VandenBroeck and Keller, Equation (23)

Percentage Accurate: 77.5% → 99.6%

Time: 20.1s

Alternatives: 23

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.5% 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: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.12 \cdot 10^{+45}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 165000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.12e+45)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 165000000.0)
       (- (* (/ 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 <= -1.12e+45) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 165000000.0) {
		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 <= (-1.12d+45)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 165000000.0d0) 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 <= -1.12e+45) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 165000000.0) {
		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 <= -1.12e+45:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 165000000.0:
		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 <= -1.12e+45)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 165000000.0)
		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 <= -1.12e+45)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 165000000.0)
		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, -1.12e+45], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 165000000.0], 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 < -1.12e45

   1. Initial program 55.2%

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

   2. Taylor expanded in F around -inf 99.7%

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

      1. +-commutative 99.7%

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

      2. unsub-neg 99.7%

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

      3. un-div-inv 99.8%

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

   4. Applied egg-rr 99.8%

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

   if -1.12e45 < F < 1.65e8

   1. Initial program 99.4%

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

      1. div-inv 99.6%

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

      2. expm1-log1p-u 73.0%

         \[\leadsto \left(-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\tan B}\right)\right)}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      3. expm1-udef 53.9%

         \[\leadsto \left(-\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\tan B}\right)} - 1\right)}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Applied egg-rr 53.9%

      \[\leadsto \left(-\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\tan B}\right)} - 1\right)}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   4. Step-by-step derivation

      1. expm1-def 73.0%

         \[\leadsto \left(-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\tan B}\right)\right)}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. expm1-log1p 99.6%

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

   5. Simplified 99.6%

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

   if 1.65e8 < F

   1. Initial program 54.7%

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

   2. Taylor expanded in B around inf 54.6%

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

   3. Taylor expanded in F around inf 99.7%

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

      1. +-commutative 99.7%

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

      2. unsub-neg 99.7%

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

      3. associate-*r/ 99.8%

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

      4. associate-/l* 99.7%

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

      5. quot-tan 99.8%

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

   5. Applied egg-rr 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.12 \cdot 10^{+45}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 165000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 2: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -22000000000:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{-7}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -22000000000.0)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.8e-7)
       (+
        (* x (/ -1.0 (tan B)))
        (* (/ F (sin B)) (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))))
       (- (/ 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 <= -22000000000.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.8e-7) {
		tmp = (x * (-1.0 / tan(B))) + ((F / sin(B)) * sqrt((1.0 / (2.0 + (x * 2.0)))));
	} 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 <= (-22000000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.8d-7) then
        tmp = (x * ((-1.0d0) / tan(b))) + ((f / sin(b)) * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))))
    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 <= -22000000000.0) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.8e-7) {
		tmp = (x * (-1.0 / Math.tan(B))) + ((F / Math.sin(B)) * Math.sqrt((1.0 / (2.0 + (x * 2.0)))));
	} 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 <= -22000000000.0:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.8e-7:
		tmp = (x * (-1.0 / math.tan(B))) + ((F / math.sin(B)) * math.sqrt((1.0 / (2.0 + (x * 2.0)))))
	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 <= -22000000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.8e-7)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(Float64(F / sin(B)) * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))));
	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 <= -22000000000.0)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.8e-7)
		tmp = (x * (-1.0 / tan(B))) + ((F / sin(B)) * sqrt((1.0 / (2.0 + (x * 2.0)))));
	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, -22000000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.8e-7], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.2e10

   1. Initial program 57.8%

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

   2. Taylor expanded in F around -inf 99.7%

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

      1. +-commutative 99.7%

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

      2. unsub-neg 99.7%

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

      3. un-div-inv 99.8%

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

   4. Applied egg-rr 99.8%

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

   if -2.2e10 < F < 1.79999999999999997e-7

   1. Initial program 99.4%

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

   2. Taylor expanded in F around 0 99.4%

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

   if 1.79999999999999997e-7 < F

   1. Initial program 56.9%

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

   2. Taylor expanded in B around inf 56.8%

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

   3. Taylor expanded in F around inf 98.6%

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

      1. +-commutative 98.6%

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

      2. unsub-neg 98.6%

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

      3. associate-*r/ 98.6%

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

      4. associate-/l* 98.6%

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

      5. quot-tan 98.6%

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

   5. Applied egg-rr 98.6%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -22000000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{-7}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 3: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -22000000000:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{-7}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\frac{\sin B}{\sqrt{\frac{1}{2 + x \cdot 2}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -22000000000.0)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.8e-7)
       (+
        (* x (/ -1.0 (tan B)))
        (/ F (/ (sin B) (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))))))
       (- (/ 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 <= -22000000000.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.8e-7) {
		tmp = (x * (-1.0 / tan(B))) + (F / (sin(B) / sqrt((1.0 / (2.0 + (x * 2.0))))));
	} 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 <= (-22000000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.8d-7) then
        tmp = (x * ((-1.0d0) / tan(b))) + (f / (sin(b) / sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))))
    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 <= -22000000000.0) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.8e-7) {
		tmp = (x * (-1.0 / Math.tan(B))) + (F / (Math.sin(B) / Math.sqrt((1.0 / (2.0 + (x * 2.0))))));
	} 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 <= -22000000000.0:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.8e-7:
		tmp = (x * (-1.0 / math.tan(B))) + (F / (math.sin(B) / math.sqrt((1.0 / (2.0 + (x * 2.0))))))
	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 <= -22000000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.8e-7)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(F / Float64(sin(B) / sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0)))))));
	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 <= -22000000000.0)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.8e-7)
		tmp = (x * (-1.0 / tan(B))) + (F / (sin(B) / sqrt((1.0 / (2.0 + (x * 2.0))))));
	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, -22000000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.8e-7], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(F / N[(N[Sin[B], $MachinePrecision] / N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.2e10

   1. Initial program 57.8%

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

   2. Taylor expanded in F around -inf 99.7%

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

      1. +-commutative 99.7%

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

      2. unsub-neg 99.7%

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

      3. un-div-inv 99.8%

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

   4. Applied egg-rr 99.8%

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

   if -2.2e10 < F < 1.79999999999999997e-7

   1. Initial program 99.4%

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

      1. associate-*l/ 99.4%

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

      2. associate-/l* 99.4%

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

      3. +-commutative 99.4%

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

      4. fma-def 99.4%

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

      5. fma-def 99.4%

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

      6. metadata-eval 99.4%

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

      7. metadata-eval 99.4%

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

   3. Applied egg-rr 99.4%

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

   4. Taylor expanded in F around 0 99.4%

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

   if 1.79999999999999997e-7 < F

   1. Initial program 56.9%

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

   2. Taylor expanded in B around inf 56.8%

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

   3. Taylor expanded in F around inf 98.6%

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

      1. +-commutative 98.6%

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

      2. unsub-neg 98.6%

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

      3. associate-*r/ 98.6%

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

      4. associate-/l* 98.6%

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

      5. quot-tan 98.6%

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

   5. Applied egg-rr 98.6%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -22000000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{-7}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\frac{\sin B}{\sqrt{\frac{1}{2 + x \cdot 2}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 4: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -22000000000:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{-7}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B \cdot \sqrt{2 + x \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -22000000000.0)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.8e-7)
       (+ (* x (/ -1.0 (tan B))) (/ F (* (sin B) (sqrt (+ 2.0 (* x 2.0))))))
       (- (/ 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 <= -22000000000.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.8e-7) {
		tmp = (x * (-1.0 / tan(B))) + (F / (sin(B) * sqrt((2.0 + (x * 2.0)))));
	} 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 <= (-22000000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.8d-7) then
        tmp = (x * ((-1.0d0) / tan(b))) + (f / (sin(b) * sqrt((2.0d0 + (x * 2.0d0)))))
    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 <= -22000000000.0) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.8e-7) {
		tmp = (x * (-1.0 / Math.tan(B))) + (F / (Math.sin(B) * Math.sqrt((2.0 + (x * 2.0)))));
	} 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 <= -22000000000.0:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.8e-7:
		tmp = (x * (-1.0 / math.tan(B))) + (F / (math.sin(B) * math.sqrt((2.0 + (x * 2.0)))))
	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 <= -22000000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.8e-7)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(F / Float64(sin(B) * sqrt(Float64(2.0 + Float64(x * 2.0))))));
	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 <= -22000000000.0)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.8e-7)
		tmp = (x * (-1.0 / tan(B))) + (F / (sin(B) * sqrt((2.0 + (x * 2.0)))));
	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, -22000000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.8e-7], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.2e10

   1. Initial program 57.8%

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

   2. Taylor expanded in F around -inf 99.7%

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

      1. +-commutative 99.7%

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

      2. unsub-neg 99.7%

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

      3. un-div-inv 99.8%

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

   4. Applied egg-rr 99.8%

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

   if -2.2e10 < F < 1.79999999999999997e-7

   1. Initial program 99.4%

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

      1. associate-*l/ 99.4%

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

      2. associate-/l* 99.4%

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

      3. +-commutative 99.4%

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

      4. fma-def 99.4%

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

      5. fma-def 99.4%

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

      6. metadata-eval 99.4%

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

      7. metadata-eval 99.4%

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

   3. Applied egg-rr 99.4%

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

   4. Taylor expanded in F around 0 99.4%

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

   if 1.79999999999999997e-7 < F

   1. Initial program 56.9%

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

   2. Taylor expanded in B around inf 56.8%

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

   3. Taylor expanded in F around inf 98.6%

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

      1. +-commutative 98.6%

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

      2. unsub-neg 98.6%

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

      3. associate-*r/ 98.6%

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

      4. associate-/l* 98.6%

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

      5. quot-tan 98.6%

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

   5. Applied egg-rr 98.6%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -22000000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{-7}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B \cdot \sqrt{2 + x \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 5: 92.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -22000000000:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 6.5 \cdot 10^{-151}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B}\\ \mathbf{elif}\;F \leq 86000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -22000000000.0)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 6.5e-151)
       (+ (* x (/ -1.0 (tan B))) (* (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))) (/ F B)))
       (if (<= F 86000.0)
         (- (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5)) (/ x B))
         (- (/ 1.0 (sin B)) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -22000000000.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 6.5e-151) {
		tmp = (x * (-1.0 / tan(B))) + (sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B));
	} else if (F <= 86000.0) {
		tmp = ((F / sin(B)) * pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - (x / B);
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-22000000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 6.5d-151) then
        tmp = (x * ((-1.0d0) / tan(b))) + (sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))) * (f / b))
    else if (f <= 86000.0d0) then
        tmp = ((f / sin(b)) * ((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0))) - (x / b)
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -22000000000.0) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 6.5e-151) {
		tmp = (x * (-1.0 / Math.tan(B))) + (Math.sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B));
	} else if (F <= 86000.0) {
		tmp = ((F / Math.sin(B)) * Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - (x / B);
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -22000000000.0:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 6.5e-151:
		tmp = (x * (-1.0 / math.tan(B))) + (math.sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B))
	elif F <= 86000.0:
		tmp = ((F / math.sin(B)) * math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - (x / B)
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -22000000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 6.5e-151)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0)))) * Float64(F / B)));
	elseif (F <= 86000.0)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -22000000000.0)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 6.5e-151)
		tmp = (x * (-1.0 / tan(B))) + (sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B));
	elseif (F <= 86000.0)
		tmp = ((F / sin(B)) * ((((F * F) + 2.0) + (x * 2.0)) ^ -0.5)) - (x / B);
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -22000000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 6.5e-151], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 86000.0], 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] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.2e10

   1. Initial program 57.8%

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

   2. Taylor expanded in F around -inf 99.7%

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

      1. +-commutative 99.7%

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

      2. unsub-neg 99.7%

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

      3. un-div-inv 99.8%

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

   4. Applied egg-rr 99.8%

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

   if -2.2e10 < F < 6.4999999999999994e-151

   1. Initial program 99.4%

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

   2. Taylor expanded in B around 0 88.9%

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

   3. Taylor expanded in F around 0 88.9%

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

   if 6.4999999999999994e-151 < F < 86000

   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. Taylor expanded in B around 0 85.1%

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

   if 86000 < F

   1. Initial program 54.7%

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

   2. Taylor expanded in B around inf 54.6%

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

   3. Taylor expanded in F around inf 99.7%

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

      1. +-commutative 99.7%

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

      2. unsub-neg 99.7%

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

      3. associate-*r/ 99.8%

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

      4. associate-/l* 99.7%

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

      5. quot-tan 99.8%

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

   5. Applied egg-rr 99.8%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -22000000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 6.5 \cdot 10^{-151}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B}\\ \mathbf{elif}\;F \leq 86000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 6: 92.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -600:\\ \;\;\;\;\frac{-1}{\sin B} - t_1\\ \mathbf{elif}\;F \leq 1.25 \cdot 10^{-152}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + t_0 \cdot \frac{F}{B}\\ \mathbf{elif}\;F \leq 2200000:\\ \;\;\;\;\frac{F}{\sin B} \cdot t_0 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5)) (t_1 (/ x (tan B))))
   (if (<= F -600.0)
     (- (/ -1.0 (sin B)) t_1)
     (if (<= F 1.25e-152)
       (+ (/ -1.0 (/ (tan B) x)) (* t_0 (/ F B)))
       (if (<= F 2200000.0)
         (- (* (/ F (sin B)) t_0) (/ x B))
         (- (/ 1.0 (sin B)) t_1))))))

C

double code(double F, double B, double x) {
	double t_0 = pow((((F * F) + 2.0) + (x * 2.0)), -0.5);
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -600.0) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= 1.25e-152) {
		tmp = (-1.0 / (tan(B) / x)) + (t_0 * (F / B));
	} else if (F <= 2200000.0) {
		tmp = ((F / sin(B)) * t_0) - (x / B);
	} else {
		tmp = (1.0 / sin(B)) - t_1;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0)
    t_1 = x / tan(b)
    if (f <= (-600.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_1
    else if (f <= 1.25d-152) then
        tmp = ((-1.0d0) / (tan(b) / x)) + (t_0 * (f / b))
    else if (f <= 2200000.0d0) then
        tmp = ((f / sin(b)) * t_0) - (x / b)
    else
        tmp = (1.0d0 / sin(b)) - t_1
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5);
	double t_1 = x / Math.tan(B);
	double tmp;
	if (F <= -600.0) {
		tmp = (-1.0 / Math.sin(B)) - t_1;
	} else if (F <= 1.25e-152) {
		tmp = (-1.0 / (Math.tan(B) / x)) + (t_0 * (F / B));
	} else if (F <= 2200000.0) {
		tmp = ((F / Math.sin(B)) * t_0) - (x / B);
	} else {
		tmp = (1.0 / Math.sin(B)) - t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)
	t_1 = x / math.tan(B)
	tmp = 0
	if F <= -600.0:
		tmp = (-1.0 / math.sin(B)) - t_1
	elif F <= 1.25e-152:
		tmp = (-1.0 / (math.tan(B) / x)) + (t_0 * (F / B))
	elif F <= 2200000.0:
		tmp = ((F / math.sin(B)) * t_0) - (x / B)
	else:
		tmp = (1.0 / math.sin(B)) - t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -600.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= 1.25e-152)
		tmp = Float64(Float64(-1.0 / Float64(tan(B) / x)) + Float64(t_0 * Float64(F / B)));
	elseif (F <= 2200000.0)
		tmp = Float64(Float64(Float64(F / sin(B)) * t_0) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (((F * F) + 2.0) + (x * 2.0)) ^ -0.5;
	t_1 = x / tan(B);
	tmp = 0.0;
	if (F <= -600.0)
		tmp = (-1.0 / sin(B)) - t_1;
	elseif (F <= 1.25e-152)
		tmp = (-1.0 / (tan(B) / x)) + (t_0 * (F / B));
	elseif (F <= 2200000.0)
		tmp = ((F / sin(B)) * t_0) - (x / B);
	else
		tmp = (1.0 / sin(B)) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[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, -600.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, 1.25e-152], N[(N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(F / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2200000.0], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -600

   1. Initial program 58.4%

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

   2. Taylor expanded in F around -inf 99.7%

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

      1. +-commutative 99.7%

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

      2. unsub-neg 99.7%

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

      3. un-div-inv 99.8%

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

   4. Applied egg-rr 99.8%

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

   if -600 < F < 1.2499999999999999e-152

   1. Initial program 99.4%

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

   2. Taylor expanded in B around 0 88.7%

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

      1. div-inv 89.0%

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

      2. clear-num 88.8%

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

   4. Applied egg-rr 88.8%

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

   if 1.2499999999999999e-152 < F < 2.2e6

   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. Taylor expanded in B around 0 85.1%

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

   if 2.2e6 < F

   1. Initial program 54.7%

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

   2. Taylor expanded in B around inf 54.6%

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

   3. Taylor expanded in F around inf 99.7%

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

      1. +-commutative 99.7%

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

      2. unsub-neg 99.7%

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

      3. associate-*r/ 99.8%

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

      4. associate-/l* 99.7%

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

      5. quot-tan 99.8%

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

   5. Applied egg-rr 99.8%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -600:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.25 \cdot 10^{-152}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B}\\ \mathbf{elif}\;F \leq 2200000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 7: 91.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -22000000000:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{-7}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -22000000000.0)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.8e-7)
       (+ (* x (/ -1.0 (tan B))) (* (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))) (/ F 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 <= -22000000000.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.8e-7) {
		tmp = (x * (-1.0 / tan(B))) + (sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / 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 <= (-22000000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.8d-7) then
        tmp = (x * ((-1.0d0) / tan(b))) + (sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))) * (f / 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 <= -22000000000.0) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.8e-7) {
		tmp = (x * (-1.0 / Math.tan(B))) + (Math.sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / 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 <= -22000000000.0:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.8e-7:
		tmp = (x * (-1.0 / math.tan(B))) + (math.sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / 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 <= -22000000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.8e-7)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0)))) * Float64(F / 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 <= -22000000000.0)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.8e-7)
		tmp = (x * (-1.0 / tan(B))) + (sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / 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, -22000000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.8e-7], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.2e10

   1. Initial program 57.8%

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

   2. Taylor expanded in F around -inf 99.7%

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

      1. +-commutative 99.7%

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

      2. unsub-neg 99.7%

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

      3. un-div-inv 99.8%

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

   4. Applied egg-rr 99.8%

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

   if -2.2e10 < F < 1.79999999999999997e-7

   1. Initial program 99.4%

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

   2. Taylor expanded in B around 0 81.2%

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

   3. Taylor expanded in F around 0 81.2%

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

   if 1.79999999999999997e-7 < F

   1. Initial program 56.9%

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

   2. Taylor expanded in B around inf 56.8%

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

   3. Taylor expanded in F around inf 98.6%

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

      1. +-commutative 98.6%

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

      2. unsub-neg 98.6%

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

      3. associate-*r/ 98.6%

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

      4. associate-/l* 98.6%

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

      5. quot-tan 98.6%

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

   5. Applied egg-rr 98.6%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -22000000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{-7}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 8: 84.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.1 \cdot 10^{-108}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 7.8 \cdot 10^{-60}:\\ \;\;\;\;\frac{x}{\sin B} \cdot \left(-\cos B\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.1e-108)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 7.8e-60)
       (* (/ x (sin B)) (- (cos B)))
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.1e-108) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 7.8e-60) {
		tmp = (x / sin(B)) * -cos(B);
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.1d-108)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 7.8d-60) then
        tmp = (x / sin(b)) * -cos(b)
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.1e-108) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 7.8e-60) {
		tmp = (x / Math.sin(B)) * -Math.cos(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.1e-108:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 7.8e-60:
		tmp = (x / math.sin(B)) * -math.cos(B)
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.1e-108)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 7.8e-60)
		tmp = Float64(Float64(x / sin(B)) * Float64(-cos(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.1e-108)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 7.8e-60)
		tmp = (x / sin(B)) * -cos(B);
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.1e-108], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 7.8e-60], N[(N[(x / N[Sin[B], $MachinePrecision]), $MachinePrecision] * (-N[Cos[B], $MachinePrecision])), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.1000000000000001e-108

   1. Initial program 66.4%

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

   2. Taylor expanded in F around -inf 91.3%

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

      1. +-commutative 91.3%

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

      2. unsub-neg 91.3%

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

      3. un-div-inv 91.4%

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

   4. Applied egg-rr 91.4%

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

   if -1.1000000000000001e-108 < F < 7.8000000000000004e-60

   1. Initial program 99.4%

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

   2. Taylor expanded in B around inf 99.4%

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

   3. Taylor expanded in F around inf 34.0%

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

   4. Taylor expanded in x around inf 74.1%

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

      1. mul-1-neg 74.1%

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

      2. associate-*l/ 74.1%

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

      3. distribute-rgt-neg-in 74.1%

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

   6. Simplified 74.1%

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

   if 7.8000000000000004e-60 < F

   1. Initial program 60.6%

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

   2. Taylor expanded in B around inf 60.5%

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

   3. Taylor expanded in F around inf 94.6%

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

      1. +-commutative 94.6%

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

      2. unsub-neg 94.6%

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

      3. associate-*r/ 94.7%

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

      4. associate-/l* 94.6%

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

      5. quot-tan 94.7%

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

   5. Applied egg-rr 94.7%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.1 \cdot 10^{-108}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 7.8 \cdot 10^{-60}:\\ \;\;\;\;\frac{x}{\sin B} \cdot \left(-\cos B\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 9: 70.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -22000000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.7 \cdot 10^{-38}:\\ \;\;\;\;\frac{x}{\sin B} \cdot \left(-\cos B\right)\\ \mathbf{elif}\;F \leq 6.1 \cdot 10^{+175} \lor \neg \left(F \leq 7 \cdot 10^{+219}\right):\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -22000000000.0)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 2.7e-38)
     (* (/ x (sin B)) (- (cos B)))
     (if (or (<= F 6.1e+175) (not (<= F 7e+219)))
       (- (/ 1.0 (sin B)) (/ x B))
       (+ (* x (/ -1.0 (tan B))) (/ 1.0 B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -22000000000.0) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 2.7e-38) {
		tmp = (x / sin(B)) * -cos(B);
	} else if ((F <= 6.1e+175) || !(F <= 7e+219)) {
		tmp = (1.0 / sin(B)) - (x / B);
	} else {
		tmp = (x * (-1.0 / tan(B))) + (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 (f <= (-22000000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 2.7d-38) then
        tmp = (x / sin(b)) * -cos(b)
    else if ((f <= 6.1d+175) .or. (.not. (f <= 7d+219))) then
        tmp = (1.0d0 / sin(b)) - (x / b)
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -22000000000.0) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 2.7e-38) {
		tmp = (x / Math.sin(B)) * -Math.cos(B);
	} else if ((F <= 6.1e+175) || !(F <= 7e+219)) {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -22000000000.0:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 2.7e-38:
		tmp = (x / math.sin(B)) * -math.cos(B)
	elif (F <= 6.1e+175) or not (F <= 7e+219):
		tmp = (1.0 / math.sin(B)) - (x / B)
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -22000000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 2.7e-38)
		tmp = Float64(Float64(x / sin(B)) * Float64(-cos(B)));
	elseif ((F <= 6.1e+175) || !(F <= 7e+219))
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -22000000000.0)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 2.7e-38)
		tmp = (x / sin(B)) * -cos(B);
	elseif ((F <= 6.1e+175) || ~((F <= 7e+219)))
		tmp = (1.0 / sin(B)) - (x / B);
	else
		tmp = (x * (-1.0 / tan(B))) + (1.0 / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -22000000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.7e-38], N[(N[(x / N[Sin[B], $MachinePrecision]), $MachinePrecision] * (-N[Cos[B], $MachinePrecision])), $MachinePrecision], If[Or[LessEqual[F, 6.1e+175], N[Not[LessEqual[F, 7e+219]], $MachinePrecision]], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.2e10

   1. Initial program 57.8%

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

   2. Taylor expanded in F around -inf 99.7%

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

   3. Taylor expanded in B around 0 74.8%

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

   if -2.2e10 < F < 2.70000000000000005e-38

   1. Initial program 99.4%

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

   2. Taylor expanded in B around inf 99.4%

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

   3. Taylor expanded in F around inf 39.0%

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

   4. Taylor expanded in x around inf 70.8%

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

      1. mul-1-neg 70.8%

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

      2. associate-*l/ 70.8%

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

      3. distribute-rgt-neg-in 70.8%

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

   6. Simplified 70.8%

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

   if 2.70000000000000005e-38 < F < 6.0999999999999997e175 or 7.0000000000000002e219 < F

   1. Initial program 60.1%

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

   2. Taylor expanded in B around inf 60.0%

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

   3. Taylor expanded in F around inf 94.8%

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

   4. Taylor expanded in B around 0 87.3%

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

   if 6.0999999999999997e175 < F < 7.0000000000000002e219

   1. Initial program 54.7%

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

   2. Taylor expanded in B around 0 54.7%

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

   3. Taylor expanded in F around inf 92.3%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -22000000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.7 \cdot 10^{-38}:\\ \;\;\;\;\frac{x}{\sin B} \cdot \left(-\cos B\right)\\ \mathbf{elif}\;F \leq 6.1 \cdot 10^{+175} \lor \neg \left(F \leq 7 \cdot 10^{+219}\right):\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \end{array} \]

Alternative 10: 70.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -22000000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.7 \cdot 10^{-38}:\\ \;\;\;\;\frac{-x}{\frac{\sin B}{\cos B}}\\ \mathbf{elif}\;F \leq 1.22 \cdot 10^{+176} \lor \neg \left(F \leq 9.8 \cdot 10^{+219}\right):\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -22000000000.0)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 2.7e-38)
     (/ (- x) (/ (sin B) (cos B)))
     (if (or (<= F 1.22e+176) (not (<= F 9.8e+219)))
       (- (/ 1.0 (sin B)) (/ x B))
       (+ (* x (/ -1.0 (tan B))) (/ 1.0 B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -22000000000.0) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 2.7e-38) {
		tmp = -x / (sin(B) / cos(B));
	} else if ((F <= 1.22e+176) || !(F <= 9.8e+219)) {
		tmp = (1.0 / sin(B)) - (x / B);
	} else {
		tmp = (x * (-1.0 / tan(B))) + (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 (f <= (-22000000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 2.7d-38) then
        tmp = -x / (sin(b) / cos(b))
    else if ((f <= 1.22d+176) .or. (.not. (f <= 9.8d+219))) then
        tmp = (1.0d0 / sin(b)) - (x / b)
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -22000000000.0) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 2.7e-38) {
		tmp = -x / (Math.sin(B) / Math.cos(B));
	} else if ((F <= 1.22e+176) || !(F <= 9.8e+219)) {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -22000000000.0:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 2.7e-38:
		tmp = -x / (math.sin(B) / math.cos(B))
	elif (F <= 1.22e+176) or not (F <= 9.8e+219):
		tmp = (1.0 / math.sin(B)) - (x / B)
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -22000000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 2.7e-38)
		tmp = Float64(Float64(-x) / Float64(sin(B) / cos(B)));
	elseif ((F <= 1.22e+176) || !(F <= 9.8e+219))
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -22000000000.0)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 2.7e-38)
		tmp = -x / (sin(B) / cos(B));
	elseif ((F <= 1.22e+176) || ~((F <= 9.8e+219)))
		tmp = (1.0 / sin(B)) - (x / B);
	else
		tmp = (x * (-1.0 / tan(B))) + (1.0 / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -22000000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.7e-38], N[((-x) / N[(N[Sin[B], $MachinePrecision] / N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[F, 1.22e+176], N[Not[LessEqual[F, 9.8e+219]], $MachinePrecision]], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.2e10

   1. Initial program 57.8%

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

   2. Taylor expanded in F around -inf 99.7%

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

   3. Taylor expanded in B around 0 74.8%

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

   if -2.2e10 < F < 2.70000000000000005e-38

   1. Initial program 99.4%

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

   2. Taylor expanded in F around -inf 39.3%

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

   3. Taylor expanded in x around inf 70.8%

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

      1. mul-1-neg 70.8%

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

      2. associate-/l* 70.8%

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

      3. distribute-neg-frac 70.8%

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

   5. Simplified 70.8%

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

   if 2.70000000000000005e-38 < F < 1.2199999999999999e176 or 9.80000000000000007e219 < F

   1. Initial program 60.1%

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

   2. Taylor expanded in B around inf 60.0%

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

   3. Taylor expanded in F around inf 94.8%

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

   4. Taylor expanded in B around 0 87.3%

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

   if 1.2199999999999999e176 < F < 9.80000000000000007e219

   1. Initial program 54.7%

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

   2. Taylor expanded in B around 0 54.7%

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

   3. Taylor expanded in F around inf 92.3%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -22000000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.7 \cdot 10^{-38}:\\ \;\;\;\;\frac{-x}{\frac{\sin B}{\cos B}}\\ \mathbf{elif}\;F \leq 1.22 \cdot 10^{+176} \lor \neg \left(F \leq 9.8 \cdot 10^{+219}\right):\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \end{array} \]

Alternative 11: 76.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.1 \cdot 10^{-108}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.7 \cdot 10^{-38}:\\ \;\;\;\;\frac{x}{\sin B} \cdot \left(-\cos B\right)\\ \mathbf{elif}\;F \leq 2.25 \cdot 10^{+182} \lor \neg \left(F \leq 9.5 \cdot 10^{+219}\right):\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.1e-108)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 2.7e-38)
     (* (/ x (sin B)) (- (cos B)))
     (if (or (<= F 2.25e+182) (not (<= F 9.5e+219)))
       (- (/ 1.0 (sin B)) (/ x B))
       (+ (* x (/ -1.0 (tan B))) (/ 1.0 B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.1e-108) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 2.7e-38) {
		tmp = (x / sin(B)) * -cos(B);
	} else if ((F <= 2.25e+182) || !(F <= 9.5e+219)) {
		tmp = (1.0 / sin(B)) - (x / B);
	} else {
		tmp = (x * (-1.0 / tan(B))) + (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 (f <= (-1.1d-108)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 2.7d-38) then
        tmp = (x / sin(b)) * -cos(b)
    else if ((f <= 2.25d+182) .or. (.not. (f <= 9.5d+219))) then
        tmp = (1.0d0 / sin(b)) - (x / b)
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.1e-108) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 2.7e-38) {
		tmp = (x / Math.sin(B)) * -Math.cos(B);
	} else if ((F <= 2.25e+182) || !(F <= 9.5e+219)) {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.1e-108:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 2.7e-38:
		tmp = (x / math.sin(B)) * -math.cos(B)
	elif (F <= 2.25e+182) or not (F <= 9.5e+219):
		tmp = (1.0 / math.sin(B)) - (x / B)
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.1e-108)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 2.7e-38)
		tmp = Float64(Float64(x / sin(B)) * Float64(-cos(B)));
	elseif ((F <= 2.25e+182) || !(F <= 9.5e+219))
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.1e-108)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 2.7e-38)
		tmp = (x / sin(B)) * -cos(B);
	elseif ((F <= 2.25e+182) || ~((F <= 9.5e+219)))
		tmp = (1.0 / sin(B)) - (x / B);
	else
		tmp = (x * (-1.0 / tan(B))) + (1.0 / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.1e-108], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.7e-38], N[(N[(x / N[Sin[B], $MachinePrecision]), $MachinePrecision] * (-N[Cos[B], $MachinePrecision])), $MachinePrecision], If[Or[LessEqual[F, 2.25e+182], N[Not[LessEqual[F, 9.5e+219]], $MachinePrecision]], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.1000000000000001e-108

   1. Initial program 66.4%

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

   2. Taylor expanded in F around -inf 91.3%

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

      1. +-commutative 91.3%

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

      2. unsub-neg 91.3%

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

      3. un-div-inv 91.4%

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

   4. Applied egg-rr 91.4%

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

   if -1.1000000000000001e-108 < F < 2.70000000000000005e-38

   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. Taylor expanded in B around inf 99.4%

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

   3. Taylor expanded in F around inf 35.2%

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

   4. Taylor expanded in x around inf 73.8%

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

      1. mul-1-neg 73.8%

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

      2. associate-*l/ 73.9%

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

      3. distribute-rgt-neg-in 73.9%

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

   6. Simplified 73.9%

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

   if 2.70000000000000005e-38 < F < 2.25000000000000015e182 or 9.49999999999999959e219 < F

   1. Initial program 60.1%

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

   2. Taylor expanded in B around inf 60.0%

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

   3. Taylor expanded in F around inf 94.8%

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

   4. Taylor expanded in B around 0 87.3%

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

   if 2.25000000000000015e182 < F < 9.49999999999999959e219

   1. Initial program 54.7%

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

   2. Taylor expanded in B around 0 54.7%

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

   3. Taylor expanded in F around inf 92.3%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.1 \cdot 10^{-108}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.7 \cdot 10^{-38}:\\ \;\;\;\;\frac{x}{\sin B} \cdot \left(-\cos B\right)\\ \mathbf{elif}\;F \leq 2.25 \cdot 10^{+182} \lor \neg \left(F \leq 9.5 \cdot 10^{+219}\right):\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \end{array} \]

Alternative 12: 64.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - \frac{x}{B}\\ t_1 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -0.0045:\\ \;\;\;\;t_1 + \frac{-1}{B}\\ \mathbf{elif}\;F \leq -3.3 \cdot 10^{-224}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 1.92 \cdot 10^{-193}:\\ \;\;\;\;t_1 + \frac{F}{B} \cdot \frac{1}{F}\\ \mathbf{elif}\;F \leq 3.1 \cdot 10^{-66}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 6.8 \cdot 10^{+57} \lor \neg \left(F \leq 5.1 \cdot 10^{+177}\right) \land F \leq 2.2 \cdot 10^{+220}:\\ \;\;\;\;t_1 + \frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (* (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))) (/ F B)) (/ x B)))
        (t_1 (* x (/ -1.0 (tan B)))))
   (if (<= F -0.0045)
     (+ t_1 (/ -1.0 B))
     (if (<= F -3.3e-224)
       t_0
       (if (<= F 1.92e-193)
         (+ t_1 (* (/ F B) (/ 1.0 F)))
         (if (<= F 3.1e-66)
           t_0
           (if (or (<= F 6.8e+57) (and (not (<= F 5.1e+177)) (<= F 2.2e+220)))
             (+ t_1 (/ 1.0 B))
             (- (/ 1.0 (sin B)) (/ x B)))))))))

C

double code(double F, double B, double x) {
	double t_0 = (sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - (x / B);
	double t_1 = x * (-1.0 / tan(B));
	double tmp;
	if (F <= -0.0045) {
		tmp = t_1 + (-1.0 / B);
	} else if (F <= -3.3e-224) {
		tmp = t_0;
	} else if (F <= 1.92e-193) {
		tmp = t_1 + ((F / B) * (1.0 / F));
	} else if (F <= 3.1e-66) {
		tmp = t_0;
	} else if ((F <= 6.8e+57) || (!(F <= 5.1e+177) && (F <= 2.2e+220))) {
		tmp = t_1 + (1.0 / B);
	} else {
		tmp = (1.0 / sin(B)) - (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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))) * (f / b)) - (x / b)
    t_1 = x * ((-1.0d0) / tan(b))
    if (f <= (-0.0045d0)) then
        tmp = t_1 + ((-1.0d0) / b)
    else if (f <= (-3.3d-224)) then
        tmp = t_0
    else if (f <= 1.92d-193) then
        tmp = t_1 + ((f / b) * (1.0d0 / f))
    else if (f <= 3.1d-66) then
        tmp = t_0
    else if ((f <= 6.8d+57) .or. (.not. (f <= 5.1d+177)) .and. (f <= 2.2d+220)) then
        tmp = t_1 + (1.0d0 / b)
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (Math.sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - (x / B);
	double t_1 = x * (-1.0 / Math.tan(B));
	double tmp;
	if (F <= -0.0045) {
		tmp = t_1 + (-1.0 / B);
	} else if (F <= -3.3e-224) {
		tmp = t_0;
	} else if (F <= 1.92e-193) {
		tmp = t_1 + ((F / B) * (1.0 / F));
	} else if (F <= 3.1e-66) {
		tmp = t_0;
	} else if ((F <= 6.8e+57) || (!(F <= 5.1e+177) && (F <= 2.2e+220))) {
		tmp = t_1 + (1.0 / B);
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (math.sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - (x / B)
	t_1 = x * (-1.0 / math.tan(B))
	tmp = 0
	if F <= -0.0045:
		tmp = t_1 + (-1.0 / B)
	elif F <= -3.3e-224:
		tmp = t_0
	elif F <= 1.92e-193:
		tmp = t_1 + ((F / B) * (1.0 / F))
	elif F <= 3.1e-66:
		tmp = t_0
	elif (F <= 6.8e+57) or (not (F <= 5.1e+177) and (F <= 2.2e+220)):
		tmp = t_1 + (1.0 / B)
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0)))) * Float64(F / B)) - Float64(x / B))
	t_1 = Float64(x * Float64(-1.0 / tan(B)))
	tmp = 0.0
	if (F <= -0.0045)
		tmp = Float64(t_1 + Float64(-1.0 / B));
	elseif (F <= -3.3e-224)
		tmp = t_0;
	elseif (F <= 1.92e-193)
		tmp = Float64(t_1 + Float64(Float64(F / B) * Float64(1.0 / F)));
	elseif (F <= 3.1e-66)
		tmp = t_0;
	elseif ((F <= 6.8e+57) || (!(F <= 5.1e+177) && (F <= 2.2e+220)))
		tmp = Float64(t_1 + Float64(1.0 / B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - (x / B);
	t_1 = x * (-1.0 / tan(B));
	tmp = 0.0;
	if (F <= -0.0045)
		tmp = t_1 + (-1.0 / B);
	elseif (F <= -3.3e-224)
		tmp = t_0;
	elseif (F <= 1.92e-193)
		tmp = t_1 + ((F / B) * (1.0 / F));
	elseif (F <= 3.1e-66)
		tmp = t_0;
	elseif ((F <= 6.8e+57) || (~((F <= 5.1e+177)) && (F <= 2.2e+220)))
		tmp = t_1 + (1.0 / B);
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.0045], N[(t$95$1 + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -3.3e-224], t$95$0, If[LessEqual[F, 1.92e-193], N[(t$95$1 + N[(N[(F / B), $MachinePrecision] * N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.1e-66], t$95$0, If[Or[LessEqual[F, 6.8e+57], And[N[Not[LessEqual[F, 5.1e+177]], $MachinePrecision], LessEqual[F, 2.2e+220]]], N[(t$95$1 + N[(1.0 / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -0.00449999999999999966

   1. Initial program 58.4%

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

   2. Taylor expanded in F around -inf 99.7%

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

   3. Taylor expanded in B around 0 74.7%

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

   if -0.00449999999999999966 < F < -3.3000000000000001e-224 or 1.92e-193 < F < 3.0999999999999997e-66

   1. Initial program 99.3%

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

   2. Taylor expanded in B around 0 76.0%

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

   3. Taylor expanded in B around 0 59.1%

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

   4. Taylor expanded in F around 0 59.1%

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

   if -3.3000000000000001e-224 < F < 1.92e-193

   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. Taylor expanded in B around 0 93.5%

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

   3. Taylor expanded in F around inf 71.5%

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

   if 3.0999999999999997e-66 < F < 6.79999999999999984e57 or 5.1000000000000004e177 < F < 2.19999999999999989e220

   1. Initial program 84.2%

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

   2. Taylor expanded in B around 0 74.2%

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

   3. Taylor expanded in F around inf 83.5%

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

   if 6.79999999999999984e57 < F < 5.1000000000000004e177 or 2.19999999999999989e220 < F

   1. Initial program 43.4%

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

   2. Taylor expanded in B around inf 43.4%

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

   3. Taylor expanded in F around inf 99.7%

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

   4. Taylor expanded in B around 0 90.7%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{1}{\sin B} \]
3. Recombined 5 regimes into one program.

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.0045:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq -3.3 \cdot 10^{-224}:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.92 \cdot 10^{-193}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B} \cdot \frac{1}{F}\\ \mathbf{elif}\;F \leq 3.1 \cdot 10^{-66}:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 6.8 \cdot 10^{+57} \lor \neg \left(F \leq 5.1 \cdot 10^{+177}\right) \land F \leq 2.2 \cdot 10^{+220}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 13: 62.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -280000000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.35 \cdot 10^{-278}:\\ \;\;\;\;t_0 + \frac{F}{B} \cdot \frac{-1}{F}\\ \mathbf{elif}\;F \leq 3.1 \cdot 10^{-37} \lor \neg \left(F \leq 5.6 \cdot 10^{+177}\right) \land F \leq 7.2 \cdot 10^{+219}:\\ \;\;\;\;t_0 + \frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B)))))
   (if (<= F -280000000000.0)
     (- (/ -1.0 (sin B)) (/ x B))
     (if (<= F 2.35e-278)
       (+ t_0 (* (/ F B) (/ -1.0 F)))
       (if (or (<= F 3.1e-37) (and (not (<= F 5.6e+177)) (<= F 7.2e+219)))
         (+ t_0 (/ 1.0 B))
         (- (/ 1.0 (sin B)) (/ x B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double tmp;
	if (F <= -280000000000.0) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 2.35e-278) {
		tmp = t_0 + ((F / B) * (-1.0 / F));
	} else if ((F <= 3.1e-37) || (!(F <= 5.6e+177) && (F <= 7.2e+219))) {
		tmp = t_0 + (1.0 / B);
	} else {
		tmp = (1.0 / sin(B)) - (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) :: t_0
    real(8) :: tmp
    t_0 = x * ((-1.0d0) / tan(b))
    if (f <= (-280000000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 2.35d-278) then
        tmp = t_0 + ((f / b) * ((-1.0d0) / f))
    else if ((f <= 3.1d-37) .or. (.not. (f <= 5.6d+177)) .and. (f <= 7.2d+219)) then
        tmp = t_0 + (1.0d0 / b)
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / Math.tan(B));
	double tmp;
	if (F <= -280000000000.0) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 2.35e-278) {
		tmp = t_0 + ((F / B) * (-1.0 / F));
	} else if ((F <= 3.1e-37) || (!(F <= 5.6e+177) && (F <= 7.2e+219))) {
		tmp = t_0 + (1.0 / B);
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * (-1.0 / math.tan(B))
	tmp = 0
	if F <= -280000000000.0:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 2.35e-278:
		tmp = t_0 + ((F / B) * (-1.0 / F))
	elif (F <= 3.1e-37) or (not (F <= 5.6e+177) and (F <= 7.2e+219)):
		tmp = t_0 + (1.0 / B)
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	tmp = 0.0
	if (F <= -280000000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 2.35e-278)
		tmp = Float64(t_0 + Float64(Float64(F / B) * Float64(-1.0 / F)));
	elseif ((F <= 3.1e-37) || (!(F <= 5.6e+177) && (F <= 7.2e+219)))
		tmp = Float64(t_0 + Float64(1.0 / B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x * (-1.0 / tan(B));
	tmp = 0.0;
	if (F <= -280000000000.0)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 2.35e-278)
		tmp = t_0 + ((F / B) * (-1.0 / F));
	elseif ((F <= 3.1e-37) || (~((F <= 5.6e+177)) && (F <= 7.2e+219)))
		tmp = t_0 + (1.0 / B);
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -280000000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.35e-278], N[(t$95$0 + N[(N[(F / B), $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[F, 3.1e-37], And[N[Not[LessEqual[F, 5.6e+177]], $MachinePrecision], LessEqual[F, 7.2e+219]]], N[(t$95$0 + N[(1.0 / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.8e11

   1. Initial program 57.8%

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

   2. Taylor expanded in F around -inf 99.7%

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

   3. Taylor expanded in B around 0 74.8%

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

   if -2.8e11 < F < 2.3499999999999998e-278

   1. Initial program 99.4%

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

   2. Taylor expanded in B around 0 88.8%

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

   3. Taylor expanded in F around -inf 57.0%

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

   if 2.3499999999999998e-278 < F < 3.09999999999999993e-37 or 5.60000000000000004e177 < F < 7.20000000000000012e219

   1. Initial program 87.6%

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

   2. Taylor expanded in B around 0 64.5%

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

   3. Taylor expanded in F around inf 55.3%

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

   if 3.09999999999999993e-37 < F < 5.60000000000000004e177 or 7.20000000000000012e219 < F

   1. Initial program 59.5%

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

   2. Taylor expanded in B around inf 59.5%

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

   3. Taylor expanded in F around inf 96.0%

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

   4. Taylor expanded in B around 0 88.4%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -280000000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.35 \cdot 10^{-278}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B} \cdot \frac{-1}{F}\\ \mathbf{elif}\;F \leq 3.1 \cdot 10^{-37} \lor \neg \left(F \leq 5.6 \cdot 10^{+177}\right) \land F \leq 7.2 \cdot 10^{+219}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 14: 62.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -2.1 \cdot 10^{-199}:\\ \;\;\;\;t_0 + \frac{-1}{B}\\ \mathbf{elif}\;F \leq 2.2 \cdot 10^{+45}:\\ \;\;\;\;t_0 + \frac{F}{B} \cdot \frac{1}{F}\\ \mathbf{elif}\;F \leq 2.25 \cdot 10^{+182} \lor \neg \left(F \leq 7 \cdot 10^{+219}\right):\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B)))))
   (if (<= F -2.1e-199)
     (+ t_0 (/ -1.0 B))
     (if (<= F 2.2e+45)
       (+ t_0 (* (/ F B) (/ 1.0 F)))
       (if (or (<= F 2.25e+182) (not (<= F 7e+219)))
         (- (/ 1.0 (sin B)) (/ x B))
         (+ t_0 (/ 1.0 B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double tmp;
	if (F <= -2.1e-199) {
		tmp = t_0 + (-1.0 / B);
	} else if (F <= 2.2e+45) {
		tmp = t_0 + ((F / B) * (1.0 / F));
	} else if ((F <= 2.25e+182) || !(F <= 7e+219)) {
		tmp = (1.0 / sin(B)) - (x / B);
	} else {
		tmp = t_0 + (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) :: t_0
    real(8) :: tmp
    t_0 = x * ((-1.0d0) / tan(b))
    if (f <= (-2.1d-199)) then
        tmp = t_0 + ((-1.0d0) / b)
    else if (f <= 2.2d+45) then
        tmp = t_0 + ((f / b) * (1.0d0 / f))
    else if ((f <= 2.25d+182) .or. (.not. (f <= 7d+219))) then
        tmp = (1.0d0 / sin(b)) - (x / b)
    else
        tmp = t_0 + (1.0d0 / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / Math.tan(B));
	double tmp;
	if (F <= -2.1e-199) {
		tmp = t_0 + (-1.0 / B);
	} else if (F <= 2.2e+45) {
		tmp = t_0 + ((F / B) * (1.0 / F));
	} else if ((F <= 2.25e+182) || !(F <= 7e+219)) {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	} else {
		tmp = t_0 + (1.0 / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * (-1.0 / math.tan(B))
	tmp = 0
	if F <= -2.1e-199:
		tmp = t_0 + (-1.0 / B)
	elif F <= 2.2e+45:
		tmp = t_0 + ((F / B) * (1.0 / F))
	elif (F <= 2.25e+182) or not (F <= 7e+219):
		tmp = (1.0 / math.sin(B)) - (x / B)
	else:
		tmp = t_0 + (1.0 / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	tmp = 0.0
	if (F <= -2.1e-199)
		tmp = Float64(t_0 + Float64(-1.0 / B));
	elseif (F <= 2.2e+45)
		tmp = Float64(t_0 + Float64(Float64(F / B) * Float64(1.0 / F)));
	elseif ((F <= 2.25e+182) || !(F <= 7e+219))
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	else
		tmp = Float64(t_0 + Float64(1.0 / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x * (-1.0 / tan(B));
	tmp = 0.0;
	if (F <= -2.1e-199)
		tmp = t_0 + (-1.0 / B);
	elseif (F <= 2.2e+45)
		tmp = t_0 + ((F / B) * (1.0 / F));
	elseif ((F <= 2.25e+182) || ~((F <= 7e+219)))
		tmp = (1.0 / sin(B)) - (x / B);
	else
		tmp = t_0 + (1.0 / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.1e-199], N[(t$95$0 + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.2e+45], N[(t$95$0 + N[(N[(F / B), $MachinePrecision] * N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[F, 2.25e+182], N[Not[LessEqual[F, 7e+219]], $MachinePrecision]], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.10000000000000002e-199

   1. Initial program 71.7%

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

   2. Taylor expanded in F around -inf 80.8%

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

   3. Taylor expanded in B around 0 65.7%

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

   if -2.10000000000000002e-199 < F < 2.2e45

   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. Taylor expanded in B around 0 81.8%

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

   3. Taylor expanded in F around inf 60.5%

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

   if 2.2e45 < F < 2.25000000000000015e182 or 7.0000000000000002e219 < F

   1. Initial program 43.4%

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

   2. Taylor expanded in B around inf 43.4%

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

   3. Taylor expanded in F around inf 99.7%

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

   4. Taylor expanded in B around 0 90.7%

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

   if 2.25000000000000015e182 < F < 7.0000000000000002e219

   1. Initial program 54.7%

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

   2. Taylor expanded in B around 0 54.7%

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

   3. Taylor expanded in F around inf 92.3%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.1 \cdot 10^{-199}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq 2.2 \cdot 10^{+45}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B} \cdot \frac{1}{F}\\ \mathbf{elif}\;F \leq 2.25 \cdot 10^{+182} \lor \neg \left(F \leq 7 \cdot 10^{+219}\right):\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \end{array} \]

Alternative 15: 61.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq 2.7 \cdot 10^{-38}:\\ \;\;\;\;t_0 + \frac{-1}{B}\\ \mathbf{elif}\;F \leq 9.5 \cdot 10^{+179} \lor \neg \left(F \leq 1.35 \cdot 10^{+220}\right):\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B)))))
   (if (<= F 2.7e-38)
     (+ t_0 (/ -1.0 B))
     (if (or (<= F 9.5e+179) (not (<= F 1.35e+220)))
       (- (/ 1.0 (sin B)) (/ x B))
       (+ t_0 (/ 1.0 B))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double tmp;
	if (F <= 2.7e-38) {
		tmp = t_0 + (-1.0 / B);
	} else if ((F <= 9.5e+179) || !(F <= 1.35e+220)) {
		tmp = (1.0 / sin(B)) - (x / B);
	} else {
		tmp = t_0 + (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) :: t_0
    real(8) :: tmp
    t_0 = x * ((-1.0d0) / tan(b))
    if (f <= 2.7d-38) then
        tmp = t_0 + ((-1.0d0) / b)
    else if ((f <= 9.5d+179) .or. (.not. (f <= 1.35d+220))) then
        tmp = (1.0d0 / sin(b)) - (x / b)
    else
        tmp = t_0 + (1.0d0 / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / Math.tan(B));
	double tmp;
	if (F <= 2.7e-38) {
		tmp = t_0 + (-1.0 / B);
	} else if ((F <= 9.5e+179) || !(F <= 1.35e+220)) {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	} else {
		tmp = t_0 + (1.0 / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * (-1.0 / math.tan(B))
	tmp = 0
	if F <= 2.7e-38:
		tmp = t_0 + (-1.0 / B)
	elif (F <= 9.5e+179) or not (F <= 1.35e+220):
		tmp = (1.0 / math.sin(B)) - (x / B)
	else:
		tmp = t_0 + (1.0 / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	tmp = 0.0
	if (F <= 2.7e-38)
		tmp = Float64(t_0 + Float64(-1.0 / B));
	elseif ((F <= 9.5e+179) || !(F <= 1.35e+220))
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	else
		tmp = Float64(t_0 + Float64(1.0 / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x * (-1.0 / tan(B));
	tmp = 0.0;
	if (F <= 2.7e-38)
		tmp = t_0 + (-1.0 / B);
	elseif ((F <= 9.5e+179) || ~((F <= 1.35e+220)))
		tmp = (1.0 / sin(B)) - (x / B);
	else
		tmp = t_0 + (1.0 / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, 2.7e-38], N[(t$95$0 + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[F, 9.5e+179], N[Not[LessEqual[F, 1.35e+220]], $MachinePrecision]], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < 2.70000000000000005e-38

   1. Initial program 82.4%

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

   2. Taylor expanded in F around -inf 64.0%

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

   3. Taylor expanded in B around 0 58.9%

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

   if 2.70000000000000005e-38 < F < 9.5e179 or 1.3499999999999999e220 < F

   1. Initial program 60.1%

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

   2. Taylor expanded in B around inf 60.0%

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

   3. Taylor expanded in F around inf 94.8%

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

   4. Taylor expanded in B around 0 87.3%

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

   if 9.5e179 < F < 1.3499999999999999e220

   1. Initial program 54.7%

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

   2. Taylor expanded in B around 0 54.7%

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

   3. Taylor expanded in F around inf 92.3%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 2.7 \cdot 10^{-38}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq 9.5 \cdot 10^{+179} \lor \neg \left(F \leq 1.35 \cdot 10^{+220}\right):\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \end{array} \]

Alternative 16: 57.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.4 \cdot 10^{-99}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{-60}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.4e-99)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 5.2e-60) (/ (- x) B) (- (/ 1.0 (sin B)) (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.4e-99) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 5.2e-60) {
		tmp = -x / B;
	} else {
		tmp = (1.0 / sin(B)) - (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 <= (-3.4d-99)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 5.2d-60) then
        tmp = -x / b
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.4e-99) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 5.2e-60) {
		tmp = -x / B;
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.4e-99:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 5.2e-60:
		tmp = -x / B
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.4e-99)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 5.2e-60)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -3.4e-99)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 5.2e-60)
		tmp = -x / B;
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.4e-99], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.2e-60], N[((-x) / B), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.40000000000000007e-99

   1. Initial program 64.8%

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

   2. Taylor expanded in F around -inf 92.0%

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

   3. Taylor expanded in B around 0 70.1%

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

   if -3.40000000000000007e-99 < F < 5.1999999999999995e-60

   1. Initial program 99.4%

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

   2. Taylor expanded in F around -inf 35.9%

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

   3. Taylor expanded in B around 0 19.1%

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

      1. associate-*r/ 19.1%

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

      2. distribute-lft-in 19.1%

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

      3. metadata-eval 19.1%

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

      4. neg-mul-1 19.1%

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

   5. Simplified 19.1%

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

   6. Taylor expanded in x around inf 38.8%

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

      1. associate-*r/ 38.8%

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

      2. mul-1-neg 38.8%

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

   8. Simplified 38.8%

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

   if 5.1999999999999995e-60 < F

   1. Initial program 60.6%

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

   2. Taylor expanded in B around inf 60.5%

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

   3. Taylor expanded in F around inf 94.6%

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

   4. Taylor expanded in B around 0 78.0%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.4 \cdot 10^{-99}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{-60}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 17: 61.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 2.7 \cdot 10^{-38}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F 2.7e-38)
   (+ (* x (/ -1.0 (tan B))) (/ -1.0 B))
   (- (/ 1.0 (sin B)) (/ x B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= 2.7e-38) {
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / B);
	} else {
		tmp = (1.0 / sin(B)) - (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 <= 2.7d-38) then
        tmp = (x * ((-1.0d0) / tan(b))) + ((-1.0d0) / b)
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= 2.7e-38) {
		tmp = (x * (-1.0 / Math.tan(B))) + (-1.0 / B);
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= 2.7e-38:
		tmp = (x * (-1.0 / math.tan(B))) + (-1.0 / B)
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= 2.7e-38)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(-1.0 / B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= 2.7e-38)
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / B);
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, 2.7e-38], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 2.70000000000000005e-38

   1. Initial program 82.4%

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

   2. Taylor expanded in F around -inf 64.0%

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

   3. Taylor expanded in B around 0 58.9%

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

   if 2.70000000000000005e-38 < F

   1. Initial program 59.3%

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

   2. Taylor expanded in B around inf 59.2%

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

   3. Taylor expanded in F around inf 95.5%

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

   4. Taylor expanded in B around 0 80.5%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 2.7 \cdot 10^{-38}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 18: 50.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.8 \cdot 10^{-100}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.2 \cdot 10^{-48}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.8e-100)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 3.2e-48) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.8e-100) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 3.2e-48) {
		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 <= (-3.8d-100)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 3.2d-48) 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 <= -3.8e-100) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 3.2e-48) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.8e-100:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 3.2e-48:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.8e-100)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 3.2e-48)
		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 <= -3.8e-100)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 3.2e-48)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.8e-100], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.2e-48], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.79999999999999997e-100

   1. Initial program 64.8%

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

   2. Taylor expanded in F around -inf 92.0%

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

   3. Taylor expanded in B around 0 70.1%

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

   if -3.79999999999999997e-100 < F < 3.1999999999999998e-48

   1. Initial program 99.4%

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

   2. Taylor expanded in F around -inf 36.2%

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

   3. Taylor expanded in B around 0 18.8%

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

      1. associate-*r/ 18.8%

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

      2. distribute-lft-in 18.8%

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

      3. metadata-eval 18.8%

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

      4. neg-mul-1 18.8%

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

   5. Simplified 18.8%

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

   6. Taylor expanded in x around inf 38.0%

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

      1. associate-*r/ 38.0%

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

      2. mul-1-neg 38.0%

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

   8. Simplified 38.0%

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

   if 3.1999999999999998e-48 < F

   1. Initial program 59.7%

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

   2. Taylor expanded in B around 0 46.8%

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

   3. Taylor expanded in B around 0 31.4%

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

   4. Taylor expanded in F around inf 55.8%

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

      1. div-sub 55.8%

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

   6. Simplified 55.8%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.8 \cdot 10^{-100}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.2 \cdot 10^{-48}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 19: 37.2% accurate, 35.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.8 \cdot 10^{+18}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq 2.8 \cdot 10^{-55}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.8e+18)
   (/ -1.0 B)
   (if (<= F 2.8e-55) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.8e+18) {
		tmp = -1.0 / B;
	} else if (F <= 2.8e-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 <= (-2.8d+18)) then
        tmp = (-1.0d0) / b
    else if (f <= 2.8d-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 <= -2.8e+18) {
		tmp = -1.0 / B;
	} else if (F <= 2.8e-55) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.8e+18:
		tmp = -1.0 / B
	elif F <= 2.8e-55:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.8e+18)
		tmp = Float64(-1.0 / B);
	elseif (F <= 2.8e-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 <= -2.8e+18)
		tmp = -1.0 / B;
	elseif (F <= 2.8e-55)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.8e+18], N[(-1.0 / B), $MachinePrecision], If[LessEqual[F, 2.8e-55], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.8e18

   1. Initial program 56.5%

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

   2. Taylor expanded in F around -inf 99.7%

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

   3. Taylor expanded in B around 0 52.6%

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

      1. associate-*r/ 52.6%

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

      2. distribute-lft-in 52.6%

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

      3. metadata-eval 52.6%

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

      4. neg-mul-1 52.6%

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

   5. Simplified 52.6%

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

   6. Taylor expanded in x around 0 34.7%

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

   if -2.8e18 < F < 2.79999999999999984e-55

   1. Initial program 99.4%

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

   2. Taylor expanded in F around -inf 39.9%

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

   3. Taylor expanded in B around 0 23.3%

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

      1. associate-*r/ 23.3%

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

      2. distribute-lft-in 23.3%

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

      3. metadata-eval 23.3%

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

      4. neg-mul-1 23.3%

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

   5. Simplified 23.3%

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

   6. Taylor expanded in x around inf 39.2%

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

      1. associate-*r/ 39.2%

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

      2. mul-1-neg 39.2%

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

   8. Simplified 39.2%

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

   if 2.79999999999999984e-55 < F

   1. Initial program 59.7%

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

   2. Taylor expanded in B around 0 46.8%

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

   3. Taylor expanded in B around 0 31.4%

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

   4. Taylor expanded in F around inf 55.8%

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

      1. div-sub 55.8%

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

   6. Simplified 55.8%

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

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.8 \cdot 10^{+18}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq 2.8 \cdot 10^{-55}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 20: 43.1% accurate, 35.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3 \cdot 10^{-99}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 5.6 \cdot 10^{-45}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3e-99)
   (/ (- -1.0 x) B)
   (if (<= F 5.6e-45) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3e-99) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 5.6e-45) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-3d-99)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 5.6d-45) then
        tmp = -x / b
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -3e-99) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 5.6e-45) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3e-99:
		tmp = (-1.0 - x) / B
	elif F <= 5.6e-45:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3e-99)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 5.6e-45)
		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 <= -3e-99)
		tmp = (-1.0 - x) / B;
	elseif (F <= 5.6e-45)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3e-99], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 5.6e-45], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.00000000000000006e-99

   1. Initial program 64.8%

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

   2. Taylor expanded in F around -inf 92.0%

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

   3. Taylor expanded in B around 0 51.5%

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

      1. associate-*r/ 51.5%

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

      2. distribute-lft-in 51.5%

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

      3. metadata-eval 51.5%

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

      4. neg-mul-1 51.5%

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

   5. Simplified 51.5%

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

   if -3.00000000000000006e-99 < F < 5.6000000000000003e-45

   1. Initial program 99.4%

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

   2. Taylor expanded in F around -inf 36.2%

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

   3. Taylor expanded in B around 0 18.8%

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

      1. associate-*r/ 18.8%

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

      2. distribute-lft-in 18.8%

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

      3. metadata-eval 18.8%

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

      4. neg-mul-1 18.8%

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

   5. Simplified 18.8%

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

   6. Taylor expanded in x around inf 38.0%

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

      1. associate-*r/ 38.0%

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

      2. mul-1-neg 38.0%

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

   8. Simplified 38.0%

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

   if 5.6000000000000003e-45 < F

   1. Initial program 59.7%

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

   2. Taylor expanded in B around 0 46.8%

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

   3. Taylor expanded in B around 0 31.4%

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

   4. Taylor expanded in F around inf 55.8%

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

      1. div-sub 55.8%

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

   6. Simplified 55.8%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3 \cdot 10^{-99}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 5.6 \cdot 10^{-45}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 21: 43.0% accurate, 35.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{-99}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 7.8 \cdot 10^{-47}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2e-99)
   (- (/ -1.0 B) (/ x B))
   (if (<= F 7.8e-47) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2e-99) {
		tmp = (-1.0 / B) - (x / B);
	} else if (F <= 7.8e-47) {
		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 <= (-2d-99)) then
        tmp = ((-1.0d0) / b) - (x / b)
    else if (f <= 7.8d-47) 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 <= -2e-99) {
		tmp = (-1.0 / B) - (x / B);
	} else if (F <= 7.8e-47) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2e-99:
		tmp = (-1.0 / B) - (x / B)
	elif F <= 7.8e-47:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2e-99)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / B));
	elseif (F <= 7.8e-47)
		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 <= -2e-99)
		tmp = (-1.0 / B) - (x / B);
	elseif (F <= 7.8e-47)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2e-99], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7.8e-47], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2e-99

   1. Initial program 64.8%

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

   2. Taylor expanded in F around -inf 92.0%

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

   3. Taylor expanded in B around 0 51.5%

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

      1. associate-*r/ 51.5%

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

      2. distribute-lft-in 51.5%

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

      3. metadata-eval 51.5%

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

      4. neg-mul-1 51.5%

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

   5. Simplified 51.5%

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

   6. Taylor expanded in x around 0 51.6%

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

      1. sub-neg 51.6%

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

      2. mul-1-neg 51.6%

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

      3. +-commutative 51.6%

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

      4. distribute-neg-frac 51.6%

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

      5. metadata-eval 51.6%

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

      6. sub-neg 51.6%

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

   8. Simplified 51.6%

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

   if -2e-99 < F < 7.79999999999999956e-47

   1. Initial program 99.4%

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

   2. Taylor expanded in F around -inf 36.2%

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

   3. Taylor expanded in B around 0 18.8%

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

      1. associate-*r/ 18.8%

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

      2. distribute-lft-in 18.8%

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

      3. metadata-eval 18.8%

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

      4. neg-mul-1 18.8%

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

   5. Simplified 18.8%

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

   6. Taylor expanded in x around inf 38.0%

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

      1. associate-*r/ 38.0%

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

      2. mul-1-neg 38.0%

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

   8. Simplified 38.0%

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

   if 7.79999999999999956e-47 < F

   1. Initial program 59.7%

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

   2. Taylor expanded in B around 0 46.8%

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

   3. Taylor expanded in B around 0 31.4%

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

   4. Taylor expanded in F around inf 55.8%

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

      1. div-sub 55.8%

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

   6. Simplified 55.8%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{-99}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 7.8 \cdot 10^{-47}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 22: 29.7% accurate, 53.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -4.1 \cdot 10^{+17}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4.1e+17) (/ -1.0 B) (/ (- x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.1e+17) {
		tmp = -1.0 / B;
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-4.1d+17)) then
        tmp = (-1.0d0) / b
    else
        tmp = -x / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.1e+17) {
		tmp = -1.0 / B;
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -4.1e+17:
		tmp = -1.0 / B
	else:
		tmp = -x / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -4.1e+17)
		tmp = Float64(-1.0 / B);
	else
		tmp = Float64(Float64(-x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -4.1e+17)
		tmp = -1.0 / B;
	else
		tmp = -x / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -4.1e+17], N[(-1.0 / B), $MachinePrecision], N[((-x) / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -4.1e17

   1. Initial program 56.5%

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

   2. Taylor expanded in F around -inf 99.7%

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

   3. Taylor expanded in B around 0 52.6%

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

      1. associate-*r/ 52.6%

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

      2. distribute-lft-in 52.6%

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

      3. metadata-eval 52.6%

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

      4. neg-mul-1 52.6%

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

   5. Simplified 52.6%

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

   6. Taylor expanded in x around 0 34.7%

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

   if -4.1e17 < F

   1. Initial program 80.9%

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

   2. Taylor expanded in F around -inf 41.7%

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

   3. Taylor expanded in B around 0 26.0%

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

      1. associate-*r/ 26.0%

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

      2. distribute-lft-in 26.0%

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

      3. metadata-eval 26.0%

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

      4. neg-mul-1 26.0%

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

   5. Simplified 26.0%

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

   6. Taylor expanded in x around inf 34.9%

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

      1. associate-*r/ 34.9%

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

      2. mul-1-neg 34.9%

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

   8. Simplified 34.9%

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

4. Final simplification 34.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.1 \cdot 10^{+17}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \]

Alternative 23: 10.2% 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 74.6%

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

2. Taylor expanded in F around -inf 56.9%

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

3. Taylor expanded in B around 0 33.0%

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

   1. associate-*r/ 33.0%

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

   2. distribute-lft-in 33.0%

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

   3. metadata-eval 33.0%

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

   4. neg-mul-1 33.0%

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

5. Simplified 33.0%

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

6. Taylor expanded in x around 0 12.0%

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

7. Final simplification 12.0%

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

Reproduce

herbie shell --seed 2023334 
(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))))))