VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.9% → 99.7%

Time: 24.1s

Alternatives: 25

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: 76.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -85000000000:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 180000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -85000000000.0)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 180000000.0)
       (- (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* x 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 <= -85000000000.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 180000000.0) {
		tmp = ((F / sin(B)) * pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - 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 <= (-85000000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 180000000.0d0) then
        tmp = ((f / sin(b)) * ((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0))) - 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 <= -85000000000.0) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 180000000.0) {
		tmp = ((F / Math.sin(B)) * Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - 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 <= -85000000000.0:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 180000000.0:
		tmp = ((F / math.sin(B)) * math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - 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 <= -85000000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 180000000.0)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5)) - 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 <= -85000000000.0)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 180000000.0)
		tmp = ((F / sin(B)) * ((((F * F) + 2.0) + (x * 2.0)) ^ -0.5)) - 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, -85000000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 180000000.0], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $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 < -8.5e10

   1. Initial program 58.9%

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

   3. Taylor expanded in F around -inf 99.7%

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

      1. div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

   if -8.5e10 < F < 1.8e8

   1. Initial program 99.6%

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

      1. div-inv 34.7%

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

   4. Applied egg-rr 99.7%

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

   if 1.8e8 < F

   1. Initial program 54.2%

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

      1. div-inv 48.2%

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

   4. Applied egg-rr 54.3%

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

   5. Taylor expanded in F around inf 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -85000000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 180000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.8:\\ \;\;\;\;\frac{-1}{\sin B} - t\_1\\ \mathbf{elif}\;F \leq 1.85:\\ \;\;\;\;F \cdot \left(t\_0 \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\right) - t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))) (t_1 (/ x (tan B))))
   (if (<= F -1.8)
     (- (/ -1.0 (sin B)) t_1)
     (if (<= F 1.85)
       (- (* F (* t_0 (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))))) t_1)
       (- t_0 t_1)))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / sin(B);
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -1.8) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= 1.85) {
		tmp = (F * (t_0 * sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_1;
	} else {
		tmp = t_0 - 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 = 1.0d0 / sin(b)
    t_1 = x / tan(b)
    if (f <= (-1.8d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_1
    else if (f <= 1.85d0) then
        tmp = (f * (t_0 * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))))) - t_1
    else
        tmp = t_0 - t_1
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = 1.0 / Math.sin(B);
	double t_1 = x / Math.tan(B);
	double tmp;
	if (F <= -1.8) {
		tmp = (-1.0 / Math.sin(B)) - t_1;
	} else if (F <= 1.85) {
		tmp = (F * (t_0 * Math.sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_1;
	} else {
		tmp = t_0 - t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = 1.0 / math.sin(B)
	t_1 = x / math.tan(B)
	tmp = 0
	if F <= -1.8:
		tmp = (-1.0 / math.sin(B)) - t_1
	elif F <= 1.85:
		tmp = (F * (t_0 * math.sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_1
	else:
		tmp = t_0 - t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / sin(B))
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.8)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= 1.85)
		tmp = Float64(Float64(F * Float64(t_0 * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0)))))) - t_1);
	else
		tmp = Float64(t_0 - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = 1.0 / sin(B);
	t_1 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.8)
		tmp = (-1.0 / sin(B)) - t_1;
	elseif (F <= 1.85)
		tmp = (F * (t_0 * sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_1;
	else
		tmp = t_0 - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.8], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, 1.85], N[(N[(F * N[(t$95$0 * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(t$95$0 - t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.80000000000000004

   1. Initial program 61.1%

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

   3. Taylor expanded in F around -inf 98.4%

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

      1. div-inv 98.4%

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

   5. Applied egg-rr 98.4%

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

   if -1.80000000000000004 < F < 1.8500000000000001

   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. Taylor expanded in F around 0 99.4%

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

   if 1.8500000000000001 < F

   1. Initial program 57.0%

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

      1. div-inv 46.8%

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

   4. Applied egg-rr 57.0%

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

   5. Taylor expanded in F around inf 97.8%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.8:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.85:\\ \;\;\;\;F \cdot \left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\right) - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.45:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.45)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.4)
       (- (* F (/ (sqrt 0.5) (sin B))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.45) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.4) {
		tmp = (F * (sqrt(0.5) / sin(B))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.45d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.4d0) then
        tmp = (f * (sqrt(0.5d0) / sin(b))) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.45) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.4) {
		tmp = (F * (Math.sqrt(0.5) / Math.sin(B))) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.45:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.4:
		tmp = (F * (math.sqrt(0.5) / math.sin(B))) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.45)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.4)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / sin(B))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.45)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.4)
		tmp = (F * (sqrt(0.5) / sin(B))) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.45], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.4], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.44999999999999996

   1. Initial program 61.1%

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

   3. Taylor expanded in F around -inf 98.4%

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

      1. div-inv 98.4%

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

   5. Applied egg-rr 98.4%

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

   if -1.44999999999999996 < F < 1.3999999999999999

   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. Taylor expanded in F around 0 99.4%

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

   6. Taylor expanded in x around 0 99.2%

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

   if 1.3999999999999999 < F

   1. Initial program 57.0%

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

      1. div-inv 46.8%

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

   4. Applied egg-rr 57.0%

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

   5. Taylor expanded in F around inf 97.8%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.45:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3900000:\\ \;\;\;\;\frac{-1}{\sin B} - t\_1\\ \mathbf{elif}\;F \leq -2.4 \cdot 10^{-175}:\\ \;\;\;\;\frac{F}{\sin B} \cdot t\_0 - \frac{x}{B}\\ \mathbf{elif}\;F \leq 900:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + t\_0 \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5)) (t_1 (/ x (tan B))))
   (if (<= F -3900000.0)
     (- (/ -1.0 (sin B)) t_1)
     (if (<= F -2.4e-175)
       (- (* (/ F (sin B)) t_0) (/ x B))
       (if (<= F 900.0)
         (+ (* x (/ -1.0 (tan B))) (* t_0 (/ F B)))
         (- (/ 1.0 (sin B)) t_1))))))

C

double code(double F, double B, double x) {
	double t_0 = pow((((F * F) + 2.0) + (x * 2.0)), -0.5);
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -3900000.0) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= -2.4e-175) {
		tmp = ((F / sin(B)) * t_0) - (x / B);
	} else if (F <= 900.0) {
		tmp = (x * (-1.0 / tan(B))) + (t_0 * (F / 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 = (((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0)
    t_1 = x / tan(b)
    if (f <= (-3900000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_1
    else if (f <= (-2.4d-175)) then
        tmp = ((f / sin(b)) * t_0) - (x / b)
    else if (f <= 900.0d0) then
        tmp = (x * ((-1.0d0) / tan(b))) + (t_0 * (f / 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.pow((((F * F) + 2.0) + (x * 2.0)), -0.5);
	double t_1 = x / Math.tan(B);
	double tmp;
	if (F <= -3900000.0) {
		tmp = (-1.0 / Math.sin(B)) - t_1;
	} else if (F <= -2.4e-175) {
		tmp = ((F / Math.sin(B)) * t_0) - (x / B);
	} else if (F <= 900.0) {
		tmp = (x * (-1.0 / Math.tan(B))) + (t_0 * (F / B));
	} else {
		tmp = (1.0 / Math.sin(B)) - t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)
	t_1 = x / math.tan(B)
	tmp = 0
	if F <= -3900000.0:
		tmp = (-1.0 / math.sin(B)) - t_1
	elif F <= -2.4e-175:
		tmp = ((F / math.sin(B)) * t_0) - (x / B)
	elif F <= 900.0:
		tmp = (x * (-1.0 / math.tan(B))) + (t_0 * (F / B))
	else:
		tmp = (1.0 / math.sin(B)) - t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -3900000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= -2.4e-175)
		tmp = Float64(Float64(Float64(F / sin(B)) * t_0) - Float64(x / B));
	elseif (F <= 900.0)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(t_0 * Float64(F / 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 = (((F * F) + 2.0) + (x * 2.0)) ^ -0.5;
	t_1 = x / tan(B);
	tmp = 0.0;
	if (F <= -3900000.0)
		tmp = (-1.0 / sin(B)) - t_1;
	elseif (F <= -2.4e-175)
		tmp = ((F / sin(B)) * t_0) - (x / B);
	elseif (F <= 900.0)
		tmp = (x * (-1.0 / tan(B))) + (t_0 * (F / B));
	else
		tmp = (1.0 / sin(B)) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -3900000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, -2.4e-175], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 900.0], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(F / B), $MachinePrecision]), $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 < -3.9e6

   1. Initial program 60.0%

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

   3. Taylor expanded in F around -inf 99.6%

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

      1. div-inv 99.7%

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

   5. Applied egg-rr 99.7%

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

   if -3.9e6 < F < -2.4e-175

   1. Initial program 99.5%

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

   3. Taylor expanded in B around 0 92.9%

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

   if -2.4e-175 < F < 900

   1. Initial program 99.6%

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

   3. Taylor expanded in B around 0 90.1%

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

   if 900 < F

   1. Initial program 55.7%

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

      1. div-inv 46.7%

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

   4. Applied egg-rr 55.7%

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

   5. Taylor expanded in F around inf 98.6%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3900000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -2.4 \cdot 10^{-175}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 900:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -7000000:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq -3.1 \cdot 10^{-175}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.175:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -7000000.0)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F -3.1e-175)
       (- (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5)) (/ x B))
       (if (<= F 0.175)
         (- (* (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))) (/ F 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 <= -7000000.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= -3.1e-175) {
		tmp = ((F / sin(B)) * pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - (x / B);
	} else if (F <= 0.175) {
		tmp = (sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-7000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= (-3.1d-175)) then
        tmp = ((f / sin(b)) * ((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0))) - (x / b)
    else if (f <= 0.175d0) then
        tmp = (sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))) * (f / b)) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -7000000.0) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= -3.1e-175) {
		tmp = ((F / Math.sin(B)) * Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - (x / B);
	} else if (F <= 0.175) {
		tmp = (Math.sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -7000000.0:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= -3.1e-175:
		tmp = ((F / math.sin(B)) * math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - (x / B)
	elif F <= 0.175:
		tmp = (math.sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -7000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= -3.1e-175)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5)) - Float64(x / B));
	elseif (F <= 0.175)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0)))) * Float64(F / B)) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -7000000.0)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= -3.1e-175)
		tmp = ((F / sin(B)) * ((((F * F) + 2.0) + (x * 2.0)) ^ -0.5)) - (x / B);
	elseif (F <= 0.175)
		tmp = (sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -7000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -3.1e-175], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.175], N[(N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -7e6

   1. Initial program 60.0%

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

   3. Taylor expanded in F around -inf 99.6%

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

      1. div-inv 99.7%

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

   5. Applied egg-rr 99.7%

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

   if -7e6 < F < -3.09999999999999999e-175

   1. Initial program 99.5%

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

   3. Taylor expanded in B around 0 92.9%

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

   if -3.09999999999999999e-175 < F < 0.17499999999999999

   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 B around 0 90.0%

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

   6. Taylor expanded in F around 0 89.7%

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

   if 0.17499999999999999 < F

   1. Initial program 57.0%

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

      1. div-inv 46.8%

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

   4. Applied egg-rr 57.0%

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

   5. Taylor expanded in F around inf 97.8%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -3.1 \cdot 10^{-175}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.175:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F}{\frac{\sin B}{\sqrt{0.5}}}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2.16 \cdot 10^{-7}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -3.1 \cdot 10^{-175}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 8.8 \cdot 10^{-75}:\\ \;\;\;\;\frac{\frac{F}{B}}{F} - t\_1\\ \mathbf{elif}\;F \leq 8 \cdot 10^{-37}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 1.08 \cdot 10^{+39}:\\ \;\;\;\;\frac{1}{B} - t\_1\\ \mathbf{elif}\;F \leq 1.45 \cdot 10^{+108}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \frac{1}{F} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{F \cdot B} - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ F (/ (sin B) (sqrt 0.5)))) (t_1 (/ x (tan B))))
   (if (<= F -2.16e-7)
     (- (/ -1.0 (sin B)) (/ x B))
     (if (<= F -3.1e-175)
       t_0
       (if (<= F 8.8e-75)
         (- (/ (/ F B) F) t_1)
         (if (<= F 8e-37)
           t_0
           (if (<= F 1.08e+39)
             (- (/ 1.0 B) t_1)
             (if (<= F 1.45e+108)
               (- (* (/ F (sin B)) (/ 1.0 F)) (/ x B))
               (- (/ F (* F B)) t_1)))))))))

C

double code(double F, double B, double x) {
	double t_0 = F / (sin(B) / sqrt(0.5));
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -2.16e-7) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -3.1e-175) {
		tmp = t_0;
	} else if (F <= 8.8e-75) {
		tmp = ((F / B) / F) - t_1;
	} else if (F <= 8e-37) {
		tmp = t_0;
	} else if (F <= 1.08e+39) {
		tmp = (1.0 / B) - t_1;
	} else if (F <= 1.45e+108) {
		tmp = ((F / sin(B)) * (1.0 / F)) - (x / B);
	} else {
		tmp = (F / (F * B)) - t_1;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = f / (sin(b) / sqrt(0.5d0))
    t_1 = x / tan(b)
    if (f <= (-2.16d-7)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-3.1d-175)) then
        tmp = t_0
    else if (f <= 8.8d-75) then
        tmp = ((f / b) / f) - t_1
    else if (f <= 8d-37) then
        tmp = t_0
    else if (f <= 1.08d+39) then
        tmp = (1.0d0 / b) - t_1
    else if (f <= 1.45d+108) then
        tmp = ((f / sin(b)) * (1.0d0 / f)) - (x / b)
    else
        tmp = (f / (f * b)) - t_1
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = F / (Math.sin(B) / Math.sqrt(0.5));
	double t_1 = x / Math.tan(B);
	double tmp;
	if (F <= -2.16e-7) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -3.1e-175) {
		tmp = t_0;
	} else if (F <= 8.8e-75) {
		tmp = ((F / B) / F) - t_1;
	} else if (F <= 8e-37) {
		tmp = t_0;
	} else if (F <= 1.08e+39) {
		tmp = (1.0 / B) - t_1;
	} else if (F <= 1.45e+108) {
		tmp = ((F / Math.sin(B)) * (1.0 / F)) - (x / B);
	} else {
		tmp = (F / (F * B)) - t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = F / (math.sin(B) / math.sqrt(0.5))
	t_1 = x / math.tan(B)
	tmp = 0
	if F <= -2.16e-7:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -3.1e-175:
		tmp = t_0
	elif F <= 8.8e-75:
		tmp = ((F / B) / F) - t_1
	elif F <= 8e-37:
		tmp = t_0
	elif F <= 1.08e+39:
		tmp = (1.0 / B) - t_1
	elif F <= 1.45e+108:
		tmp = ((F / math.sin(B)) * (1.0 / F)) - (x / B)
	else:
		tmp = (F / (F * B)) - t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(F / Float64(sin(B) / sqrt(0.5)))
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -2.16e-7)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -3.1e-175)
		tmp = t_0;
	elseif (F <= 8.8e-75)
		tmp = Float64(Float64(Float64(F / B) / F) - t_1);
	elseif (F <= 8e-37)
		tmp = t_0;
	elseif (F <= 1.08e+39)
		tmp = Float64(Float64(1.0 / B) - t_1);
	elseif (F <= 1.45e+108)
		tmp = Float64(Float64(Float64(F / sin(B)) * Float64(1.0 / F)) - Float64(x / B));
	else
		tmp = Float64(Float64(F / Float64(F * B)) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = F / (sin(B) / sqrt(0.5));
	t_1 = x / tan(B);
	tmp = 0.0;
	if (F <= -2.16e-7)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -3.1e-175)
		tmp = t_0;
	elseif (F <= 8.8e-75)
		tmp = ((F / B) / F) - t_1;
	elseif (F <= 8e-37)
		tmp = t_0;
	elseif (F <= 1.08e+39)
		tmp = (1.0 / B) - t_1;
	elseif (F <= 1.45e+108)
		tmp = ((F / sin(B)) * (1.0 / F)) - (x / B);
	else
		tmp = (F / (F * B)) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(F / N[(N[Sin[B], $MachinePrecision] / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.16e-7], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -3.1e-175], t$95$0, If[LessEqual[F, 8.8e-75], N[(N[(N[(F / B), $MachinePrecision] / F), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, 8e-37], t$95$0, If[LessEqual[F, 1.08e+39], N[(N[(1.0 / B), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, 1.45e+108], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(1.0 / F), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(F / N[(F * B), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if F < -2.16000000000000002e-7

   1. Initial program 61.6%

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

   3. Taylor expanded in F around -inf 98.4%

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

   4. Taylor expanded in B around 0 75.6%

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

   if -2.16000000000000002e-7 < F < -3.09999999999999999e-175 or 8.80000000000000022e-75 < F < 8.00000000000000053e-37

   1. Initial program 99.4%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in F around 0 99.2%

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

   6. Taylor expanded in x around 0 99.3%

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

   7. Taylor expanded in F around inf 68.4%

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{0.5}}{\sin B}} \]
   8. Step-by-step derivation

      1. associate-*l/ 68.4%

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

      2. associate-/r/ 68.3%

         \[\leadsto \color{blue}{\frac{F}{\frac{\sin B}{\sqrt{0.5}}}} \]

   9. Simplified 68.3%

      \[\leadsto \color{blue}{\frac{F}{\frac{\sin B}{\sqrt{0.5}}}} \]

   if -3.09999999999999999e-175 < F < 8.80000000000000022e-75

   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 B around 0 94.1%

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

   6. Taylor expanded in F around inf 47.6%

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

      1. *-commutative 47.6%

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

   8. Simplified 47.6%

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

      1. un-div-inv 47.6%

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

      2. *-commutative 47.6%

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

      3. associate-/r* 61.1%

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

   10. Applied egg-rr 61.1%

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

   if 8.00000000000000053e-37 < F < 1.07999999999999998e39

   1. Initial program 85.5%

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

   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 B around 0 79.1%

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

   6. Taylor expanded in F around inf 60.2%

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

      1. *-commutative 60.2%

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

   8. Simplified 60.2%

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

   9. Taylor expanded in F around 0 60.3%

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

   if 1.07999999999999998e39 < F < 1.45000000000000004e108

   1. Initial program 99.8%

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

   3. Taylor expanded in F around inf 99.8%

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

   4. Taylor expanded in B around 0 89.8%

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

   if 1.45000000000000004e108 < F

   1. Initial program 38.6%

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

   3. Simplified 59.6%

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

   5. Taylor expanded in B around 0 57.4%

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

   6. Taylor expanded in F around inf 68.5%

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

      1. *-commutative 68.5%

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

   8. Simplified 68.5%

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

      1. un-div-inv 68.6%

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

   10. Applied egg-rr 68.6%

       \[\leadsto \color{blue}{\frac{F}{F \cdot B}} - \frac{x}{\tan B} \]
3. Recombined 6 regimes into one program.

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.16 \cdot 10^{-7}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -3.1 \cdot 10^{-175}:\\ \;\;\;\;\frac{F}{\frac{\sin B}{\sqrt{0.5}}}\\ \mathbf{elif}\;F \leq 8.8 \cdot 10^{-75}:\\ \;\;\;\;\frac{\frac{F}{B}}{F} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 8 \cdot 10^{-37}:\\ \;\;\;\;\frac{F}{\frac{\sin B}{\sqrt{0.5}}}\\ \mathbf{elif}\;F \leq 1.08 \cdot 10^{+39}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.45 \cdot 10^{+108}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \frac{1}{F} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{F \cdot B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2.16 \cdot 10^{-7}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -2.8 \cdot 10^{-175}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 5.3 \cdot 10^{-71}:\\ \;\;\;\;\frac{\frac{F}{B}}{F} - t\_0\\ \mathbf{elif}\;F \leq 2.75 \cdot 10^{-36}:\\ \;\;\;\;\frac{F}{\frac{\sin B}{\sqrt{0.5}}}\\ \mathbf{elif}\;F \leq 1.08 \cdot 10^{+39}:\\ \;\;\;\;\frac{1}{B} - t\_0\\ \mathbf{elif}\;F \leq 5.7 \cdot 10^{+107}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \frac{1}{F} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{F \cdot B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -2.16e-7)
     (- (/ -1.0 (sin B)) (/ x B))
     (if (<= F -2.8e-175)
       (/ (* F (sqrt 0.5)) (sin B))
       (if (<= F 5.3e-71)
         (- (/ (/ F B) F) t_0)
         (if (<= F 2.75e-36)
           (/ F (/ (sin B) (sqrt 0.5)))
           (if (<= F 1.08e+39)
             (- (/ 1.0 B) t_0)
             (if (<= F 5.7e+107)
               (- (* (/ F (sin B)) (/ 1.0 F)) (/ x B))
               (- (/ F (* F B)) t_0)))))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -2.16e-7) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -2.8e-175) {
		tmp = (F * sqrt(0.5)) / sin(B);
	} else if (F <= 5.3e-71) {
		tmp = ((F / B) / F) - t_0;
	} else if (F <= 2.75e-36) {
		tmp = F / (sin(B) / sqrt(0.5));
	} else if (F <= 1.08e+39) {
		tmp = (1.0 / B) - t_0;
	} else if (F <= 5.7e+107) {
		tmp = ((F / sin(B)) * (1.0 / F)) - (x / B);
	} else {
		tmp = (F / (F * 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 <= (-2.16d-7)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-2.8d-175)) then
        tmp = (f * sqrt(0.5d0)) / sin(b)
    else if (f <= 5.3d-71) then
        tmp = ((f / b) / f) - t_0
    else if (f <= 2.75d-36) then
        tmp = f / (sin(b) / sqrt(0.5d0))
    else if (f <= 1.08d+39) then
        tmp = (1.0d0 / b) - t_0
    else if (f <= 5.7d+107) then
        tmp = ((f / sin(b)) * (1.0d0 / f)) - (x / b)
    else
        tmp = (f / (f * 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 <= -2.16e-7) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -2.8e-175) {
		tmp = (F * Math.sqrt(0.5)) / Math.sin(B);
	} else if (F <= 5.3e-71) {
		tmp = ((F / B) / F) - t_0;
	} else if (F <= 2.75e-36) {
		tmp = F / (Math.sin(B) / Math.sqrt(0.5));
	} else if (F <= 1.08e+39) {
		tmp = (1.0 / B) - t_0;
	} else if (F <= 5.7e+107) {
		tmp = ((F / Math.sin(B)) * (1.0 / F)) - (x / B);
	} else {
		tmp = (F / (F * B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -2.16e-7:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -2.8e-175:
		tmp = (F * math.sqrt(0.5)) / math.sin(B)
	elif F <= 5.3e-71:
		tmp = ((F / B) / F) - t_0
	elif F <= 2.75e-36:
		tmp = F / (math.sin(B) / math.sqrt(0.5))
	elif F <= 1.08e+39:
		tmp = (1.0 / B) - t_0
	elif F <= 5.7e+107:
		tmp = ((F / math.sin(B)) * (1.0 / F)) - (x / B)
	else:
		tmp = (F / (F * B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -2.16e-7)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -2.8e-175)
		tmp = Float64(Float64(F * sqrt(0.5)) / sin(B));
	elseif (F <= 5.3e-71)
		tmp = Float64(Float64(Float64(F / B) / F) - t_0);
	elseif (F <= 2.75e-36)
		tmp = Float64(F / Float64(sin(B) / sqrt(0.5)));
	elseif (F <= 1.08e+39)
		tmp = Float64(Float64(1.0 / B) - t_0);
	elseif (F <= 5.7e+107)
		tmp = Float64(Float64(Float64(F / sin(B)) * Float64(1.0 / F)) - Float64(x / B));
	else
		tmp = Float64(Float64(F / Float64(F * 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 <= -2.16e-7)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -2.8e-175)
		tmp = (F * sqrt(0.5)) / sin(B);
	elseif (F <= 5.3e-71)
		tmp = ((F / B) / F) - t_0;
	elseif (F <= 2.75e-36)
		tmp = F / (sin(B) / sqrt(0.5));
	elseif (F <= 1.08e+39)
		tmp = (1.0 / B) - t_0;
	elseif (F <= 5.7e+107)
		tmp = ((F / sin(B)) * (1.0 / F)) - (x / B);
	else
		tmp = (F / (F * 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, -2.16e-7], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -2.8e-175], N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.3e-71], N[(N[(N[(F / B), $MachinePrecision] / F), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 2.75e-36], N[(F / N[(N[Sin[B], $MachinePrecision] / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.08e+39], N[(N[(1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 5.7e+107], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(1.0 / F), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(F / N[(F * B), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if F < -2.16000000000000002e-7

   1. Initial program 61.6%

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

   3. Taylor expanded in F around -inf 98.4%

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

   4. Taylor expanded in B around 0 75.6%

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

   if -2.16000000000000002e-7 < F < -2.8e-175

   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 99.5%

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

   5. Taylor expanded in F around 0 99.3%

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

   6. Taylor expanded in x around 0 99.3%

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

   7. Taylor expanded in F around inf 68.0%

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

   if -2.8e-175 < F < 5.29999999999999999e-71

   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 B around 0 94.1%

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

   6. Taylor expanded in F around inf 47.6%

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

      1. *-commutative 47.6%

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

   8. Simplified 47.6%

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

      1. un-div-inv 47.6%

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

      2. *-commutative 47.6%

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

      3. associate-/r* 61.1%

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

   10. Applied egg-rr 61.1%

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

   if 5.29999999999999999e-71 < F < 2.74999999999999992e-36

   1. Initial program 99.1%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in F around 0 99.0%

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

   6. Taylor expanded in x around 0 99.3%

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

   7. Taylor expanded in F around inf 71.0%

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{0.5}}{\sin B}} \]
   8. Step-by-step derivation

      1. associate-*l/ 71.0%

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

      2. associate-/r/ 71.1%

         \[\leadsto \color{blue}{\frac{F}{\frac{\sin B}{\sqrt{0.5}}}} \]

   9. Simplified 71.1%

      \[\leadsto \color{blue}{\frac{F}{\frac{\sin B}{\sqrt{0.5}}}} \]

   if 2.74999999999999992e-36 < F < 1.07999999999999998e39

   1. Initial program 85.5%

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

   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 B around 0 79.1%

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

   6. Taylor expanded in F around inf 60.2%

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

      1. *-commutative 60.2%

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

   8. Simplified 60.2%

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

   9. Taylor expanded in F around 0 60.3%

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

   if 1.07999999999999998e39 < F < 5.69999999999999972e107

   1. Initial program 99.8%

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

   3. Taylor expanded in F around inf 99.8%

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

   4. Taylor expanded in B around 0 89.8%

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

   if 5.69999999999999972e107 < F

   1. Initial program 38.6%

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

   3. Simplified 59.6%

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

   5. Taylor expanded in B around 0 57.4%

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

   6. Taylor expanded in F around inf 68.5%

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

      1. *-commutative 68.5%

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

   8. Simplified 68.5%

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

      1. un-div-inv 68.6%

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

   10. Applied egg-rr 68.6%

       \[\leadsto \color{blue}{\frac{F}{F \cdot B}} - \frac{x}{\tan B} \]
3. Recombined 7 regimes into one program.

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.16 \cdot 10^{-7}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -2.8 \cdot 10^{-175}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 5.3 \cdot 10^{-71}:\\ \;\;\;\;\frac{\frac{F}{B}}{F} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.75 \cdot 10^{-36}:\\ \;\;\;\;\frac{F}{\frac{\sin B}{\sqrt{0.5}}}\\ \mathbf{elif}\;F \leq 1.08 \cdot 10^{+39}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 5.7 \cdot 10^{+107}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \frac{1}{F} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{F \cdot B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.21:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 0.17:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{2 + x \cdot 2}}}{B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -0.21)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 0.17)
       (- (* F (/ (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))) 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 <= -0.21) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 0.17) {
		tmp = (F * (sqrt((1.0 / (2.0 + (x * 2.0)))) / B)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-0.21d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 0.17d0) then
        tmp = (f * (sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))) / b)) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -0.21) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 0.17) {
		tmp = (F * (Math.sqrt((1.0 / (2.0 + (x * 2.0)))) / B)) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -0.21:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 0.17:
		tmp = (F * (math.sqrt((1.0 / (2.0 + (x * 2.0)))) / B)) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -0.21)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 0.17)
		tmp = Float64(Float64(F * Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0)))) / B)) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -0.21)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 0.17)
		tmp = (F * (sqrt((1.0 / (2.0 + (x * 2.0)))) / B)) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.21], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.17], N[(N[(F * N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / B), $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 < -0.209999999999999992

   1. Initial program 61.1%

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

   3. Taylor expanded in F around -inf 98.4%

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

      1. div-inv 98.4%

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

   5. Applied egg-rr 98.4%

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

   if -0.209999999999999992 < F < 0.170000000000000012

   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. Taylor expanded in B around 0 83.3%

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

   6. Taylor expanded in F around 0 83.0%

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

      1. associate-*l/ 83.1%

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

      2. *-lft-identity 83.1%

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

   8. Simplified 83.1%

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

   if 0.170000000000000012 < F

   1. Initial program 57.0%

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

      1. div-inv 46.8%

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

   4. Applied egg-rr 57.0%

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

   5. Taylor expanded in F around inf 97.8%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.21:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.17:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{2 + x \cdot 2}}}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 92.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.35:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 0.26:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -0.35)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 0.26)
       (- (* (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))) (/ F 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 <= -0.35) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 0.26) {
		tmp = (sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-0.35d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 0.26d0) then
        tmp = (sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))) * (f / b)) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -0.35) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 0.26) {
		tmp = (Math.sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -0.35:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 0.26:
		tmp = (math.sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -0.35)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 0.26)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0)))) * Float64(F / B)) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -0.35)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 0.26)
		tmp = (sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.35], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.26], N[(N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / B), $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 < -0.34999999999999998

   1. Initial program 61.1%

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

   3. Taylor expanded in F around -inf 98.4%

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

      1. div-inv 98.4%

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

   5. Applied egg-rr 98.4%

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

   if -0.34999999999999998 < F < 0.26000000000000001

   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. Taylor expanded in B around 0 83.3%

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

   6. Taylor expanded in F around 0 83.1%

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

   if 0.26000000000000001 < F

   1. Initial program 57.0%

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

      1. div-inv 46.8%

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

   4. Applied egg-rr 57.0%

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

   5. Taylor expanded in F around inf 97.8%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.35:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.26:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 83.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3.8 \cdot 10^{-33}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq -2.8 \cdot 10^{-175}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 2.4 \cdot 10^{-49}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -3.8e-33)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F -2.8e-175)
       (/ (* F (sqrt 0.5)) (sin B))
       (if (<= F 2.4e-49)
         (/ (* x (cos B)) (- (sin B)))
         (- (/ 1.0 (sin B)) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -3.8e-33) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= -2.8e-175) {
		tmp = (F * sqrt(0.5)) / sin(B);
	} else if (F <= 2.4e-49) {
		tmp = (x * cos(B)) / -sin(B);
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-3.8d-33)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= (-2.8d-175)) then
        tmp = (f * sqrt(0.5d0)) / sin(b)
    else if (f <= 2.4d-49) then
        tmp = (x * cos(b)) / -sin(b)
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -3.8e-33) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= -2.8e-175) {
		tmp = (F * Math.sqrt(0.5)) / Math.sin(B);
	} else if (F <= 2.4e-49) {
		tmp = (x * Math.cos(B)) / -Math.sin(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -3.8e-33:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= -2.8e-175:
		tmp = (F * math.sqrt(0.5)) / math.sin(B)
	elif F <= 2.4e-49:
		tmp = (x * math.cos(B)) / -math.sin(B)
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -3.8e-33)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= -2.8e-175)
		tmp = Float64(Float64(F * sqrt(0.5)) / sin(B));
	elseif (F <= 2.4e-49)
		tmp = Float64(Float64(x * cos(B)) / Float64(-sin(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -3.8e-33)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= -2.8e-175)
		tmp = (F * sqrt(0.5)) / sin(B);
	elseif (F <= 2.4e-49)
		tmp = (x * cos(B)) / -sin(B);
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -3.8e-33], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -2.8e-175], N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.4e-49], N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.79999999999999994e-33

   1. Initial program 65.9%

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

   3. Taylor expanded in F around -inf 93.1%

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

      1. div-inv 93.2%

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

   5. Applied egg-rr 93.2%

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

   if -3.79999999999999994e-33 < F < -2.8e-175

   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 99.5%

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

   5. Taylor expanded in F around 0 99.4%

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

   6. Taylor expanded in x around 0 99.5%

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

   7. Taylor expanded in F around inf 73.7%

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

   if -2.8e-175 < F < 2.39999999999999992e-49

   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 F around -inf 37.4%

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

   4. Taylor expanded in x around inf 85.9%

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

   if 2.39999999999999992e-49 < F

   1. Initial program 62.0%

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

      1. div-inv 43.9%

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

   4. Applied egg-rr 62.1%

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

   5. Taylor expanded in F around inf 89.5%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.8 \cdot 10^{-33}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -2.8 \cdot 10^{-175}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 2.4 \cdot 10^{-49}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 70.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.16 \cdot 10^{-7}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -3.2 \cdot 10^{-175}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 9.5 \cdot 10^{-51}:\\ \;\;\;\;x \cdot \frac{\cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.16e-7)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F -3.2e-175)
     (/ (* F (sqrt 0.5)) (sin B))
     (if (<= F 9.5e-51)
       (* x (/ (cos B) (- (sin B))))
       (- (/ 1.0 B) (/ x (tan B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.16e-7) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -3.2e-175) {
		tmp = (F * sqrt(0.5)) / sin(B);
	} else if (F <= 9.5e-51) {
		tmp = x * (cos(B) / -sin(B));
	} else {
		tmp = (1.0 / B) - (x / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-2.16d-7)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-3.2d-175)) then
        tmp = (f * sqrt(0.5d0)) / sin(b)
    else if (f <= 9.5d-51) then
        tmp = x * (cos(b) / -sin(b))
    else
        tmp = (1.0d0 / b) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.16e-7) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -3.2e-175) {
		tmp = (F * Math.sqrt(0.5)) / Math.sin(B);
	} else if (F <= 9.5e-51) {
		tmp = x * (Math.cos(B) / -Math.sin(B));
	} else {
		tmp = (1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.16e-7:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -3.2e-175:
		tmp = (F * math.sqrt(0.5)) / math.sin(B)
	elif F <= 9.5e-51:
		tmp = x * (math.cos(B) / -math.sin(B))
	else:
		tmp = (1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.16e-7)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -3.2e-175)
		tmp = Float64(Float64(F * sqrt(0.5)) / sin(B));
	elseif (F <= 9.5e-51)
		tmp = Float64(x * Float64(cos(B) / Float64(-sin(B))));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.16e-7)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -3.2e-175)
		tmp = (F * sqrt(0.5)) / sin(B);
	elseif (F <= 9.5e-51)
		tmp = x * (cos(B) / -sin(B));
	else
		tmp = (1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.16e-7], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -3.2e-175], N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 9.5e-51], N[(x * N[(N[Cos[B], $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.16000000000000002e-7

   1. Initial program 61.6%

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

   3. Taylor expanded in F around -inf 98.4%

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

   4. Taylor expanded in B around 0 75.6%

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

   if -2.16000000000000002e-7 < F < -3.2e-175

   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 99.5%

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

   5. Taylor expanded in F around 0 99.3%

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

   6. Taylor expanded in x around 0 99.3%

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

   7. Taylor expanded in F around inf 68.0%

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

   if -3.2e-175 < F < 9.4999999999999998e-51

   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 F around -inf 37.4%

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

   4. Taylor expanded in x around inf 85.9%

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

      1. mul-1-neg 85.9%

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

      2. associate-/l* 85.9%

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

      3. distribute-rgt-neg-in 85.9%

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

   6. Simplified 85.9%

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

   if 9.4999999999999998e-51 < F

   1. Initial program 62.0%

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

   3. Simplified 76.7%

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

   5. Taylor expanded in B around 0 60.7%

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

   6. Taylor expanded in F around inf 59.8%

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

      1. *-commutative 59.8%

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

   8. Simplified 59.8%

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

   9. Taylor expanded in F around 0 59.9%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.16 \cdot 10^{-7}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -3.2 \cdot 10^{-175}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 9.5 \cdot 10^{-51}:\\ \;\;\;\;x \cdot \frac{\cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 70.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.16 \cdot 10^{-7}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.9 \cdot 10^{-175}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 4.8 \cdot 10^{-50}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.16e-7)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F -1.9e-175)
     (/ (* F (sqrt 0.5)) (sin B))
     (if (<= F 4.8e-50)
       (/ (* x (cos B)) (- (sin B)))
       (- (/ 1.0 B) (/ x (tan B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.16e-7) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -1.9e-175) {
		tmp = (F * sqrt(0.5)) / sin(B);
	} else if (F <= 4.8e-50) {
		tmp = (x * cos(B)) / -sin(B);
	} else {
		tmp = (1.0 / B) - (x / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-2.16d-7)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-1.9d-175)) then
        tmp = (f * sqrt(0.5d0)) / sin(b)
    else if (f <= 4.8d-50) then
        tmp = (x * cos(b)) / -sin(b)
    else
        tmp = (1.0d0 / b) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.16e-7) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -1.9e-175) {
		tmp = (F * Math.sqrt(0.5)) / Math.sin(B);
	} else if (F <= 4.8e-50) {
		tmp = (x * Math.cos(B)) / -Math.sin(B);
	} else {
		tmp = (1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.16e-7:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -1.9e-175:
		tmp = (F * math.sqrt(0.5)) / math.sin(B)
	elif F <= 4.8e-50:
		tmp = (x * math.cos(B)) / -math.sin(B)
	else:
		tmp = (1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.16e-7)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -1.9e-175)
		tmp = Float64(Float64(F * sqrt(0.5)) / sin(B));
	elseif (F <= 4.8e-50)
		tmp = Float64(Float64(x * cos(B)) / Float64(-sin(B)));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.16e-7)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -1.9e-175)
		tmp = (F * sqrt(0.5)) / sin(B);
	elseif (F <= 4.8e-50)
		tmp = (x * cos(B)) / -sin(B);
	else
		tmp = (1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.16e-7], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.9e-175], N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.8e-50], N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.16000000000000002e-7

   1. Initial program 61.6%

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

   3. Taylor expanded in F around -inf 98.4%

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

   4. Taylor expanded in B around 0 75.6%

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

   if -2.16000000000000002e-7 < F < -1.9e-175

   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 99.5%

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

   5. Taylor expanded in F around 0 99.3%

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

   6. Taylor expanded in x around 0 99.3%

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

   7. Taylor expanded in F around inf 68.0%

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

   if -1.9e-175 < F < 4.80000000000000004e-50

   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 F around -inf 37.4%

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

   4. Taylor expanded in x around inf 85.9%

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

   if 4.80000000000000004e-50 < F

   1. Initial program 62.0%

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

   3. Simplified 76.7%

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

   5. Taylor expanded in B around 0 60.7%

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

   6. Taylor expanded in F around inf 59.8%

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

      1. *-commutative 59.8%

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

   8. Simplified 59.8%

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

   9. Taylor expanded in F around 0 59.9%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.16 \cdot 10^{-7}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.9 \cdot 10^{-175}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 4.8 \cdot 10^{-50}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 76.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -6 \cdot 10^{-37}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq -3.2 \cdot 10^{-175}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 2.7 \cdot 10^{-50}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -6e-37)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F -3.2e-175)
       (/ (* F (sqrt 0.5)) (sin B))
       (if (<= F 2.7e-50) (/ (* x (cos B)) (- (sin B))) (- (/ 1.0 B) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -6e-37) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= -3.2e-175) {
		tmp = (F * sqrt(0.5)) / sin(B);
	} else if (F <= 2.7e-50) {
		tmp = (x * cos(B)) / -sin(B);
	} else {
		tmp = (1.0 / B) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-6d-37)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= (-3.2d-175)) then
        tmp = (f * sqrt(0.5d0)) / sin(b)
    else if (f <= 2.7d-50) then
        tmp = (x * cos(b)) / -sin(b)
    else
        tmp = (1.0d0 / b) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -6e-37) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= -3.2e-175) {
		tmp = (F * Math.sqrt(0.5)) / Math.sin(B);
	} else if (F <= 2.7e-50) {
		tmp = (x * Math.cos(B)) / -Math.sin(B);
	} else {
		tmp = (1.0 / B) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -6e-37:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= -3.2e-175:
		tmp = (F * math.sqrt(0.5)) / math.sin(B)
	elif F <= 2.7e-50:
		tmp = (x * math.cos(B)) / -math.sin(B)
	else:
		tmp = (1.0 / B) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -6e-37)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= -3.2e-175)
		tmp = Float64(Float64(F * sqrt(0.5)) / sin(B));
	elseif (F <= 2.7e-50)
		tmp = Float64(Float64(x * cos(B)) / Float64(-sin(B)));
	else
		tmp = Float64(Float64(1.0 / B) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -6e-37)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= -3.2e-175)
		tmp = (F * sqrt(0.5)) / sin(B);
	elseif (F <= 2.7e-50)
		tmp = (x * cos(B)) / -sin(B);
	else
		tmp = (1.0 / B) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -6e-37], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -3.2e-175], N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.7e-50], N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -6e-37

   1. Initial program 65.9%

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

   3. Taylor expanded in F around -inf 93.1%

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

      1. div-inv 93.2%

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

   5. Applied egg-rr 93.2%

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

   if -6e-37 < F < -3.2e-175

   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 99.5%

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

   5. Taylor expanded in F around 0 99.4%

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

   6. Taylor expanded in x around 0 99.5%

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

   7. Taylor expanded in F around inf 73.7%

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

   if -3.2e-175 < F < 2.7e-50

   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 F around -inf 37.4%

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

   4. Taylor expanded in x around inf 85.9%

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

   if 2.7e-50 < F

   1. Initial program 62.0%

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

   3. Simplified 76.7%

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

   5. Taylor expanded in B around 0 60.7%

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

   6. Taylor expanded in F around inf 59.8%

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

      1. *-commutative 59.8%

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

   8. Simplified 59.8%

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

   9. Taylor expanded in F around 0 59.9%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6 \cdot 10^{-37}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -3.2 \cdot 10^{-175}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 2.7 \cdot 10^{-50}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 92.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.11:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 0.26:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -0.11)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 0.26) (- (* F (/ (sqrt 0.5) 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 <= -0.11) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 0.26) {
		tmp = (F * (sqrt(0.5) / B)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-0.11d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 0.26d0) then
        tmp = (f * (sqrt(0.5d0) / b)) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -0.11) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 0.26) {
		tmp = (F * (Math.sqrt(0.5) / B)) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -0.11:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 0.26:
		tmp = (F * (math.sqrt(0.5) / B)) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -0.11)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 0.26)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / B)) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -0.11)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 0.26)
		tmp = (F * (sqrt(0.5) / B)) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.11], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.26], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / B), $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 < -0.110000000000000001

   1. Initial program 61.1%

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

   3. Taylor expanded in F around -inf 98.4%

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

      1. div-inv 98.4%

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

   5. Applied egg-rr 98.4%

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

   if -0.110000000000000001 < F < 0.26000000000000001

   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. Taylor expanded in B around 0 83.3%

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

   6. Taylor expanded in F around 0 83.0%

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

   7. Taylor expanded in x around 0 82.9%

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

      1. associate-/l* 82.9%

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

   9. Simplified 82.9%

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

   if 0.26000000000000001 < F

   1. Initial program 57.0%

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

      1. div-inv 46.8%

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

   4. Applied egg-rr 57.0%

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

   5. Taylor expanded in F around inf 97.8%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.11:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.26:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 92.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.21:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 0.078:\\ \;\;\;\;\sqrt{0.5} \cdot \frac{F}{B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -0.21)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 0.078)
       (- (* (sqrt 0.5) (/ F 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 <= -0.21) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 0.078) {
		tmp = (sqrt(0.5) * (F / B)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-0.21d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 0.078d0) then
        tmp = (sqrt(0.5d0) * (f / b)) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -0.21) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 0.078) {
		tmp = (Math.sqrt(0.5) * (F / B)) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -0.21:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 0.078:
		tmp = (math.sqrt(0.5) * (F / B)) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -0.21)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 0.078)
		tmp = Float64(Float64(sqrt(0.5) * Float64(F / B)) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -0.21)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 0.078)
		tmp = (sqrt(0.5) * (F / B)) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.21], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.078], N[(N[(N[Sqrt[0.5], $MachinePrecision] * N[(F / B), $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 < -0.209999999999999992

   1. Initial program 61.1%

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

   3. Taylor expanded in F around -inf 98.4%

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

      1. div-inv 98.4%

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

   5. Applied egg-rr 98.4%

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

   if -0.209999999999999992 < F < 0.0779999999999999999

   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. Taylor expanded in B around 0 83.3%

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

   6. Taylor expanded in F around 0 83.0%

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

   7. Taylor expanded in x around 0 82.9%

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

      1. associate-/l* 82.9%

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

   9. Simplified 82.9%

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

       1. sub-neg 82.9%

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

       2. clear-num 82.9%

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

       3. un-div-inv 82.8%

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

       4. distribute-neg-frac 82.8%

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

   11. Applied egg-rr 82.8%

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

       1. +-commutative 82.8%

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

       2. associate-/r/ 82.9%

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

   13. Simplified 82.9%

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

   if 0.0779999999999999999 < F

   1. Initial program 57.0%

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

      1. div-inv 46.8%

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

   4. Applied egg-rr 57.0%

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

   5. Taylor expanded in F around inf 97.8%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.21:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.078:\\ \;\;\;\;\sqrt{0.5} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 64.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F \cdot \sqrt{0.5} - x}{B}\\ t_1 := \frac{F}{\sin B} \cdot \frac{1}{F} - \frac{x}{B}\\ t_2 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.11:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.5 \cdot 10^{-275}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{-60}:\\ \;\;\;\;\frac{\frac{F}{B}}{F} - t\_2\\ \mathbf{elif}\;F \leq 0.18:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 1.225 \cdot 10^{+26}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;F \leq 3.35 \cdot 10^{+39}:\\ \;\;\;\;\frac{1}{B} - t\_2\\ \mathbf{elif}\;F \leq 1.45 \cdot 10^{+108}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{F \cdot B} - t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- (* F (sqrt 0.5)) x) B))
        (t_1 (- (* (/ F (sin B)) (/ 1.0 F)) (/ x B)))
        (t_2 (/ x (tan B))))
   (if (<= F -0.11)
     (- (/ -1.0 (sin B)) (/ x B))
     (if (<= F 2.5e-275)
       t_0
       (if (<= F 1.8e-60)
         (- (/ (/ F B) F) t_2)
         (if (<= F 0.18)
           t_0
           (if (<= F 1.225e+26)
             t_1
             (if (<= F 3.35e+39)
               (- (/ 1.0 B) t_2)
               (if (<= F 1.45e+108) t_1 (- (/ F (* F B)) t_2))))))))))

C

double code(double F, double B, double x) {
	double t_0 = ((F * sqrt(0.5)) - x) / B;
	double t_1 = ((F / sin(B)) * (1.0 / F)) - (x / B);
	double t_2 = x / tan(B);
	double tmp;
	if (F <= -0.11) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 2.5e-275) {
		tmp = t_0;
	} else if (F <= 1.8e-60) {
		tmp = ((F / B) / F) - t_2;
	} else if (F <= 0.18) {
		tmp = t_0;
	} else if (F <= 1.225e+26) {
		tmp = t_1;
	} else if (F <= 3.35e+39) {
		tmp = (1.0 / B) - t_2;
	} else if (F <= 1.45e+108) {
		tmp = t_1;
	} else {
		tmp = (F / (F * B)) - t_2;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = ((f * sqrt(0.5d0)) - x) / b
    t_1 = ((f / sin(b)) * (1.0d0 / f)) - (x / b)
    t_2 = x / tan(b)
    if (f <= (-0.11d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 2.5d-275) then
        tmp = t_0
    else if (f <= 1.8d-60) then
        tmp = ((f / b) / f) - t_2
    else if (f <= 0.18d0) then
        tmp = t_0
    else if (f <= 1.225d+26) then
        tmp = t_1
    else if (f <= 3.35d+39) then
        tmp = (1.0d0 / b) - t_2
    else if (f <= 1.45d+108) then
        tmp = t_1
    else
        tmp = (f / (f * b)) - t_2
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = ((F * Math.sqrt(0.5)) - x) / B;
	double t_1 = ((F / Math.sin(B)) * (1.0 / F)) - (x / B);
	double t_2 = x / Math.tan(B);
	double tmp;
	if (F <= -0.11) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 2.5e-275) {
		tmp = t_0;
	} else if (F <= 1.8e-60) {
		tmp = ((F / B) / F) - t_2;
	} else if (F <= 0.18) {
		tmp = t_0;
	} else if (F <= 1.225e+26) {
		tmp = t_1;
	} else if (F <= 3.35e+39) {
		tmp = (1.0 / B) - t_2;
	} else if (F <= 1.45e+108) {
		tmp = t_1;
	} else {
		tmp = (F / (F * B)) - t_2;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = ((F * math.sqrt(0.5)) - x) / B
	t_1 = ((F / math.sin(B)) * (1.0 / F)) - (x / B)
	t_2 = x / math.tan(B)
	tmp = 0
	if F <= -0.11:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 2.5e-275:
		tmp = t_0
	elif F <= 1.8e-60:
		tmp = ((F / B) / F) - t_2
	elif F <= 0.18:
		tmp = t_0
	elif F <= 1.225e+26:
		tmp = t_1
	elif F <= 3.35e+39:
		tmp = (1.0 / B) - t_2
	elif F <= 1.45e+108:
		tmp = t_1
	else:
		tmp = (F / (F * B)) - t_2
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(Float64(F * sqrt(0.5)) - x) / B)
	t_1 = Float64(Float64(Float64(F / sin(B)) * Float64(1.0 / F)) - Float64(x / B))
	t_2 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -0.11)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 2.5e-275)
		tmp = t_0;
	elseif (F <= 1.8e-60)
		tmp = Float64(Float64(Float64(F / B) / F) - t_2);
	elseif (F <= 0.18)
		tmp = t_0;
	elseif (F <= 1.225e+26)
		tmp = t_1;
	elseif (F <= 3.35e+39)
		tmp = Float64(Float64(1.0 / B) - t_2);
	elseif (F <= 1.45e+108)
		tmp = t_1;
	else
		tmp = Float64(Float64(F / Float64(F * B)) - t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = ((F * sqrt(0.5)) - x) / B;
	t_1 = ((F / sin(B)) * (1.0 / F)) - (x / B);
	t_2 = x / tan(B);
	tmp = 0.0;
	if (F <= -0.11)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 2.5e-275)
		tmp = t_0;
	elseif (F <= 1.8e-60)
		tmp = ((F / B) / F) - t_2;
	elseif (F <= 0.18)
		tmp = t_0;
	elseif (F <= 1.225e+26)
		tmp = t_1;
	elseif (F <= 3.35e+39)
		tmp = (1.0 / B) - t_2;
	elseif (F <= 1.45e+108)
		tmp = t_1;
	else
		tmp = (F / (F * B)) - t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(1.0 / F), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.11], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.5e-275], t$95$0, If[LessEqual[F, 1.8e-60], N[(N[(N[(F / B), $MachinePrecision] / F), $MachinePrecision] - t$95$2), $MachinePrecision], If[LessEqual[F, 0.18], t$95$0, If[LessEqual[F, 1.225e+26], t$95$1, If[LessEqual[F, 3.35e+39], N[(N[(1.0 / B), $MachinePrecision] - t$95$2), $MachinePrecision], If[LessEqual[F, 1.45e+108], t$95$1, N[(N[(F / N[(F * B), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if F < -0.110000000000000001

   1. Initial program 61.1%

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

   3. Taylor expanded in F around -inf 98.4%

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

   4. Taylor expanded in B around 0 75.3%

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

   if -0.110000000000000001 < F < 2.49999999999999992e-275 or 1.8e-60 < F < 0.17999999999999999

   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. Taylor expanded in B around 0 82.4%

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

   6. Taylor expanded in F around 0 82.0%

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

   7. Taylor expanded in x around 0 81.7%

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

      1. associate-/l* 81.7%

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

   9. Simplified 81.7%

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

   10. Taylor expanded in B around 0 62.5%

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

   if 2.49999999999999992e-275 < F < 1.8e-60

   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. Taylor expanded in B around 0 85.3%

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

   6. Taylor expanded in F around inf 49.4%

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

      1. *-commutative 49.4%

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

   8. Simplified 49.4%

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

      1. un-div-inv 49.4%

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

      2. *-commutative 49.4%

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

      3. associate-/r* 61.3%

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

   10. Applied egg-rr 61.3%

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

   if 0.17999999999999999 < F < 1.22499999999999993e26 or 3.34999999999999975e39 < F < 1.45000000000000004e108

   1. Initial program 99.5%

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

   3. Taylor expanded in F around inf 92.8%

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

   4. Taylor expanded in B around 0 82.2%

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

   if 1.22499999999999993e26 < F < 3.34999999999999975e39

   1. Initial program 36.4%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in B around 0 99.5%

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

   6. Taylor expanded in F around inf 99.5%

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

      1. *-commutative 99.5%

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

   8. Simplified 99.5%

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

   9. Taylor expanded in F around 0 100.0%

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

   if 1.45000000000000004e108 < F

   1. Initial program 38.6%

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

   3. Simplified 59.6%

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

   5. Taylor expanded in B around 0 57.4%

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

   6. Taylor expanded in F around inf 68.5%

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

      1. *-commutative 68.5%

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

   8. Simplified 68.5%

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

      1. un-div-inv 68.6%

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

   10. Applied egg-rr 68.6%

       \[\leadsto \color{blue}{\frac{F}{F \cdot B}} - \frac{x}{\tan B} \]
3. Recombined 6 regimes into one program.

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.11:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.5 \cdot 10^{-275}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{-60}:\\ \;\;\;\;\frac{\frac{F}{B}}{F} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.18:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{elif}\;F \leq 1.225 \cdot 10^{+26}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \frac{1}{F} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.35 \cdot 10^{+39}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.45 \cdot 10^{+108}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \frac{1}{F} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{F \cdot B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 64.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ t_1 := \frac{F}{\sin B} \cdot \frac{1}{F} - \frac{x}{B}\\ \mathbf{if}\;F \leq -0.038:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 8.5 \cdot 10^{-279}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{elif}\;F \leq 1.46 \cdot 10^{-61}:\\ \;\;\;\;\frac{\frac{F}{B}}{F} - t\_0\\ \mathbf{elif}\;F \leq 0.4:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{elif}\;F \leq 1.225 \cdot 10^{+26}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;F \leq 1.08 \cdot 10^{+39}:\\ \;\;\;\;\frac{1}{B} - t\_0\\ \mathbf{elif}\;F \leq 4.3 \cdot 10^{+107}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{F \cdot B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))) (t_1 (- (* (/ F (sin B)) (/ 1.0 F)) (/ x B))))
   (if (<= F -0.038)
     (- (/ -1.0 (sin B)) (/ x B))
     (if (<= F 8.5e-279)
       (/ (- (* F (sqrt 0.5)) x) B)
       (if (<= F 1.46e-61)
         (- (/ (/ F B) F) t_0)
         (if (<= F 0.4)
           (/ (- (* F (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))) x) B)
           (if (<= F 1.225e+26)
             t_1
             (if (<= F 1.08e+39)
               (- (/ 1.0 B) t_0)
               (if (<= F 4.3e+107) t_1 (- (/ F (* F B)) t_0))))))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double t_1 = ((F / sin(B)) * (1.0 / F)) - (x / B);
	double tmp;
	if (F <= -0.038) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 8.5e-279) {
		tmp = ((F * sqrt(0.5)) - x) / B;
	} else if (F <= 1.46e-61) {
		tmp = ((F / B) / F) - t_0;
	} else if (F <= 0.4) {
		tmp = ((F * sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B;
	} else if (F <= 1.225e+26) {
		tmp = t_1;
	} else if (F <= 1.08e+39) {
		tmp = (1.0 / B) - t_0;
	} else if (F <= 4.3e+107) {
		tmp = t_1;
	} else {
		tmp = (F / (F * 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) :: t_1
    real(8) :: tmp
    t_0 = x / tan(b)
    t_1 = ((f / sin(b)) * (1.0d0 / f)) - (x / b)
    if (f <= (-0.038d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 8.5d-279) then
        tmp = ((f * sqrt(0.5d0)) - x) / b
    else if (f <= 1.46d-61) then
        tmp = ((f / b) / f) - t_0
    else if (f <= 0.4d0) then
        tmp = ((f * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))) - x) / b
    else if (f <= 1.225d+26) then
        tmp = t_1
    else if (f <= 1.08d+39) then
        tmp = (1.0d0 / b) - t_0
    else if (f <= 4.3d+107) then
        tmp = t_1
    else
        tmp = (f / (f * 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 t_1 = ((F / Math.sin(B)) * (1.0 / F)) - (x / B);
	double tmp;
	if (F <= -0.038) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 8.5e-279) {
		tmp = ((F * Math.sqrt(0.5)) - x) / B;
	} else if (F <= 1.46e-61) {
		tmp = ((F / B) / F) - t_0;
	} else if (F <= 0.4) {
		tmp = ((F * Math.sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B;
	} else if (F <= 1.225e+26) {
		tmp = t_1;
	} else if (F <= 1.08e+39) {
		tmp = (1.0 / B) - t_0;
	} else if (F <= 4.3e+107) {
		tmp = t_1;
	} else {
		tmp = (F / (F * B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	t_1 = ((F / math.sin(B)) * (1.0 / F)) - (x / B)
	tmp = 0
	if F <= -0.038:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 8.5e-279:
		tmp = ((F * math.sqrt(0.5)) - x) / B
	elif F <= 1.46e-61:
		tmp = ((F / B) / F) - t_0
	elif F <= 0.4:
		tmp = ((F * math.sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B
	elif F <= 1.225e+26:
		tmp = t_1
	elif F <= 1.08e+39:
		tmp = (1.0 / B) - t_0
	elif F <= 4.3e+107:
		tmp = t_1
	else:
		tmp = (F / (F * B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	t_1 = Float64(Float64(Float64(F / sin(B)) * Float64(1.0 / F)) - Float64(x / B))
	tmp = 0.0
	if (F <= -0.038)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 8.5e-279)
		tmp = Float64(Float64(Float64(F * sqrt(0.5)) - x) / B);
	elseif (F <= 1.46e-61)
		tmp = Float64(Float64(Float64(F / B) / F) - t_0);
	elseif (F <= 0.4)
		tmp = Float64(Float64(Float64(F * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))) - x) / B);
	elseif (F <= 1.225e+26)
		tmp = t_1;
	elseif (F <= 1.08e+39)
		tmp = Float64(Float64(1.0 / B) - t_0);
	elseif (F <= 4.3e+107)
		tmp = t_1;
	else
		tmp = Float64(Float64(F / Float64(F * B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	t_1 = ((F / sin(B)) * (1.0 / F)) - (x / B);
	tmp = 0.0;
	if (F <= -0.038)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 8.5e-279)
		tmp = ((F * sqrt(0.5)) - x) / B;
	elseif (F <= 1.46e-61)
		tmp = ((F / B) / F) - t_0;
	elseif (F <= 0.4)
		tmp = ((F * sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B;
	elseif (F <= 1.225e+26)
		tmp = t_1;
	elseif (F <= 1.08e+39)
		tmp = (1.0 / B) - t_0;
	elseif (F <= 4.3e+107)
		tmp = t_1;
	else
		tmp = (F / (F * B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(1.0 / F), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.038], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 8.5e-279], N[(N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.46e-61], N[(N[(N[(F / B), $MachinePrecision] / F), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.4], 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, 1.225e+26], t$95$1, If[LessEqual[F, 1.08e+39], N[(N[(1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 4.3e+107], t$95$1, N[(N[(F / N[(F * B), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if F < -0.0379999999999999991

   1. Initial program 61.1%

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

   3. Taylor expanded in F around -inf 98.4%

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

   4. Taylor expanded in B around 0 75.3%

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

   if -0.0379999999999999991 < F < 8.5000000000000002e-279

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in B around 0 82.8%

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

   6. Taylor expanded in F around 0 82.7%

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

   7. Taylor expanded in x around 0 82.7%

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

      1. associate-/l* 82.7%

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

   9. Simplified 82.7%

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

   10. Taylor expanded in B around 0 63.1%

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

   if 8.5000000000000002e-279 < F < 1.46e-61

   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. Taylor expanded in B around 0 85.3%

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

   6. Taylor expanded in F around inf 49.4%

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

      1. *-commutative 49.4%

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

   8. Simplified 49.4%

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

      1. un-div-inv 49.4%

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

      2. *-commutative 49.4%

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

      3. associate-/r* 61.3%

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

   10. Applied egg-rr 61.3%

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

   if 1.46e-61 < F < 0.40000000000000002

   1. Initial program 99.1%

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

   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 B around 0 79.3%

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

   6. Taylor expanded in F around 0 77.1%

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

   7. Taylor expanded in B around 0 60.8%

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

   if 0.40000000000000002 < F < 1.22499999999999993e26 or 1.07999999999999998e39 < F < 4.3e107

   1. Initial program 99.5%

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

   3. Taylor expanded in F around inf 92.8%

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

   4. Taylor expanded in B around 0 82.2%

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

   if 1.22499999999999993e26 < F < 1.07999999999999998e39

   1. Initial program 36.4%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in B around 0 99.5%

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

   6. Taylor expanded in F around inf 99.5%

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

      1. *-commutative 99.5%

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

   8. Simplified 99.5%

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

   9. Taylor expanded in F around 0 100.0%

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

   if 4.3e107 < F

   1. Initial program 38.6%

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

   3. Simplified 59.6%

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

   5. Taylor expanded in B around 0 57.4%

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

   6. Taylor expanded in F around inf 68.5%

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

      1. *-commutative 68.5%

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

   8. Simplified 68.5%

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

      1. un-div-inv 68.6%

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

   10. Applied egg-rr 68.6%

       \[\leadsto \color{blue}{\frac{F}{F \cdot B}} - \frac{x}{\tan B} \]
3. Recombined 7 regimes into one program.

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.038:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 8.5 \cdot 10^{-279}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{elif}\;F \leq 1.46 \cdot 10^{-61}:\\ \;\;\;\;\frac{\frac{F}{B}}{F} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.4:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{elif}\;F \leq 1.225 \cdot 10^{+26}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \frac{1}{F} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.08 \cdot 10^{+39}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 4.3 \cdot 10^{+107}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \frac{1}{F} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{F \cdot B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 55.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-163} \lor \neg \left(x \leq 1.65 \cdot 10^{-94}\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (or (<= x -2e-163) (not (<= x 1.65e-94)))
   (- (/ 1.0 B) (/ x (tan B)))
   (/ (* F (sqrt 0.5)) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if ((x <= -2e-163) || !(x <= 1.65e-94)) {
		tmp = (1.0 / B) - (x / tan(B));
	} else {
		tmp = (F * sqrt(0.5)) / 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 ((x <= (-2d-163)) .or. (.not. (x <= 1.65d-94))) then
        tmp = (1.0d0 / b) - (x / tan(b))
    else
        tmp = (f * sqrt(0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if ((x <= -2e-163) || !(x <= 1.65e-94)) {
		tmp = (1.0 / B) - (x / Math.tan(B));
	} else {
		tmp = (F * Math.sqrt(0.5)) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if (x <= -2e-163) or not (x <= 1.65e-94):
		tmp = (1.0 / B) - (x / math.tan(B))
	else:
		tmp = (F * math.sqrt(0.5)) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if ((x <= -2e-163) || !(x <= 1.65e-94))
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(F * sqrt(0.5)) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if ((x <= -2e-163) || ~((x <= 1.65e-94)))
		tmp = (1.0 / B) - (x / tan(B));
	else
		tmp = (F * sqrt(0.5)) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[Or[LessEqual[x, -2e-163], N[Not[LessEqual[x, 1.65e-94]], $MachinePrecision]], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.99999999999999985e-163 or 1.6500000000000001e-94 < x

   1. Initial program 78.7%

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

   3. Simplified 89.3%

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

   5. Taylor expanded in B around 0 83.5%

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

   6. Taylor expanded in F around inf 67.2%

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

      1. *-commutative 67.2%

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

   8. Simplified 67.2%

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

   9. Taylor expanded in F around 0 72.1%

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

   if -1.99999999999999985e-163 < x < 1.6500000000000001e-94

   1. Initial program 74.3%

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

   3. Simplified 78.6%

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

   5. Taylor expanded in B around 0 44.8%

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

   6. Taylor expanded in F around 0 33.7%

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

   7. Taylor expanded in x around 0 33.7%

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

      1. associate-/l* 33.7%

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

   9. Simplified 33.7%

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

   10. Taylor expanded in F around inf 25.1%

       \[\leadsto \color{blue}{\frac{F \cdot \sqrt{0.5}}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-163} \lor \neg \left(x \leq 1.65 \cdot 10^{-94}\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 58.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-93} \lor \neg \left(x \leq 0.000122\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (or (<= x -5.8e-93) (not (<= x 0.000122)))
   (- (/ 1.0 B) (/ x (tan B)))
   (/ (- (* F (sqrt 0.5)) x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if ((x <= -5.8e-93) || !(x <= 0.000122)) {
		tmp = (1.0 / B) - (x / tan(B));
	} else {
		tmp = ((F * sqrt(0.5)) - 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 ((x <= (-5.8d-93)) .or. (.not. (x <= 0.000122d0))) then
        tmp = (1.0d0 / b) - (x / tan(b))
    else
        tmp = ((f * sqrt(0.5d0)) - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if ((x <= -5.8e-93) || !(x <= 0.000122)) {
		tmp = (1.0 / B) - (x / Math.tan(B));
	} else {
		tmp = ((F * Math.sqrt(0.5)) - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if (x <= -5.8e-93) or not (x <= 0.000122):
		tmp = (1.0 / B) - (x / math.tan(B))
	else:
		tmp = ((F * math.sqrt(0.5)) - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if ((x <= -5.8e-93) || !(x <= 0.000122))
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(Float64(F * sqrt(0.5)) - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if ((x <= -5.8e-93) || ~((x <= 0.000122)))
		tmp = (1.0 / B) - (x / tan(B));
	else
		tmp = ((F * sqrt(0.5)) - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[Or[LessEqual[x, -5.8e-93], N[Not[LessEqual[x, 0.000122]], $MachinePrecision]], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.7999999999999997e-93 or 1.21999999999999997e-4 < x

   1. Initial program 80.9%

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

   3. Simplified 94.3%

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

   5. Taylor expanded in B around 0 91.7%

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

   6. Taylor expanded in F around inf 83.3%

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

      1. *-commutative 83.3%

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

   8. Simplified 83.3%

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

   9. Taylor expanded in F around 0 89.9%

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

   if -5.7999999999999997e-93 < x < 1.21999999999999997e-4

   1. Initial program 73.8%

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

   3. Simplified 77.5%

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

   5. Taylor expanded in B around 0 49.7%

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

   6. Taylor expanded in F around 0 38.8%

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

   7. Taylor expanded in x around 0 38.6%

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

      1. associate-/l* 38.6%

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

   9. Simplified 38.6%

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

   10. Taylor expanded in B around 0 32.1%

       \[\leadsto \color{blue}{\frac{F \cdot \sqrt{0.5} - x}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-93} \lor \neg \left(x \leq 0.000122\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 63.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.26:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 9.5 \cdot 10^{-197}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.26)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 9.5e-197)
     (/ (- (* F (sqrt 0.5)) x) B)
     (- (/ 1.0 B) (/ x (tan B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.26) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 9.5e-197) {
		tmp = ((F * sqrt(0.5)) - x) / B;
	} else {
		tmp = (1.0 / B) - (x / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-0.26d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 9.5d-197) then
        tmp = ((f * sqrt(0.5d0)) - x) / b
    else
        tmp = (1.0d0 / b) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.26) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 9.5e-197) {
		tmp = ((F * Math.sqrt(0.5)) - x) / B;
	} else {
		tmp = (1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -0.26:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 9.5e-197:
		tmp = ((F * math.sqrt(0.5)) - x) / B
	else:
		tmp = (1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.26)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 9.5e-197)
		tmp = Float64(Float64(Float64(F * sqrt(0.5)) - x) / B);
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -0.26)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 9.5e-197)
		tmp = ((F * sqrt(0.5)) - x) / B;
	else
		tmp = (1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.26], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 9.5e-197], N[(N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.26000000000000001

   1. Initial program 61.1%

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

   3. Taylor expanded in F around -inf 98.4%

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

   4. Taylor expanded in B around 0 75.3%

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

   if -0.26000000000000001 < F < 9.5000000000000003e-197

   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 B around 0 83.7%

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

   6. Taylor expanded in F around 0 83.6%

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

   7. Taylor expanded in x around 0 83.6%

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

      1. associate-/l* 83.7%

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

   9. Simplified 83.7%

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

   10. Taylor expanded in B around 0 62.1%

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

   if 9.5000000000000003e-197 < F

   1. Initial program 72.5%

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

   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 B around 0 67.1%

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

   6. Taylor expanded in F around inf 58.3%

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

      1. *-commutative 58.3%

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

   8. Simplified 58.3%

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

   9. Taylor expanded in F around 0 60.3%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.26:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 9.5 \cdot 10^{-197}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 43.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.18 \cdot 10^{-48}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq -5.6 \cdot 10^{-144}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B}\\ \mathbf{elif}\;F \leq 4.8 \cdot 10^{-51}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.18e-48)
   (/ (- -1.0 x) B)
   (if (<= F -5.6e-144)
     (* F (/ (sqrt 0.5) B))
     (if (<= F 4.8e-51) (/ (- x) B) (/ (- 1.0 x) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.18e-48) {
		tmp = (-1.0 - x) / B;
	} else if (F <= -5.6e-144) {
		tmp = F * (sqrt(0.5) / B);
	} else if (F <= 4.8e-51) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.18d-48)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= (-5.6d-144)) then
        tmp = f * (sqrt(0.5d0) / b)
    else if (f <= 4.8d-51) then
        tmp = -x / b
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.18e-48) {
		tmp = (-1.0 - x) / B;
	} else if (F <= -5.6e-144) {
		tmp = F * (Math.sqrt(0.5) / B);
	} else if (F <= 4.8e-51) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.18e-48:
		tmp = (-1.0 - x) / B
	elif F <= -5.6e-144:
		tmp = F * (math.sqrt(0.5) / B)
	elif F <= 4.8e-51:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.18e-48)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= -5.6e-144)
		tmp = Float64(F * Float64(sqrt(0.5) / B));
	elseif (F <= 4.8e-51)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.18e-48)
		tmp = (-1.0 - x) / B;
	elseif (F <= -5.6e-144)
		tmp = F * (sqrt(0.5) / B);
	elseif (F <= 4.8e-51)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.18e-48], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, -5.6e-144], N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.8e-51], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.18000000000000007e-48

   1. Initial program 67.7%

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

   3. Taylor expanded in F around -inf 89.4%

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

   4. Taylor expanded in B around 0 47.3%

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

      1. mul-1-neg 47.3%

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

      2. distribute-neg-frac2 47.3%

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

   6. Simplified 47.3%

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

   if -1.18000000000000007e-48 < F < -5.59999999999999995e-144

   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 99.6%

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

   5. Taylor expanded in B around 0 82.5%

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

   6. Taylor expanded in F around 0 82.5%

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

   7. Taylor expanded in x around 0 82.5%

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

      1. associate-/l* 82.5%

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

   9. Simplified 82.5%

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

   10. Taylor expanded in F around inf 52.8%

       \[\leadsto \color{blue}{\frac{F \cdot \sqrt{0.5}}{B}} \]
   11. Step-by-step derivation

       1. associate-/l* 52.8%

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

   12. Simplified 52.8%

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

   if -5.59999999999999995e-144 < F < 4.8e-51

   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 F around -inf 36.4%

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

   4. Taylor expanded in B around 0 22.3%

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

      1. mul-1-neg 22.3%

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

      2. distribute-neg-frac2 22.3%

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

   6. Simplified 22.3%

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

   7. Taylor expanded in x around inf 44.5%

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

      1. associate-*r/ 44.5%

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

      2. neg-mul-1 44.5%

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

   9. Simplified 44.5%

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

   if 4.8e-51 < F

   1. Initial program 62.0%

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

   3. Simplified 76.7%

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

   5. Taylor expanded in B around 0 60.7%

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

   6. Taylor expanded in F around inf 59.8%

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

      1. *-commutative 59.8%

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

   8. Simplified 59.8%

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

   9. Taylor expanded in B around 0 44.1%

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

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.18 \cdot 10^{-48}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq -5.6 \cdot 10^{-144}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B}\\ \mathbf{elif}\;F \leq 4.8 \cdot 10^{-51}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 37.5% accurate, 21.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -5.2:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq 1.45 \cdot 10^{-55}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -5.2) (/ -1.0 B) (if (<= F 1.45e-55) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -5.2) {
		tmp = -1.0 / B;
	} else if (F <= 1.45e-55) {
		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 <= (-5.2d0)) then
        tmp = (-1.0d0) / b
    else if (f <= 1.45d-55) 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 <= -5.2) {
		tmp = -1.0 / B;
	} else if (F <= 1.45e-55) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -5.2:
		tmp = -1.0 / B
	elif F <= 1.45e-55:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -5.2)
		tmp = Float64(-1.0 / B);
	elseif (F <= 1.45e-55)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -5.2)
		tmp = -1.0 / B;
	elseif (F <= 1.45e-55)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -5.2], N[(-1.0 / B), $MachinePrecision], If[LessEqual[F, 1.45e-55], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.20000000000000018

   1. Initial program 61.1%

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

   3. Taylor expanded in F around -inf 98.4%

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

   4. Taylor expanded in B around 0 49.6%

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

      1. mul-1-neg 49.6%

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

      2. distribute-neg-frac2 49.6%

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

   6. Simplified 49.6%

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

   7. Taylor expanded in x around 0 30.6%

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

   if -5.20000000000000018 < F < 1.45e-55

   1. Initial program 99.6%

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

   3. Taylor expanded in F around -inf 34.5%

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

   4. Taylor expanded in B around 0 23.5%

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

      1. mul-1-neg 23.5%

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

      2. distribute-neg-frac2 23.5%

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

   6. Simplified 23.5%

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

   7. Taylor expanded in x around inf 39.2%

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

      1. associate-*r/ 39.2%

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

      2. neg-mul-1 39.2%

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

   9. Simplified 39.2%

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

   if 1.45e-55 < F

   1. Initial program 62.0%

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

   3. Simplified 76.7%

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

   5. Taylor expanded in B around 0 60.7%

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

   6. Taylor expanded in F around inf 59.8%

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

      1. *-commutative 59.8%

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

   8. Simplified 59.8%

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

   9. Taylor expanded in B around 0 44.1%

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

4. Final simplification 38.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5.2:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq 1.45 \cdot 10^{-55}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 43.8% accurate, 21.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.55 \cdot 10^{-71}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.1 \cdot 10^{-56}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.55e-71)
   (/ (- -1.0 x) B)
   (if (<= F 1.1e-56) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.55e-71) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.1e-56) {
		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 <= (-3.55d-71)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 1.1d-56) 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 <= -3.55e-71) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.1e-56) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.55e-71:
		tmp = (-1.0 - x) / B
	elif F <= 1.1e-56:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.55e-71)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 1.1e-56)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -3.55e-71)
		tmp = (-1.0 - x) / B;
	elseif (F <= 1.1e-56)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.55e-71], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.1e-56], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.55000000000000004e-71

   1. Initial program 69.1%

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

   3. Taylor expanded in F around -inf 86.0%

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

   4. Taylor expanded in B around 0 45.6%

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

      1. mul-1-neg 45.6%

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

      2. distribute-neg-frac2 45.6%

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

   6. Simplified 45.6%

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

   if -3.55000000000000004e-71 < F < 1.10000000000000002e-56

   1. Initial program 99.6%

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

   3. Taylor expanded in F around -inf 33.5%

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

   4. Taylor expanded in B around 0 21.9%

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

      1. mul-1-neg 21.9%

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

      2. distribute-neg-frac2 21.9%

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

   6. Simplified 21.9%

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

   7. Taylor expanded in x around inf 41.8%

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

      1. associate-*r/ 41.8%

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

      2. neg-mul-1 41.8%

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

   9. Simplified 41.8%

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

   if 1.10000000000000002e-56 < F

   1. Initial program 62.0%

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

   3. Simplified 76.7%

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

   5. Taylor expanded in B around 0 60.7%

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

   6. Taylor expanded in F around inf 59.8%

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

      1. *-commutative 59.8%

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

   8. Simplified 59.8%

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

   9. Taylor expanded in B around 0 44.1%

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

4. Final simplification 43.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.55 \cdot 10^{-71}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.1 \cdot 10^{-56}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 31.4% accurate, 23.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-175} \lor \neg \left(x \leq 3.4 \cdot 10^{-68}\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (or (<= x -5.5e-175) (not (<= x 3.4e-68))) (/ (- x) B) (/ -1.0 B)))

C

double code(double F, double B, double x) {
	double tmp;
	if ((x <= -5.5e-175) || !(x <= 3.4e-68)) {
		tmp = -x / B;
	} else {
		tmp = -1.0 / 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 ((x <= (-5.5d-175)) .or. (.not. (x <= 3.4d-68))) then
        tmp = -x / b
    else
        tmp = (-1.0d0) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if ((x <= -5.5e-175) || !(x <= 3.4e-68)) {
		tmp = -x / B;
	} else {
		tmp = -1.0 / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if (x <= -5.5e-175) or not (x <= 3.4e-68):
		tmp = -x / B
	else:
		tmp = -1.0 / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if ((x <= -5.5e-175) || !(x <= 3.4e-68))
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(-1.0 / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if ((x <= -5.5e-175) || ~((x <= 3.4e-68)))
		tmp = -x / B;
	else
		tmp = -1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[Or[LessEqual[x, -5.5e-175], N[Not[LessEqual[x, 3.4e-68]], $MachinePrecision]], N[((-x) / B), $MachinePrecision], N[(-1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.50000000000000054e-175 or 3.40000000000000018e-68 < x

   1. Initial program 79.0%

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

   3. Taylor expanded in F around -inf 69.8%

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

   4. Taylor expanded in B around 0 41.0%

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

      1. mul-1-neg 41.0%

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

      2. distribute-neg-frac2 41.0%

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

   6. Simplified 41.0%

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

   7. Taylor expanded in x around inf 45.5%

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

      1. associate-*r/ 45.5%

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

      2. neg-mul-1 45.5%

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

   9. Simplified 45.5%

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

   if -5.50000000000000054e-175 < x < 3.40000000000000018e-68

   1. Initial program 74.2%

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

   3. Taylor expanded in F around -inf 34.7%

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

   4. Taylor expanded in B around 0 20.0%

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

      1. mul-1-neg 20.0%

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

      2. distribute-neg-frac2 20.0%

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

   6. Simplified 20.0%

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

   7. Taylor expanded in x around 0 20.0%

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

4. Final simplification 35.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-175} \lor \neg \left(x \leq 3.4 \cdot 10^{-68}\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 10.5% 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 77.1%

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

3. Taylor expanded in F around -inf 56.2%

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

4. Taylor expanded in B around 0 32.9%

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

   1. mul-1-neg 32.9%

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

   2. distribute-neg-frac2 32.9%

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

6. Simplified 32.9%

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

7. Taylor expanded in x around 0 12.0%

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

8. Final simplification 12.0%

   \[\leadsto \frac{-1}{B} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024073 
(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))))))