35e+37], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.58e-108], N[(N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 6.2e-6], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(x / 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 < -3.34999999999999984e37

   1. Initial program 58.6%

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

   3. Simplified 73.9%

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

   5. Taylor expanded in F around -inf 99.9%

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

   if -3.34999999999999984e37 < F < 1.5799999999999999e-108

   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. neg-mul-1 99.7%

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

      3. clear-num 99.6%

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

      4. un-div-inv 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. metadata-eval 99.6%

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

      2. metadata-eval 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in B around 0 89.9%

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

   if 1.5799999999999999e-108 < F < 6.1999999999999999e-6

   1. Initial program 99.7%

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

   3. Taylor expanded in B around 0 98.6%

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

      1. associate-*r/ 98.6%

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

      2. neg-mul-1 98.6%

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

   5. Simplified 98.6%

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

   6. Taylor expanded in F around 0 98.6%

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

      1. +-commutative 98.6%

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

      2. neg-mul-1 98.6%

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

      3. unsub-neg 98.6%

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

      4. *-commutative 98.6%

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

      5. *-commutative 98.6%

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

   8. Simplified 98.6%

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

   if 6.1999999999999999e-6 < 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 83.7%

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

   5. Taylor expanded in F around inf 98.1%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.35 \cdot 10^{+37}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.58 \cdot 10^{-108}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B}\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + x \cdot 2\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_1\\ \mathbf{elif}\;F \leq 6.6 \cdot 10^{-106}:\\ \;\;\;\;\frac{F}{B \cdot \sqrt{t\_0}} - t\_1\\ \mathbf{elif}\;F \leq 4.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{t\_0}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (+ 2.0 (* x 2.0))) (t_1 (/ x (tan B))))
   (if (<= F -3.5e-11)
     (- (/ -1.0 (sin B)) t_1)
     (if (<= F 6.6e-106)
       (- (/ F (* B (sqrt t_0))) t_1)
       (if (<= F 4.6e-6)
         (- (* (/ F (sin B)) (sqrt (/ 1.0 t_0))) (/ x B))
         (- (/ 1.0 (sin B)) t_1))))))

C

double code(double F, double B, double x) {
	double t_0 = 2.0 + (x * 2.0);
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -3.5e-11) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= 6.6e-106) {
		tmp = (F / (B * sqrt(t_0))) - t_1;
	} else if (F <= 4.6e-6) {
		tmp = ((F / sin(B)) * sqrt((1.0 / t_0))) - (x / B);
	} 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 = 2.0d0 + (x * 2.0d0)
    t_1 = x / tan(b)
    if (f <= (-3.5d-11)) then
        tmp = ((-1.0d0) / sin(b)) - t_1
    else if (f <= 6.6d-106) then
        tmp = (f / (b * sqrt(t_0))) - t_1
    else if (f <= 4.6d-6) then
        tmp = ((f / sin(b)) * sqrt((1.0d0 / t_0))) - (x / b)
    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 = 2.0 + (x * 2.0);
	double t_1 = x / Math.tan(B);
	double tmp;
	if (F <= -3.5e-11) {
		tmp = (-1.0 / Math.sin(B)) - t_1;
	} else if (F <= 6.6e-106) {
		tmp = (F / (B * Math.sqrt(t_0))) - t_1;
	} else if (F <= 4.6e-6) {
		tmp = ((F / Math.sin(B)) * Math.sqrt((1.0 / t_0))) - (x / B);
	} else {
		tmp = (1.0 / Math.sin(B)) - t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = 2.0 + (x * 2.0)
	t_1 = x / math.tan(B)
	tmp = 0
	if F <= -3.5e-11:
		tmp = (-1.0 / math.sin(B)) - t_1
	elif F <= 6.6e-106:
		tmp = (F / (B * math.sqrt(t_0))) - t_1
	elif F <= 4.6e-6:
		tmp = ((F / math.sin(B)) * math.sqrt((1.0 / t_0))) - (x / B)
	else:
		tmp = (1.0 / math.sin(B)) - t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(2.0 + Float64(x * 2.0))
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -3.5e-11)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= 6.6e-106)
		tmp = Float64(Float64(F / Float64(B * sqrt(t_0))) - t_1);
	elseif (F <= 4.6e-6)
		tmp = Float64(Float64(Float64(F / sin(B)) * sqrt(Float64(1.0 / t_0))) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = 2.0 + (x * 2.0);
	t_1 = x / tan(B);
	tmp = 0.0;
	if (F <= -3.5e-11)
		tmp = (-1.0 / sin(B)) - t_1;
	elseif (F <= 6.6e-106)
		tmp = (F / (B * sqrt(t_0))) - t_1;
	elseif (F <= 4.6e-6)
		tmp = ((F / sin(B)) * sqrt((1.0 / t_0))) - (x / B);
	else
		tmp = (1.0 / sin(B)) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -3.5e-11], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, 6.6e-106], N[(N[(F / N[(B * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, 4.6e-6], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.50000000000000019e-11

   1. Initial program 62.2%

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

   3. Simplified 76.2%

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

   5. Taylor expanded in F around -inf 98.3%

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

   if -3.50000000000000019e-11 < F < 6.60000000000000031e-106

   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 99.8%

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

      1. clear-num 99.7%

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

      2. un-div-inv 99.7%

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

      3. fma-define 99.7%

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

      4. fma-undefine 99.7%

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

      5. *-commutative 99.7%

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

      6. fma-define 99.7%

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

      7. fma-define 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in F around 0 99.7%

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

   8. Taylor expanded in B around 0 90.4%

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

   if 6.60000000000000031e-106 < F < 4.6e-6

   1. Initial program 99.7%

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

   3. Taylor expanded in B around 0 98.6%

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

      1. associate-*r/ 98.6%

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

      2. neg-mul-1 98.6%

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

   5. Simplified 98.6%

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

   6. Taylor expanded in F around 0 98.6%

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

      1. +-commutative 98.6%

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

      2. neg-mul-1 98.6%

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

      3. unsub-neg 98.6%

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

      4. *-commutative 98.6%

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

      5. *-commutative 98.6%

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

   8. Simplified 98.6%

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

   if 4.6e-6 < 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 83.7%

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

   5. Taylor expanded in F around inf 98.1%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 6.6 \cdot 10^{-106}:\\ \;\;\;\;\frac{F}{B \cdot \sqrt{2 + x \cdot 2}} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 4.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{2 + x \cdot 2}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_1\\ \mathbf{elif}\;F \leq 1.6 \cdot 10^{-104}:\\ \;\;\;\;\frac{F}{B \cdot t\_0} - t\_1\\ \mathbf{elif}\;F \leq 3.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{F}{\sin B \cdot t\_0} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (sqrt (+ 2.0 (* x 2.0)))) (t_1 (/ x (tan B))))
   (if (<= F -3.5e-11)
     (- (/ -1.0 (sin B)) t_1)
     (if (<= F 1.6e-104)
       (- (/ F (* B t_0)) t_1)
       (if (<= F 3.4e-6)
         (- (/ F (* (sin B) t_0)) (/ x B))
         (- (/ 1.0 (sin B)) t_1))))))

C

double code(double F, double B, double x) {
	double t_0 = sqrt((2.0 + (x * 2.0)));
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -3.5e-11) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= 1.6e-104) {
		tmp = (F / (B * t_0)) - t_1;
	} else if (F <= 3.4e-6) {
		tmp = (F / (sin(B) * t_0)) - (x / B);
	} 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 = sqrt((2.0d0 + (x * 2.0d0)))
    t_1 = x / tan(b)
    if (f <= (-3.5d-11)) then
        tmp = ((-1.0d0) / sin(b)) - t_1
    else if (f <= 1.6d-104) then
        tmp = (f / (b * t_0)) - t_1
    else if (f <= 3.4d-6) then
        tmp = (f / (sin(b) * t_0)) - (x / b)
    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 = Math.sqrt((2.0 + (x * 2.0)));
	double t_1 = x / Math.tan(B);
	double tmp;
	if (F <= -3.5e-11) {
		tmp = (-1.0 / Math.sin(B)) - t_1;
	} else if (F <= 1.6e-104) {
		tmp = (F / (B * t_0)) - t_1;
	} else if (F <= 3.4e-6) {
		tmp = (F / (Math.sin(B) * t_0)) - (x / B);
	} else {
		tmp = (1.0 / Math.sin(B)) - t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = math.sqrt((2.0 + (x * 2.0)))
	t_1 = x / math.tan(B)
	tmp = 0
	if F <= -3.5e-11:
		tmp = (-1.0 / math.sin(B)) - t_1
	elif F <= 1.6e-104:
		tmp = (F / (B * t_0)) - t_1
	elif F <= 3.4e-6:
		tmp = (F / (math.sin(B) * t_0)) - (x / B)
	else:
		tmp = (1.0 / math.sin(B)) - t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = sqrt(Float64(2.0 + Float64(x * 2.0)))
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -3.5e-11)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= 1.6e-104)
		tmp = Float64(Float64(F / Float64(B * t_0)) - t_1);
	elseif (F <= 3.4e-6)
		tmp = Float64(Float64(F / Float64(sin(B) * t_0)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = sqrt((2.0 + (x * 2.0)));
	t_1 = x / tan(B);
	tmp = 0.0;
	if (F <= -3.5e-11)
		tmp = (-1.0 / sin(B)) - t_1;
	elseif (F <= 1.6e-104)
		tmp = (F / (B * t_0)) - t_1;
	elseif (F <= 3.4e-6)
		tmp = (F / (sin(B) * t_0)) - (x / B);
	else
		tmp = (1.0 / sin(B)) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[Sqrt[N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -3.5e-11], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, 1.6e-104], N[(N[(F / N[(B * t$95$0), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, 3.4e-6], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.50000000000000019e-11

   1. Initial program 62.2%

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

   3. Simplified 76.2%

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

   5. Taylor expanded in F around -inf 98.3%

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

   if -3.50000000000000019e-11 < F < 1.59999999999999994e-104

   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 99.8%

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

      1. clear-num 99.7%

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

      2. un-div-inv 99.7%

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

      3. fma-define 99.7%

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

      4. fma-undefine 99.7%

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

      5. *-commutative 99.7%

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

      6. fma-define 99.7%

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

      7. fma-define 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in F around 0 99.7%

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

   8. Taylor expanded in B around 0 90.4%

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

   if 1.59999999999999994e-104 < F < 3.40000000000000006e-6

   1. Initial program 99.7%

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

   3. Simplified 99.4%

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

      1. clear-num 99.4%

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

      2. un-div-inv 99.5%

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

      3. fma-define 99.5%

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

      4. fma-undefine 99.5%

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

      5. *-commutative 99.5%

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

      6. fma-define 99.5%

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

      7. fma-define 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in F around 0 99.6%

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

   8. Taylor expanded in B around 0 98.4%

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

   if 3.40000000000000006e-6 < 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 83.7%

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

   5. Taylor expanded in F around inf 98.1%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.6 \cdot 10^{-104}:\\ \;\;\;\;\frac{F}{B \cdot \sqrt{2 + x \cdot 2}} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 3.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 91.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{F}{B \cdot \sqrt{2 + x \cdot 2}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -3.5e-11)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 6.2e-6)
       (- (/ F (* B (sqrt (+ 2.0 (* x 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 <= -3.5e-11) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 6.2e-6) {
		tmp = (F / (B * sqrt((2.0 + (x * 2.0))))) - 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 <= (-3.5d-11)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 6.2d-6) then
        tmp = (f / (b * sqrt((2.0d0 + (x * 2.0d0))))) - 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 <= -3.5e-11) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 6.2e-6) {
		tmp = (F / (B * Math.sqrt((2.0 + (x * 2.0))))) - 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 <= -3.5e-11:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 6.2e-6:
		tmp = (F / (B * math.sqrt((2.0 + (x * 2.0))))) - 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 <= -3.5e-11)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 6.2e-6)
		tmp = Float64(Float64(F / Float64(B * sqrt(Float64(2.0 + Float64(x * 2.0))))) - 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 <= -3.5e-11)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 6.2e-6)
		tmp = (F / (B * sqrt((2.0 + (x * 2.0))))) - 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, -3.5e-11], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 6.2e-6], N[(N[(F / N[(B * N[Sqrt[N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.50000000000000019e-11

   1. Initial program 62.2%

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

   3. Simplified 76.2%

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

   5. Taylor expanded in F around -inf 98.3%

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

   if -3.50000000000000019e-11 < F < 6.1999999999999999e-6

   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 99.7%

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

      1. clear-num 99.7%

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

      2. un-div-inv 99.7%

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

      3. fma-define 99.7%

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

      4. fma-undefine 99.7%

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

      5. *-commutative 99.7%

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

      6. fma-define 99.7%

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

      7. fma-define 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in F around 0 99.7%

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

   8. Taylor expanded in B around 0 86.7%

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

   if 6.1999999999999999e-6 < 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 83.7%

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

   5. Taylor expanded in F around inf 98.1%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{F}{B \cdot \sqrt{2 + x \cdot 2}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 84.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.55 \cdot 10^{-62}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{-120}:\\ \;\;\;\;\cos B \cdot \frac{x}{-\sin B}\\ \mathbf{elif}\;F \leq 5.8 \cdot 10^{-6}:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.55e-62)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 7.5e-120)
       (* (cos B) (/ x (- (sin B))))
       (if (<= F 5.8e-6)
         (- (* (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5) (/ F B)) (/ x B))
         (- (/ 1.0 (sin B)) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.55e-62) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 7.5e-120) {
		tmp = cos(B) * (x / -sin(B));
	} else if (F <= 5.8e-6) {
		tmp = (pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / 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 <= (-1.55d-62)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 7.5d-120) then
        tmp = cos(b) * (x / -sin(b))
    else if (f <= 5.8d-6) then
        tmp = ((((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)) * (f / b)) - (x / 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 <= -1.55e-62) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 7.5e-120) {
		tmp = Math.cos(B) * (x / -Math.sin(B));
	} else if (F <= 5.8e-6) {
		tmp = (Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / 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 <= -1.55e-62:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 7.5e-120:
		tmp = math.cos(B) * (x / -math.sin(B))
	elif F <= 5.8e-6:
		tmp = (math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / 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 <= -1.55e-62)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 7.5e-120)
		tmp = Float64(cos(B) * Float64(x / Float64(-sin(B))));
	elseif (F <= 5.8e-6)
		tmp = Float64(Float64((Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5) * Float64(F / B)) - Float64(x / 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 <= -1.55e-62)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 7.5e-120)
		tmp = cos(B) * (x / -sin(B));
	elseif (F <= 5.8e-6)
		tmp = ((((2.0 + (F * F)) + (x * 2.0)) ^ -0.5) * (F / B)) - (x / 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, -1.55e-62], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 7.5e-120], N[(N[Cos[B], $MachinePrecision] * N[(x / (-N[Sin[B], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.8e-6], N[(N[(N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / 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 < -1.55e-62

   1. Initial program 66.9%

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

   3. Simplified 79.1%

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

   5. Taylor expanded in F around -inf 90.3%

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

   if -1.55e-62 < F < 7.5000000000000004e-120

   1. Initial program 99.6%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in F around -inf 35.2%

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

   6. Taylor expanded in x around inf 84.0%

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

      1. *-commutative 84.0%

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

      2. associate-*r/ 84.1%

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

      3. neg-mul-1 84.1%

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

      4. distribute-rgt-neg-in 84.1%

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

      5. distribute-neg-frac2 84.1%

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

   8. Simplified 84.1%

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

   if 7.5000000000000004e-120 < F < 5.8000000000000004e-6

   1. Initial program 99.7%

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

   3. Taylor expanded in B around 0 98.6%

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

      1. associate-*r/ 98.6%

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

      2. neg-mul-1 98.6%

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

   5. Simplified 98.6%

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

   6. Taylor expanded in B around 0 67.9%

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

   if 5.8000000000000004e-6 < 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 83.7%

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

   5. Taylor expanded in F around inf 98.1%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.55 \cdot 10^{-62}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{-120}:\\ \;\;\;\;\cos B \cdot \frac{x}{-\sin B}\\ \mathbf{elif}\;F \leq 5.8 \cdot 10^{-6}:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 77.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.45 \cdot 10^{-62}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1.05 \cdot 10^{-103}:\\ \;\;\;\;\cos B \cdot \frac{x}{-\sin B}\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{-6}:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4.3 \cdot 10^{+253}:\\ \;\;\;\;F \cdot \frac{\frac{1}{F}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.45e-62)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.05e-103)
       (* (cos B) (/ x (- (sin B))))
       (if (<= F 6.2e-6)
         (- (* (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5) (/ F B)) (/ x B))
         (if (<= F 4.3e+253)
           (- (* F (/ (/ 1.0 F) (sin B))) (/ x 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 <= -1.45e-62) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.05e-103) {
		tmp = cos(B) * (x / -sin(B));
	} else if (F <= 6.2e-6) {
		tmp = (pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else if (F <= 4.3e+253) {
		tmp = (F * ((1.0 / F) / sin(B))) - (x / 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 <= (-1.45d-62)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.05d-103) then
        tmp = cos(b) * (x / -sin(b))
    else if (f <= 6.2d-6) then
        tmp = ((((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)) * (f / b)) - (x / b)
    else if (f <= 4.3d+253) then
        tmp = (f * ((1.0d0 / f) / sin(b))) - (x / 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 <= -1.45e-62) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.05e-103) {
		tmp = Math.cos(B) * (x / -Math.sin(B));
	} else if (F <= 6.2e-6) {
		tmp = (Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else if (F <= 4.3e+253) {
		tmp = (F * ((1.0 / F) / Math.sin(B))) - (x / 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 <= -1.45e-62:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.05e-103:
		tmp = math.cos(B) * (x / -math.sin(B))
	elif F <= 6.2e-6:
		tmp = (math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B)
	elif F <= 4.3e+253:
		tmp = (F * ((1.0 / F) / math.sin(B))) - (x / 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 <= -1.45e-62)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.05e-103)
		tmp = Float64(cos(B) * Float64(x / Float64(-sin(B))));
	elseif (F <= 6.2e-6)
		tmp = Float64(Float64((Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5) * Float64(F / B)) - Float64(x / B));
	elseif (F <= 4.3e+253)
		tmp = Float64(Float64(F * Float64(Float64(1.0 / F) / sin(B))) - Float64(x / 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 <= -1.45e-62)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.05e-103)
		tmp = cos(B) * (x / -sin(B));
	elseif (F <= 6.2e-6)
		tmp = ((((2.0 + (F * F)) + (x * 2.0)) ^ -0.5) * (F / B)) - (x / B);
	elseif (F <= 4.3e+253)
		tmp = (F * ((1.0 / F) / sin(B))) - (x / 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, -1.45e-62], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.05e-103], N[(N[Cos[B], $MachinePrecision] * N[(x / (-N[Sin[B], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 6.2e-6], N[(N[(N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.3e+253], N[(N[(F * N[(N[(1.0 / F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -1.44999999999999993e-62

   1. Initial program 66.9%

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

   3. Simplified 79.1%

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

   5. Taylor expanded in F around -inf 90.3%

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

   if -1.44999999999999993e-62 < F < 1.05000000000000002e-103

   1. Initial program 99.6%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in F around -inf 35.2%

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

   6. Taylor expanded in x around inf 84.0%

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

      1. *-commutative 84.0%

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

      2. associate-*r/ 84.1%

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

      3. neg-mul-1 84.1%

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

      4. distribute-rgt-neg-in 84.1%

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

      5. distribute-neg-frac2 84.1%

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

   8. Simplified 84.1%

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

   if 1.05000000000000002e-103 < F < 6.1999999999999999e-6

   1. Initial program 99.7%

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

   3. Taylor expanded in B around 0 98.6%

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

      1. associate-*r/ 98.6%

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

      2. neg-mul-1 98.6%

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

   5. Simplified 98.6%

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

   6. Taylor expanded in B around 0 67.9%

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

   if 6.1999999999999999e-6 < F < 4.2999999999999999e253

   1. Initial program 71.7%

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

   3. Simplified 82.5%

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

   5. Taylor expanded in F around inf 97.5%

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

   6. Taylor expanded in B around 0 76.3%

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

   if 4.2999999999999999e253 < F

   1. Initial program 64.4%

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

   3. Simplified 88.2%

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

   5. Taylor expanded in F around inf 99.7%

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

   6. Taylor expanded in B around 0 99.8%

      \[\leadsto \color{blue}{\frac{1}{B}} - \frac{x}{\tan B} \]
3. Recombined 5 regimes into one program.

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.45 \cdot 10^{-62}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.05 \cdot 10^{-103}:\\ \;\;\;\;\cos B \cdot \frac{x}{-\sin B}\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{-6}:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4.3 \cdot 10^{+253}:\\ \;\;\;\;F \cdot \frac{\frac{1}{F}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 70.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.35 \cdot 10^{+37}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4.8 \cdot 10^{-104}:\\ \;\;\;\;\cos B \cdot \frac{x}{-\sin B}\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{-8}:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 8.5 \cdot 10^{+252}:\\ \;\;\;\;F \cdot \frac{\frac{1}{F}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.35e+37)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 4.8e-104)
     (* (cos B) (/ x (- (sin B))))
     (if (<= F 5.2e-8)
       (- (* (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5) (/ F B)) (/ x B))
       (if (<= F 8.5e+252)
         (- (* F (/ (/ 1.0 F) (sin B))) (/ x B))
         (- (/ 1.0 B) (/ x (tan B))))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.35e+37) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 4.8e-104) {
		tmp = cos(B) * (x / -sin(B));
	} else if (F <= 5.2e-8) {
		tmp = (pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else if (F <= 8.5e+252) {
		tmp = (F * ((1.0 / F) / 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 <= (-3.35d+37)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 4.8d-104) then
        tmp = cos(b) * (x / -sin(b))
    else if (f <= 5.2d-8) then
        tmp = ((((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)) * (f / b)) - (x / b)
    else if (f <= 8.5d+252) then
        tmp = (f * ((1.0d0 / f) / 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 <= -3.35e+37) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 4.8e-104) {
		tmp = Math.cos(B) * (x / -Math.sin(B));
	} else if (F <= 5.2e-8) {
		tmp = (Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else if (F <= 8.5e+252) {
		tmp = (F * ((1.0 / F) / 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 <= -3.35e+37:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 4.8e-104:
		tmp = math.cos(B) * (x / -math.sin(B))
	elif F <= 5.2e-8:
		tmp = (math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B)
	elif F <= 8.5e+252:
		tmp = (F * ((1.0 / F) / 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 <= -3.35e+37)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 4.8e-104)
		tmp = Float64(cos(B) * Float64(x / Float64(-sin(B))));
	elseif (F <= 5.2e-8)
		tmp = Float64(Float64((Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5) * Float64(F / B)) - Float64(x / B));
	elseif (F <= 8.5e+252)
		tmp = Float64(Float64(F * Float64(Float64(1.0 / F) / 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 <= -3.35e+37)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 4.8e-104)
		tmp = cos(B) * (x / -sin(B));
	elseif (F <= 5.2e-8)
		tmp = ((((2.0 + (F * F)) + (x * 2.0)) ^ -0.5) * (F / B)) - (x / B);
	elseif (F <= 8.5e+252)
		tmp = (F * ((1.0 / F) / 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, -3.35e+37], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.8e-104], N[(N[Cos[B], $MachinePrecision] * N[(x / (-N[Sin[B], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.2e-8], N[(N[(N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 8.5e+252], N[(N[(F * N[(N[(1.0 / F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $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 5 regimes

2. if F < -3.34999999999999984e37

   1. Initial program 58.6%

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

   3. Simplified 73.9%

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

   5. Taylor expanded in F around -inf 99.9%

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

   6. Taylor expanded in B around 0 78.7%

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

   if -3.34999999999999984e37 < F < 4.8000000000000001e-104

   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 99.8%

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

   5. Taylor expanded in F around -inf 38.2%

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

   6. Taylor expanded in x around inf 77.6%

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

      1. *-commutative 77.6%

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

      2. associate-*r/ 77.7%

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

      3. neg-mul-1 77.7%

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

      4. distribute-rgt-neg-in 77.7%

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

      5. distribute-neg-frac2 77.7%

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

   8. Simplified 77.7%

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

   if 4.8000000000000001e-104 < F < 5.2000000000000002e-8

   1. Initial program 99.7%

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

   3. Taylor expanded in B around 0 98.6%

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

      1. associate-*r/ 98.6%

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

      2. neg-mul-1 98.6%

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

   5. Simplified 98.6%

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

   6. Taylor expanded in B around 0 67.9%

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

   if 5.2000000000000002e-8 < F < 8.50000000000000044e252

   1. Initial program 71.7%

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

   3. Simplified 82.5%

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

   5. Taylor expanded in F around inf 97.5%

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

   6. Taylor expanded in B around 0 76.3%

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

   if 8.50000000000000044e252 < F

   1. Initial program 64.4%

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

   3. Simplified 88.2%

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

   5. Taylor expanded in F around inf 99.7%

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

   6. Taylor expanded in B around 0 99.8%

      \[\leadsto \color{blue}{\frac{1}{B}} - \frac{x}{\tan B} \]
3. Recombined 5 regimes into one program.

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.35 \cdot 10^{+37}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4.8 \cdot 10^{-104}:\\ \;\;\;\;\cos B \cdot \frac{x}{-\sin B}\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{-8}:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 8.5 \cdot 10^{+252}:\\ \;\;\;\;F \cdot \frac{\frac{1}{F}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 70.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -8 \cdot 10^{+37}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3 \cdot 10^{-103}:\\ \;\;\;\;x \cdot \frac{\cos B}{-\sin B}\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{-6}:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3 \cdot 10^{+252}:\\ \;\;\;\;F \cdot \frac{\frac{1}{F}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -8e+37)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 3e-103)
     (* x (/ (cos B) (- (sin B))))
     (if (<= F 6.2e-6)
       (- (* (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5) (/ F B)) (/ x B))
       (if (<= F 3e+252)
         (- (* F (/ (/ 1.0 F) (sin B))) (/ x B))
         (- (/ 1.0 B) (/ x (tan B))))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -8e+37) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 3e-103) {
		tmp = x * (cos(B) / -sin(B));
	} else if (F <= 6.2e-6) {
		tmp = (pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else if (F <= 3e+252) {
		tmp = (F * ((1.0 / F) / 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 <= (-8d+37)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 3d-103) then
        tmp = x * (cos(b) / -sin(b))
    else if (f <= 6.2d-6) then
        tmp = ((((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)) * (f / b)) - (x / b)
    else if (f <= 3d+252) then
        tmp = (f * ((1.0d0 / f) / 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 <= -8e+37) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 3e-103) {
		tmp = x * (Math.cos(B) / -Math.sin(B));
	} else if (F <= 6.2e-6) {
		tmp = (Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else if (F <= 3e+252) {
		tmp = (F * ((1.0 / F) / 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 <= -8e+37:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 3e-103:
		tmp = x * (math.cos(B) / -math.sin(B))
	elif F <= 6.2e-6:
		tmp = (math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B)
	elif F <= 3e+252:
		tmp = (F * ((1.0 / F) / 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 <= -8e+37)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 3e-103)
		tmp = Float64(x * Float64(cos(B) / Float64(-sin(B))));
	elseif (F <= 6.2e-6)
		tmp = Float64(Float64((Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5) * Float64(F / B)) - Float64(x / B));
	elseif (F <= 3e+252)
		tmp = Float64(Float64(F * Float64(Float64(1.0 / F) / 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 <= -8e+37)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 3e-103)
		tmp = x * (cos(B) / -sin(B));
	elseif (F <= 6.2e-6)
		tmp = ((((2.0 + (F * F)) + (x * 2.0)) ^ -0.5) * (F / B)) - (x / B);
	elseif (F <= 3e+252)
		tmp = (F * ((1.0 / F) / 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, -8e+37], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3e-103], N[(x * N[(N[Cos[B], $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 6.2e-6], N[(N[(N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3e+252], N[(N[(F * N[(N[(1.0 / F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $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 5 regimes

2. if F < -7.99999999999999963e37

   1. Initial program 58.6%

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

   3. Simplified 73.9%

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

   5. Taylor expanded in F around -inf 99.9%

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

   6. Taylor expanded in B around 0 78.7%

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

   if -7.99999999999999963e37 < F < 3e-103

   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 99.8%

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

   5. Taylor expanded in F around -inf 38.2%

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

   6. Taylor expanded in x around inf 77.6%

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

      1. mul-1-neg 77.6%

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

      2. associate-/l* 77.7%

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

      3. distribute-rgt-neg-in 77.7%

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

      4. distribute-neg-frac2 77.7%

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

   8. Simplified 77.7%

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

   if 3e-103 < F < 6.1999999999999999e-6

   1. Initial program 99.7%

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

   3. Taylor expanded in B around 0 98.6%

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

      1. associate-*r/ 98.6%

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

      2. neg-mul-1 98.6%

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

   5. Simplified 98.6%

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

   6. Taylor expanded in B around 0 67.9%

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

   if 6.1999999999999999e-6 < F < 2.99999999999999989e252

   1. Initial program 71.7%

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

   3. Simplified 82.5%

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

   5. Taylor expanded in F around inf 97.5%

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

   6. Taylor expanded in B around 0 76.3%

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

   if 2.99999999999999989e252 < F

   1. Initial program 64.4%

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

   3. Simplified 88.2%

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

   5. Taylor expanded in F around inf 99.7%

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

   6. Taylor expanded in B around 0 99.8%

      \[\leadsto \color{blue}{\frac{1}{B}} - \frac{x}{\tan B} \]
3. Recombined 5 regimes into one program.

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -8 \cdot 10^{+37}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3 \cdot 10^{-103}:\\ \;\;\;\;x \cdot \frac{\cos B}{-\sin B}\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{-6}:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3 \cdot 10^{+252}:\\ \;\;\;\;F \cdot \frac{\frac{1}{F}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 62.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq 1.35 \cdot 10^{-272}:\\ \;\;\;\;\frac{-1}{B} - t\_0\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{-6}:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.08 \cdot 10^{+251}:\\ \;\;\;\;F \cdot \frac{\frac{1}{F}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F 1.35e-272)
     (- (/ -1.0 B) t_0)
     (if (<= F 6.2e-6)
       (- (* (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5) (/ F B)) (/ x B))
       (if (<= F 1.08e+251)
         (- (* F (/ (/ 1.0 F) (sin B))) (/ x 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 <= 1.35e-272) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= 6.2e-6) {
		tmp = (pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else if (F <= 1.08e+251) {
		tmp = (F * ((1.0 / F) / sin(B))) - (x / 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 <= 1.35d-272) then
        tmp = ((-1.0d0) / b) - t_0
    else if (f <= 6.2d-6) then
        tmp = ((((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)) * (f / b)) - (x / b)
    else if (f <= 1.08d+251) then
        tmp = (f * ((1.0d0 / f) / sin(b))) - (x / 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 <= 1.35e-272) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= 6.2e-6) {
		tmp = (Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else if (F <= 1.08e+251) {
		tmp = (F * ((1.0 / F) / Math.sin(B))) - (x / 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 <= 1.35e-272:
		tmp = (-1.0 / B) - t_0
	elif F <= 6.2e-6:
		tmp = (math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B)
	elif F <= 1.08e+251:
		tmp = (F * ((1.0 / F) / math.sin(B))) - (x / 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 <= 1.35e-272)
		tmp = Float64(Float64(-1.0 / B) - t_0);
	elseif (F <= 6.2e-6)
		tmp = Float64(Float64((Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5) * Float64(F / B)) - Float64(x / B));
	elseif (F <= 1.08e+251)
		tmp = Float64(Float64(F * Float64(Float64(1.0 / F) / sin(B))) - Float64(x / 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 <= 1.35e-272)
		tmp = (-1.0 / B) - t_0;
	elseif (F <= 6.2e-6)
		tmp = ((((2.0 + (F * F)) + (x * 2.0)) ^ -0.5) * (F / B)) - (x / B);
	elseif (F <= 1.08e+251)
		tmp = (F * ((1.0 / F) / sin(B))) - (x / 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, 1.35e-272], N[(N[(-1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 6.2e-6], N[(N[(N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.08e+251], N[(N[(F * N[(N[(1.0 / F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < 1.34999999999999996e-272

   1. Initial program 79.4%

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

   3. Simplified 87.0%

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

   5. Taylor expanded in F around -inf 71.7%

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

   6. Taylor expanded in B around 0 65.2%

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

   if 1.34999999999999996e-272 < F < 6.1999999999999999e-6

   1. Initial program 99.7%

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

   3. Taylor expanded in B around 0 82.5%

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

      1. associate-*r/ 82.5%

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

      2. neg-mul-1 82.5%

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

   5. Simplified 82.5%

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

   6. Taylor expanded in B around 0 70.0%

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

   if 6.1999999999999999e-6 < F < 1.08e251

   1. Initial program 71.7%

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

   3. Simplified 82.5%

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

   5. Taylor expanded in F around inf 97.5%

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

   6. Taylor expanded in B around 0 76.3%

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

   if 1.08e251 < F

   1. Initial program 64.4%

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

   3. Simplified 88.2%

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

   5. Taylor expanded in F around inf 99.7%

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

   6. Taylor expanded in B around 0 99.8%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.35 \cdot 10^{-272}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{-6}:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.08 \cdot 10^{+251}:\\ \;\;\;\;F \cdot \frac{\frac{1}{F}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 63.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq 6.8 \cdot 10^{-273}:\\ \;\;\;\;\frac{-1}{B} - t\_0\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{elif}\;F \leq 4.6 \cdot 10^{+253}:\\ \;\;\;\;F \cdot \frac{\frac{1}{F}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F 6.8e-273)
     (- (/ -1.0 B) t_0)
     (if (<= F 6.2e-6)
       (/ (- (* F (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))) x) B)
       (if (<= F 4.6e+253)
         (- (* F (/ (/ 1.0 F) (sin B))) (/ x 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 <= 6.8e-273) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= 6.2e-6) {
		tmp = ((F * sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B;
	} else if (F <= 4.6e+253) {
		tmp = (F * ((1.0 / F) / sin(B))) - (x / 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 <= 6.8d-273) then
        tmp = ((-1.0d0) / b) - t_0
    else if (f <= 6.2d-6) then
        tmp = ((f * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))) - x) / b
    else if (f <= 4.6d+253) then
        tmp = (f * ((1.0d0 / f) / sin(b))) - (x / 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 <= 6.8e-273) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= 6.2e-6) {
		tmp = ((F * Math.sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B;
	} else if (F <= 4.6e+253) {
		tmp = (F * ((1.0 / F) / Math.sin(B))) - (x / 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 <= 6.8e-273:
		tmp = (-1.0 / B) - t_0
	elif F <= 6.2e-6:
		tmp = ((F * math.sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B
	elif F <= 4.6e+253:
		tmp = (F * ((1.0 / F) / math.sin(B))) - (x / 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 <= 6.8e-273)
		tmp = Float64(Float64(-1.0 / B) - t_0);
	elseif (F <= 6.2e-6)
		tmp = Float64(Float64(Float64(F * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))) - x) / B);
	elseif (F <= 4.6e+253)
		tmp = Float64(Float64(F * Float64(Float64(1.0 / F) / sin(B))) - Float64(x / 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 <= 6.8e-273)
		tmp = (-1.0 / B) - t_0;
	elseif (F <= 6.2e-6)
		tmp = ((F * sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B;
	elseif (F <= 4.6e+253)
		tmp = (F * ((1.0 / F) / sin(B))) - (x / 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, 6.8e-273], N[(N[(-1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 6.2e-6], N[(N[(N[(F * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 4.6e+253], N[(N[(F * N[(N[(1.0 / F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < 6.79999999999999982e-273

   1. Initial program 79.4%

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

   3. Simplified 87.0%

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

   5. Taylor expanded in F around -inf 71.7%

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

   6. Taylor expanded in B around 0 65.2%

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

   if 6.79999999999999982e-273 < F < 6.1999999999999999e-6

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

      1. clear-num 99.6%

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

      2. un-div-inv 99.7%

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

      3. fma-define 99.7%

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

      4. fma-undefine 99.7%

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

      5. *-commutative 99.7%

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

      6. fma-define 99.7%

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

      7. fma-define 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in F around 0 99.7%

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

   8. Taylor expanded in B around 0 70.0%

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

   if 6.1999999999999999e-6 < F < 4.6e253

   1. Initial program 71.7%

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

   3. Simplified 82.5%

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

   5. Taylor expanded in F around inf 97.5%

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

   6. Taylor expanded in B around 0 76.3%

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

   if 4.6e253 < F

   1. Initial program 64.4%

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

   3. Simplified 88.2%

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

   5. Taylor expanded in F around inf 99.7%

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

   6. Taylor expanded in B around 0 99.8%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 6.8 \cdot 10^{-273}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{elif}\;F \leq 4.6 \cdot 10^{+253}:\\ \;\;\;\;F \cdot \frac{\frac{1}{F}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 59.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq 6.8 \cdot 10^{-272}:\\ \;\;\;\;\frac{-1}{B} - t\_0\\ \mathbf{elif}\;F \leq 2.5 \cdot 10^{-70}:\\ \;\;\;\;\frac{x}{-\sin B}\\ \mathbf{elif}\;F \leq 1.7 \cdot 10^{+247}:\\ \;\;\;\;F \cdot \frac{\frac{1}{F}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F 6.8e-272)
     (- (/ -1.0 B) t_0)
     (if (<= F 2.5e-70)
       (/ x (- (sin B)))
       (if (<= F 1.7e+247)
         (- (* F (/ (/ 1.0 F) (sin B))) (/ x 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 <= 6.8e-272) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= 2.5e-70) {
		tmp = x / -sin(B);
	} else if (F <= 1.7e+247) {
		tmp = (F * ((1.0 / F) / sin(B))) - (x / 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 <= 6.8d-272) then
        tmp = ((-1.0d0) / b) - t_0
    else if (f <= 2.5d-70) then
        tmp = x / -sin(b)
    else if (f <= 1.7d+247) then
        tmp = (f * ((1.0d0 / f) / sin(b))) - (x / 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 <= 6.8e-272) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= 2.5e-70) {
		tmp = x / -Math.sin(B);
	} else if (F <= 1.7e+247) {
		tmp = (F * ((1.0 / F) / Math.sin(B))) - (x / 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 <= 6.8e-272:
		tmp = (-1.0 / B) - t_0
	elif F <= 2.5e-70:
		tmp = x / -math.sin(B)
	elif F <= 1.7e+247:
		tmp = (F * ((1.0 / F) / math.sin(B))) - (x / 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 <= 6.8e-272)
		tmp = Float64(Float64(-1.0 / B) - t_0);
	elseif (F <= 2.5e-70)
		tmp = Float64(x / Float64(-sin(B)));
	elseif (F <= 1.7e+247)
		tmp = Float64(Float64(F * Float64(Float64(1.0 / F) / sin(B))) - Float64(x / 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 <= 6.8e-272)
		tmp = (-1.0 / B) - t_0;
	elseif (F <= 2.5e-70)
		tmp = x / -sin(B);
	elseif (F <= 1.7e+247)
		tmp = (F * ((1.0 / F) / sin(B))) - (x / 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, 6.8e-272], N[(N[(-1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 2.5e-70], N[(x / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 1.7e+247], N[(N[(F * N[(N[(1.0 / F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < 6.8000000000000006e-272

   1. Initial program 79.4%

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

   3. Simplified 87.0%

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

   5. Taylor expanded in F around -inf 71.7%

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

   6. Taylor expanded in B around 0 65.2%

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

   if 6.8000000000000006e-272 < F < 2.4999999999999999e-70

   1. Initial program 99.7%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in F around -inf 23.8%

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

   6. Taylor expanded in x around inf 77.4%

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

      1. *-commutative 77.4%

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

      2. associate-*r/ 77.6%

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

      3. neg-mul-1 77.6%

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

      4. distribute-rgt-neg-in 77.6%

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

      5. distribute-neg-frac2 77.6%

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

   8. Simplified 77.6%

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

   9. Taylor expanded in B around 0 57.2%

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

   if 2.4999999999999999e-70 < F < 1.6999999999999999e247

   1. Initial program 76.8%

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

   3. Simplified 85.5%

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

   5. Taylor expanded in F around inf 82.4%

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

   6. Taylor expanded in B around 0 65.0%

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

   if 1.6999999999999999e247 < F

   1. Initial program 64.4%

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

   3. Simplified 88.2%

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

   5. Taylor expanded in F around inf 99.7%

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

   6. Taylor expanded in B around 0 99.8%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 6.8 \cdot 10^{-272}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.5 \cdot 10^{-70}:\\ \;\;\;\;\frac{x}{-\sin B}\\ \mathbf{elif}\;F \leq 1.7 \cdot 10^{+247}:\\ \;\;\;\;F \cdot \frac{\frac{1}{F}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 59.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq 3.4 \cdot 10^{-272}:\\ \;\;\;\;\frac{-1}{B} - t\_0\\ \mathbf{elif}\;F \leq 2.5 \cdot 10^{-70}:\\ \;\;\;\;\frac{x}{-\sin B}\\ \mathbf{elif}\;F \leq 5.8 \cdot 10^{+253}:\\ \;\;\;\;\frac{F}{F \cdot \sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F 3.4e-272)
     (- (/ -1.0 B) t_0)
     (if (<= F 2.5e-70)
       (/ x (- (sin B)))
       (if (<= F 5.8e+253)
         (- (/ F (* F (sin B))) (/ x 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 <= 3.4e-272) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= 2.5e-70) {
		tmp = x / -sin(B);
	} else if (F <= 5.8e+253) {
		tmp = (F / (F * sin(B))) - (x / 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 <= 3.4d-272) then
        tmp = ((-1.0d0) / b) - t_0
    else if (f <= 2.5d-70) then
        tmp = x / -sin(b)
    else if (f <= 5.8d+253) then
        tmp = (f / (f * sin(b))) - (x / 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 <= 3.4e-272) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= 2.5e-70) {
		tmp = x / -Math.sin(B);
	} else if (F <= 5.8e+253) {
		tmp = (F / (F * Math.sin(B))) - (x / 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 <= 3.4e-272:
		tmp = (-1.0 / B) - t_0
	elif F <= 2.5e-70:
		tmp = x / -math.sin(B)
	elif F <= 5.8e+253:
		tmp = (F / (F * math.sin(B))) - (x / 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 <= 3.4e-272)
		tmp = Float64(Float64(-1.0 / B) - t_0);
	elseif (F <= 2.5e-70)
		tmp = Float64(x / Float64(-sin(B)));
	elseif (F <= 5.8e+253)
		tmp = Float64(Float64(F / Float64(F * sin(B))) - Float64(x / 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 <= 3.4e-272)
		tmp = (-1.0 / B) - t_0;
	elseif (F <= 2.5e-70)
		tmp = x / -sin(B);
	elseif (F <= 5.8e+253)
		tmp = (F / (F * sin(B))) - (x / 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, 3.4e-272], N[(N[(-1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 2.5e-70], N[(x / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 5.8e+253], N[(N[(F / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < 3.4000000000000003e-272

   1. Initial program 79.4%

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

   3. Simplified 87.0%

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

   5. Taylor expanded in F around -inf 71.7%

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

   6. Taylor expanded in B around 0 65.2%

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

   if 3.4000000000000003e-272 < F < 2.4999999999999999e-70

   1. Initial program 99.7%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in F around -inf 23.8%

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

   6. Taylor expanded in x around inf 77.4%

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

      1. *-commutative 77.4%

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

      2. associate-*r/ 77.6%

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

      3. neg-mul-1 77.6%

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

      4. distribute-rgt-neg-in 77.6%

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

      5. distribute-neg-frac2 77.6%

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

   8. Simplified 77.6%

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

   9. Taylor expanded in B around 0 57.2%

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

   if 2.4999999999999999e-70 < F < 5.79999999999999976e253

   1. Initial program 76.8%

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

   3. Simplified 85.5%

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

   5. Taylor expanded in F around -inf 36.6%

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

   6. Taylor expanded in B around 0 19.3%

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

      1. add-sqr-sqrt 7.1%

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

      2. sqrt-unprod 37.9%

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

      3. frac-times 37.9%

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

      4. metadata-eval 37.9%

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

      5. metadata-eval 37.9%

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

      6. frac-times 37.9%

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

      7. sqrt-unprod 35.8%

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

      8. add-sqr-sqrt 65.1%

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

      9. metadata-eval 65.1%

         \[\leadsto \frac{\color{blue}{{F}^{0}}}{\sin B} - \frac{x}{B} \]

      10. metadata-eval 65.1%

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

      11. pow-prod-up 65.0%

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

      12. pow1 65.0%

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

      13. inv-pow 65.0%

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

      14. associate-*r/ 65.0%

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

      15. associate-/l/ 65.0%

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

   8. Applied egg-rr 65.0%

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

      1. associate-*r/ 65.0%

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

      2. *-rgt-identity 65.0%

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

      3. *-commutative 65.0%

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

   10. Simplified 65.0%

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

   if 5.79999999999999976e253 < F

   1. Initial program 64.4%

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

   3. Simplified 88.2%

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

   5. Taylor expanded in F around inf 99.7%

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

   6. Taylor expanded in B around 0 99.8%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 3.4 \cdot 10^{-272}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.5 \cdot 10^{-70}:\\ \;\;\;\;\frac{x}{-\sin B}\\ \mathbf{elif}\;F \leq 5.8 \cdot 10^{+253}:\\ \;\;\;\;\frac{F}{F \cdot \sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 60.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq 4.8 \cdot 10^{-273}:\\ \;\;\;\;\frac{-1}{B} - t\_0\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{-70}:\\ \;\;\;\;\frac{x}{-\sin B}\\ \mathbf{elif}\;F \leq 1.1 \cdot 10^{+249}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F 4.8e-273)
     (- (/ -1.0 B) t_0)
     (if (<= F 1.8e-70)
       (/ x (- (sin B)))
       (if (<= F 1.1e+249) (/ (- 1.0 x) (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 <= 4.8e-273) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= 1.8e-70) {
		tmp = x / -sin(B);
	} else if (F <= 1.1e+249) {
		tmp = (1.0 - x) / 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 <= 4.8d-273) then
        tmp = ((-1.0d0) / b) - t_0
    else if (f <= 1.8d-70) then
        tmp = x / -sin(b)
    else if (f <= 1.1d+249) then
        tmp = (1.0d0 - x) / 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 <= 4.8e-273) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= 1.8e-70) {
		tmp = x / -Math.sin(B);
	} else if (F <= 1.1e+249) {
		tmp = (1.0 - x) / 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 <= 4.8e-273:
		tmp = (-1.0 / B) - t_0
	elif F <= 1.8e-70:
		tmp = x / -math.sin(B)
	elif F <= 1.1e+249:
		tmp = (1.0 - x) / 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 <= 4.8e-273)
		tmp = Float64(Float64(-1.0 / B) - t_0);
	elseif (F <= 1.8e-70)
		tmp = Float64(x / Float64(-sin(B)));
	elseif (F <= 1.1e+249)
		tmp = Float64(Float64(1.0 - x) / 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 <= 4.8e-273)
		tmp = (-1.0 / B) - t_0;
	elseif (F <= 1.8e-70)
		tmp = x / -sin(B);
	elseif (F <= 1.1e+249)
		tmp = (1.0 - x) / 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, 4.8e-273], N[(N[(-1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.8e-70], N[(x / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 1.1e+249], N[(N[(1.0 - x), $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 < 4.79999999999999963e-273

   1. Initial program 79.4%

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

   3. Simplified 87.0%

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

   5. Taylor expanded in F around -inf 71.7%

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

   6. Taylor expanded in B around 0 65.2%

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

   if 4.79999999999999963e-273 < F < 1.8000000000000001e-70

   1. Initial program 99.7%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in F around -inf 23.8%

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

   6. Taylor expanded in x around inf 77.4%

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

      1. *-commutative 77.4%

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

      2. associate-*r/ 77.6%

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

      3. neg-mul-1 77.6%

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

      4. distribute-rgt-neg-in 77.6%

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

      5. distribute-neg-frac2 77.6%

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

   8. Simplified 77.6%

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

   9. Taylor expanded in B around 0 57.2%

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

   if 1.8000000000000001e-70 < F < 1.0999999999999999e249

   1. Initial program 76.8%

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

   3. Simplified 85.5%

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

   5. Taylor expanded in F around inf 82.4%

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

      1. associate-*l/ 82.5%

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

      2. *-commutative 82.5%

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

   7. Simplified 82.5%

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

      1. associate-*r/ 82.4%

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

      2. sub-div 82.4%

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

   9. Applied egg-rr 82.4%

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

   10. Taylor expanded in B around 0 64.7%

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

   if 1.0999999999999999e249 < F

   1. Initial program 64.4%

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

   3. Simplified 88.2%

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

   5. Taylor expanded in F around inf 99.7%

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

   6. Taylor expanded in B around 0 99.8%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 4.8 \cdot 10^{-273}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{-70}:\\ \;\;\;\;\frac{x}{-\sin B}\\ \mathbf{elif}\;F \leq 1.1 \cdot 10^{+249}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 58.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;F \leq 9.5 \cdot 10^{-273}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 2.5 \cdot 10^{-70}:\\ \;\;\;\;\frac{x}{-\sin B}\\ \mathbf{elif}\;F \leq 8 \cdot 10^{+253}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ -1.0 B) (/ x (tan B)))))
   (if (<= F 9.5e-273)
     t_0
     (if (<= F 2.5e-70)
       (/ x (- (sin B)))
       (if (<= F 8e+253) (/ (- 1.0 x) (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 / B) - (x / tan(B));
	double tmp;
	if (F <= 9.5e-273) {
		tmp = t_0;
	} else if (F <= 2.5e-70) {
		tmp = x / -sin(B);
	} else if (F <= 8e+253) {
		tmp = (1.0 - x) / sin(B);
	} else {
		tmp = 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 = ((-1.0d0) / b) - (x / tan(b))
    if (f <= 9.5d-273) then
        tmp = t_0
    else if (f <= 2.5d-70) then
        tmp = x / -sin(b)
    else if (f <= 8d+253) then
        tmp = (1.0d0 - x) / sin(b)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (-1.0 / B) - (x / Math.tan(B));
	double tmp;
	if (F <= 9.5e-273) {
		tmp = t_0;
	} else if (F <= 2.5e-70) {
		tmp = x / -Math.sin(B);
	} else if (F <= 8e+253) {
		tmp = (1.0 - x) / Math.sin(B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (-1.0 / B) - (x / math.tan(B))
	tmp = 0
	if F <= 9.5e-273:
		tmp = t_0
	elif F <= 2.5e-70:
		tmp = x / -math.sin(B)
	elif F <= 8e+253:
		tmp = (1.0 - x) / math.sin(B)
	else:
		tmp = t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 / B) - Float64(x / tan(B)))
	tmp = 0.0
	if (F <= 9.5e-273)
		tmp = t_0;
	elseif (F <= 2.5e-70)
		tmp = Float64(x / Float64(-sin(B)));
	elseif (F <= 8e+253)
		tmp = Float64(Float64(1.0 - x) / sin(B));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (-1.0 / B) - (x / tan(B));
	tmp = 0.0;
	if (F <= 9.5e-273)
		tmp = t_0;
	elseif (F <= 2.5e-70)
		tmp = x / -sin(B);
	elseif (F <= 8e+253)
		tmp = (1.0 - x) / sin(B);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, 9.5e-273], t$95$0, If[LessEqual[F, 2.5e-70], N[(x / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 8e+253], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if F < 9.49999999999999925e-273 or 7.9999999999999995e253 < F

   1. Initial program 77.6%

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

   3. Simplified 87.1%

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

   5. Taylor expanded in F around -inf 73.6%

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

   6. Taylor expanded in B around 0 67.9%

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

   if 9.49999999999999925e-273 < F < 2.4999999999999999e-70

   1. Initial program 99.7%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in F around -inf 23.8%

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

   6. Taylor expanded in x around inf 77.4%

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

      1. *-commutative 77.4%

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

      2. associate-*r/ 77.6%

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

      3. neg-mul-1 77.6%

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

      4. distribute-rgt-neg-in 77.6%

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

      5. distribute-neg-frac2 77.6%

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

   8. Simplified 77.6%

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

   9. Taylor expanded in B around 0 57.2%

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

   if 2.4999999999999999e-70 < F < 7.9999999999999995e253

   1. Initial program 76.8%

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

   3. Simplified 85.5%

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

   5. Taylor expanded in F around inf 82.4%

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

      1. associate-*l/ 82.5%

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

      2. *-commutative 82.5%

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

   7. Simplified 82.5%

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

      1. associate-*r/ 82.4%

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

      2. sub-div 82.4%

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

   9. Applied egg-rr 82.4%

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

   10. Taylor expanded in B around 0 64.7%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 9.5 \cdot 10^{-273}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.5 \cdot 10^{-70}:\\ \;\;\;\;\frac{x}{-\sin B}\\ \mathbf{elif}\;F \leq 8 \cdot 10^{+253}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 44.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.8 \cdot 10^{-48}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.15 \cdot 10^{+17}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;F \leq 4.2 \cdot 10^{+196}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.8e-48)
   (/ (- -1.0 x) B)
   (if (<= F 1.15e+17)
     (/ x (- B))
     (if (<= F 4.2e+196) (/ 1.0 (sin B)) (/ (- 1.0 x) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.8e-48) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.15e+17) {
		tmp = x / -B;
	} else if (F <= 4.2e+196) {
		tmp = 1.0 / sin(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 <= (-3.8d-48)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 1.15d+17) then
        tmp = x / -b
    else if (f <= 4.2d+196) then
        tmp = 1.0d0 / sin(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 <= -3.8e-48) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.15e+17) {
		tmp = x / -B;
	} else if (F <= 4.2e+196) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.8e-48:
		tmp = (-1.0 - x) / B
	elif F <= 1.15e+17:
		tmp = x / -B
	elif F <= 4.2e+196:
		tmp = 1.0 / math.sin(B)
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.8e-48)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 1.15e+17)
		tmp = Float64(x / Float64(-B));
	elseif (F <= 4.2e+196)
		tmp = Float64(1.0 / sin(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 <= -3.8e-48)
		tmp = (-1.0 - x) / B;
	elseif (F <= 1.15e+17)
		tmp = x / -B;
	elseif (F <= 4.2e+196)
		tmp = 1.0 / sin(B);
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.8e-48], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.15e+17], N[(x / (-B)), $MachinePrecision], If[LessEqual[F, 4.2e+196], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.80000000000000002e-48

   1. Initial program 65.1%

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

   3. Simplified 78.0%

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

   5. Taylor expanded in F around -inf 92.3%

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

   6. Taylor expanded in B around 0 51.3%

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

      1. mul-1-neg 51.3%

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

      2. distribute-neg-frac2 51.3%

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

   8. Simplified 51.3%

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

   if -3.80000000000000002e-48 < F < 1.15e17

   1. Initial program 99.6%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in F around inf 27.6%

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

   6. Taylor expanded in B around 0 19.7%

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

   7. Taylor expanded in x around inf 39.7%

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

      1. associate-*r/ 39.7%

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

      2. neg-mul-1 39.7%

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

   9. Simplified 39.7%

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

   if 1.15e17 < F < 4.20000000000000029e196

   1. Initial program 78.8%

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

   3. Simplified 91.7%

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

   5. Taylor expanded in F around inf 99.7%

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

      1. associate-*l/ 99.7%

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

      2. *-commutative 99.7%

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

   7. Simplified 99.7%

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

      1. associate-*r/ 99.7%

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

      2. sub-div 99.7%

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

   9. Applied egg-rr 99.7%

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

   10. Taylor expanded in x around 0 66.2%

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

   if 4.20000000000000029e196 < F

   1. Initial program 52.8%

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

   3. Simplified 70.8%

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

   5. Taylor expanded in F around inf 99.7%

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

   6. Taylor expanded in B around 0 50.7%

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

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.8 \cdot 10^{-48}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.15 \cdot 10^{+17}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;F \leq 4.2 \cdot 10^{+196}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 52.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.5 \cdot 10^{-42}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.05 \cdot 10^{-83}:\\ \;\;\;\;\frac{x}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.5e-42)
   (/ (- -1.0 x) B)
   (if (<= F 1.05e-83) (/ x (- (sin B))) (/ (- 1.0 x) (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.5e-42) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.05e-83) {
		tmp = x / -sin(B);
	} else {
		tmp = (1.0 - x) / sin(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 <= (-1.5d-42)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 1.05d-83) then
        tmp = x / -sin(b)
    else
        tmp = (1.0d0 - x) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.5e-42) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.05e-83) {
		tmp = x / -Math.sin(B);
	} else {
		tmp = (1.0 - x) / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.5e-42:
		tmp = (-1.0 - x) / B
	elif F <= 1.05e-83:
		tmp = x / -math.sin(B)
	else:
		tmp = (1.0 - x) / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.5e-42)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 1.05e-83)
		tmp = Float64(x / Float64(-sin(B)));
	else
		tmp = Float64(Float64(1.0 - x) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.5e-42)
		tmp = (-1.0 - x) / B;
	elseif (F <= 1.05e-83)
		tmp = x / -sin(B);
	else
		tmp = (1.0 - x) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.5e-42], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.05e-83], N[(x / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.50000000000000014e-42

   1. Initial program 64.7%

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

   3. Simplified 77.7%

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

   5. Taylor expanded in F around -inf 93.4%

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

   6. Taylor expanded in B around 0 52.0%

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

      1. mul-1-neg 52.0%

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

      2. distribute-neg-frac2 52.0%

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

   8. Simplified 52.0%

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

   if -1.50000000000000014e-42 < F < 1.0499999999999999e-83

   1. Initial program 99.6%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in F around -inf 34.9%

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

   6. Taylor expanded in x around inf 80.2%

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

      1. *-commutative 80.2%

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

      2. associate-*r/ 80.3%

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

      3. neg-mul-1 80.3%

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

      4. distribute-rgt-neg-in 80.3%

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

      5. distribute-neg-frac2 80.3%

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

   8. Simplified 80.3%

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

   9. Taylor expanded in B around 0 46.7%

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

   if 1.0499999999999999e-83 < F

   1. Initial program 74.5%

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

   3. Simplified 86.0%

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

   5. Taylor expanded in F around inf 85.6%

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

      1. associate-*l/ 85.6%

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

      2. *-commutative 85.6%

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

   7. Simplified 85.6%

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

      1. associate-*r/ 85.6%

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

      2. sub-div 85.6%

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

   9. Applied egg-rr 85.6%

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

   10. Taylor expanded in B around 0 61.9%

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

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.5 \cdot 10^{-42}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.05 \cdot 10^{-83}:\\ \;\;\;\;\frac{x}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 51.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.9 \cdot 10^{-50}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 8.5 \cdot 10^{-101}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.9e-50)
   (/ (- -1.0 x) B)
   (if (<= F 8.5e-101) (/ x (- B)) (/ (- 1.0 x) (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.9e-50) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 8.5e-101) {
		tmp = x / -B;
	} else {
		tmp = (1.0 - x) / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-2.9d-50)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 8.5d-101) then
        tmp = x / -b
    else
        tmp = (1.0d0 - x) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.9e-50) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 8.5e-101) {
		tmp = x / -B;
	} else {
		tmp = (1.0 - x) / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.9e-50:
		tmp = (-1.0 - x) / B
	elif F <= 8.5e-101:
		tmp = x / -B
	else:
		tmp = (1.0 - x) / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.9e-50)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 8.5e-101)
		tmp = Float64(x / Float64(-B));
	else
		tmp = Float64(Float64(1.0 - x) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.9e-50)
		tmp = (-1.0 - x) / B;
	elseif (F <= 8.5e-101)
		tmp = x / -B;
	else
		tmp = (1.0 - x) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.9e-50], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 8.5e-101], N[(x / (-B)), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.90000000000000008e-50

   1. Initial program 65.1%

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

   3. Simplified 78.0%

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

   5. Taylor expanded in F around -inf 92.3%

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

   6. Taylor expanded in B around 0 51.3%

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

      1. mul-1-neg 51.3%

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

      2. distribute-neg-frac2 51.3%

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

   8. Simplified 51.3%

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

   if -2.90000000000000008e-50 < F < 8.49999999999999941e-101

   1. Initial program 99.6%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in F around inf 24.2%

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

   6. Taylor expanded in B around 0 18.3%

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

   7. Taylor expanded in x around inf 45.9%

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

      1. associate-*r/ 45.9%

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

      2. neg-mul-1 45.9%

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

   9. Simplified 45.9%

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

   if 8.49999999999999941e-101 < F

   1. Initial program 75.0%

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

   3. Simplified 86.3%

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

   5. Taylor expanded in F around inf 83.9%

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

      1. associate-*l/ 83.9%

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

      2. *-commutative 83.9%

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

   7. Simplified 83.9%

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

      1. associate-*r/ 83.9%

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

      2. sub-div 83.9%

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

   9. Applied egg-rr 83.9%

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

   10. Taylor expanded in B around 0 60.7%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.9 \cdot 10^{-50}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 8.5 \cdot 10^{-101}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 43.9% accurate, 21.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{-44}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{-101}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -5e-44)
   (/ (- -1.0 x) B)
   (if (<= F 2.3e-101) (/ x (- B)) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -5e-44) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 2.3e-101) {
		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 <= (-5d-44)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 2.3d-101) 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 <= -5e-44) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 2.3e-101) {
		tmp = x / -B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -5e-44:
		tmp = (-1.0 - x) / B
	elif F <= 2.3e-101:
		tmp = x / -B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -5e-44)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 2.3e-101)
		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 <= -5e-44)
		tmp = (-1.0 - x) / B;
	elseif (F <= 2.3e-101)
		tmp = x / -B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -5e-44], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 2.3e-101], N[(x / (-B)), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.00000000000000039e-44

   1. Initial program 65.1%

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

   3. Simplified 78.0%

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

   5. Taylor expanded in F around -inf 92.3%

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

   6. Taylor expanded in B around 0 51.3%

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

      1. mul-1-neg 51.3%

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

      2. distribute-neg-frac2 51.3%

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

   8. Simplified 51.3%

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

   if -5.00000000000000039e-44 < F < 2.2999999999999999e-101

   1. Initial program 99.6%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in F around inf 24.2%

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

   6. Taylor expanded in B around 0 18.3%

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

   7. Taylor expanded in x around inf 45.9%

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

      1. associate-*r/ 45.9%

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

      2. neg-mul-1 45.9%

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

   9. Simplified 45.9%

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

   if 2.2999999999999999e-101 < F

   1. Initial program 75.0%

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

   3. Simplified 86.3%

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

   5. Taylor expanded in F around inf 83.9%

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

   6. Taylor expanded in B around 0 34.5%

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

4. Final simplification 43.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{-44}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{-101}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 36.6% accurate, 32.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 3 \cdot 10^{-100}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F 3e-100) (/ x (- B)) (/ (- 1.0 x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= 3e-100) {
		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 <= 3d-100) 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 <= 3e-100) {
		tmp = x / -B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= 3e-100:
		tmp = x / -B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= 3e-100)
		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 <= 3e-100)
		tmp = x / -B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, 3e-100], N[(x / (-B)), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 3.0000000000000001e-100

   1. Initial program 83.4%

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

   3. Simplified 89.6%

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

   5. Taylor expanded in F around inf 35.7%

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

   6. Taylor expanded in B around 0 23.6%

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

   7. Taylor expanded in x around inf 38.4%

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

      1. associate-*r/ 38.4%

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

      2. neg-mul-1 38.4%

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

   9. Simplified 38.4%

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

   if 3.0000000000000001e-100 < F

   1. Initial program 75.0%

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

   3. Simplified 86.3%

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

   5. Taylor expanded in F around inf 83.9%

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

   6. Taylor expanded in B around 0 34.5%

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

4. Final simplification 37.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 3 \cdot 10^{-100}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 29.5% 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 80.3%

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

3. Simplified 88.3%

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

5. Taylor expanded in F around inf 53.8%

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

6. Taylor expanded in B around 0 27.7%

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

7. Taylor expanded in x around inf 32.7%

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

   1. associate-*r/ 32.7%

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

   2. neg-mul-1 32.7%

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

9. Simplified 32.7%

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

10. Final simplification 32.7%

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

Alternative 27: 10.0% accurate, 108.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (/ 1.0 B))

C

double code(double F, double B, double x) {
	return 1.0 / B;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = 1.0d0 / b
end function

Java

public static double code(double F, double B, double x) {
	return 1.0 / B;
}

Python

def code(F, B, x):
	return 1.0 / B

Julia

function code(F, B, x)
	return Float64(1.0 / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = 1.0 / B;
end

Wolfram

code[F_, B_, x_] := N[(1.0 / B), $MachinePrecision]

Derivation

1. Initial program 80.3%

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

3. Simplified 88.3%

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

5. Taylor expanded in F around inf 53.8%

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

6. Taylor expanded in B around 0 27.7%

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

7. Taylor expanded in x around 0 7.1%

   \[\leadsto \color{blue}{\frac{1}{B}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024132 
(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))))))