VandenBroeck and Keller, Equation (23)

Percentage Accurate: 77.1% → 99.5%

Time: 26.0s

Alternatives: 22

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: 77.1% 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: 99.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.5 \cdot 10^{+36}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.55 \cdot 10^{+25}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x \cdot \cos B}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.5e+36)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 1.55e+25)
     (-
      (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5))
      (/ (* x (cos B)) (sin B)))
     (+ (* x (/ -1.0 (tan B))) (/ 1.0 (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.5e+36) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 1.55e+25) {
		tmp = ((F / sin(B)) * pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - ((x * cos(B)) / sin(B));
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.5d+36)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 1.55d+25) then
        tmp = ((f / sin(b)) * ((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0))) - ((x * cos(b)) / sin(b))
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.5e+36) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 1.55e+25) {
		tmp = ((F / Math.sin(B)) * Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - ((x * Math.cos(B)) / Math.sin(B));
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.5e+36:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 1.55e+25:
		tmp = ((F / math.sin(B)) * math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - ((x * math.cos(B)) / math.sin(B))
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.5e+36)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 1.55e+25)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5)) - Float64(Float64(x * cos(B)) / sin(B)));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.5e+36)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 1.55e+25)
		tmp = ((F / sin(B)) * ((((F * F) + 2.0) + (x * 2.0)) ^ -0.5)) - ((x * cos(B)) / sin(B));
	else
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.5e+36], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.55e+25], 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[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.5e36

   1. Initial program 54.4%

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

   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 -1.5e36 < F < 1.5499999999999999e25

   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 x around 0 99.6%

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

   if 1.5499999999999999e25 < F

   1. Initial program 50.1%

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

   2. Taylor expanded in F around inf 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.5 \cdot 10^{+36}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.55 \cdot 10^{+25}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x \cdot \cos B}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 2: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.1 \cdot 10^{+36}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 2 \cdot 10^{+25}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.1e+36)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 2e+25)
       (- (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5)) t_0)
       (+ (* x (/ -1.0 (tan B))) (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.1e+36) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 2e+25) {
		tmp = ((F / sin(B)) * pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - t_0;
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.1d+36)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 2d+25) then
        tmp = ((f / sin(b)) * ((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0))) - t_0
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / sin(b))
    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+36) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 2e+25) {
		tmp = ((F / Math.sin(B)) * Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - t_0;
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.1e+36:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 2e+25:
		tmp = ((F / math.sin(B)) * math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - t_0
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.1e+36)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 2e+25)
		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(x * Float64(-1.0 / tan(B))) + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.1e+36)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 2e+25)
		tmp = ((F / sin(B)) * ((((F * F) + 2.0) + (x * 2.0)) ^ -0.5)) - t_0;
	else
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	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+36], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 2e+25], 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[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.1e36

   1. Initial program 54.4%

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

   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 -1.1e36 < F < 2.00000000000000018e25

   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. expm1-log1p-u 73.7%

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

      2. expm1-udef 57.1%

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

      3. div-inv 57.1%

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

   3. Applied egg-rr 57.1%

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

      1. expm1-def 73.7%

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

      2. expm1-log1p 99.5%

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

   5. Simplified 99.5%

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

   if 2.00000000000000018e25 < F

   1. Initial program 50.1%

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

   2. Taylor expanded in F around inf 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.1 \cdot 10^{+36}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2 \cdot 10^{+25}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 3: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.7 \cdot 10^{-10}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x \cdot \frac{1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.4)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 1.7e-10)
     (-
      (* (/ F (sin B)) (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))))
      (* x (/ 1.0 (tan B))))
     (+ (* x (/ -1.0 (tan B))) (/ 1.0 (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.4) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 1.7e-10) {
		tmp = ((F / sin(B)) * sqrt((1.0 / (2.0 + (x * 2.0))))) - (x * (1.0 / tan(B)));
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.4d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 1.7d-10) then
        tmp = ((f / sin(b)) * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))) - (x * (1.0d0 / tan(b)))
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.4) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 1.7e-10) {
		tmp = ((F / Math.sin(B)) * Math.sqrt((1.0 / (2.0 + (x * 2.0))))) - (x * (1.0 / Math.tan(B)));
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.4:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 1.7e-10:
		tmp = ((F / math.sin(B)) * math.sqrt((1.0 / (2.0 + (x * 2.0))))) - (x * (1.0 / math.tan(B)))
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.4)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 1.7e-10)
		tmp = Float64(Float64(Float64(F / sin(B)) * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))) - Float64(x * Float64(1.0 / tan(B))));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.4)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 1.7e-10)
		tmp = ((F / sin(B)) * sqrt((1.0 / (2.0 + (x * 2.0))))) - (x * (1.0 / tan(B)));
	else
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.4], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.7e-10], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.3999999999999999

   1. Initial program 60.4%

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

   2. Taylor expanded in F around -inf 96.4%

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

      1. +-commutative 96.4%

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

      2. unsub-neg 96.4%

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

      3. un-div-inv 96.5%

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

   4. Applied egg-rr 96.5%

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

   if -1.3999999999999999 < F < 1.70000000000000007e-10

   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 1.70000000000000007e-10 < F

   1. Initial program 53.6%

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

   2. Taylor expanded in F around inf 98.9%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.7 \cdot 10^{-10}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x \cdot \frac{1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 4: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -1.95:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.7 \cdot 10^{-10}:\\ \;\;\;\;t_0 + \frac{1}{\frac{\sin B}{F} \cdot \sqrt{2 + x \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B)))))
   (if (<= F -1.95)
     (- (/ -1.0 (sin B)) (/ x (tan B)))
     (if (<= F 1.7e-10)
       (+ t_0 (/ 1.0 (* (/ (sin B) F) (sqrt (+ 2.0 (* x 2.0))))))
       (+ t_0 (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double tmp;
	if (F <= -1.95) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 1.7e-10) {
		tmp = t_0 + (1.0 / ((sin(B) / F) * sqrt((2.0 + (x * 2.0)))));
	} else {
		tmp = t_0 + (1.0 / sin(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * ((-1.0d0) / tan(b))
    if (f <= (-1.95d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 1.7d-10) then
        tmp = t_0 + (1.0d0 / ((sin(b) / f) * sqrt((2.0d0 + (x * 2.0d0)))))
    else
        tmp = t_0 + (1.0d0 / sin(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 <= -1.95) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 1.7e-10) {
		tmp = t_0 + (1.0 / ((Math.sin(B) / F) * Math.sqrt((2.0 + (x * 2.0)))));
	} else {
		tmp = t_0 + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * (-1.0 / math.tan(B))
	tmp = 0
	if F <= -1.95:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 1.7e-10:
		tmp = t_0 + (1.0 / ((math.sin(B) / F) * math.sqrt((2.0 + (x * 2.0)))))
	else:
		tmp = t_0 + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	tmp = 0.0
	if (F <= -1.95)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 1.7e-10)
		tmp = Float64(t_0 + Float64(1.0 / Float64(Float64(sin(B) / F) * sqrt(Float64(2.0 + Float64(x * 2.0))))));
	else
		tmp = Float64(t_0 + Float64(1.0 / sin(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 <= -1.95)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 1.7e-10)
		tmp = t_0 + (1.0 / ((sin(B) / F) * sqrt((2.0 + (x * 2.0)))));
	else
		tmp = t_0 + (1.0 / sin(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, -1.95], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.7e-10], N[(t$95$0 + N[(1.0 / N[(N[(N[Sin[B], $MachinePrecision] / F), $MachinePrecision] * N[Sqrt[N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.94999999999999996

   1. Initial program 60.4%

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

   2. Taylor expanded in F around -inf 96.4%

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

      1. +-commutative 96.4%

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

      2. unsub-neg 96.4%

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

      3. un-div-inv 96.5%

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

   4. Applied egg-rr 96.5%

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

   if -1.94999999999999996 < F < 1.70000000000000007e-10

   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. clear-num 99.4%

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

      3. +-commutative 99.4%

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

      4. *-commutative 99.4%

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

      5. fma-udef 99.4%

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

      6. fma-def 99.4%

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

      7. metadata-eval 99.4%

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

      8. metadata-eval 99.4%

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

      9. fma-def 99.4%

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

      10. fma-udef 99.4%

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

      11. *-commutative 99.4%

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

      12. fma-def 99.4%

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

      13. fma-def 99.4%

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

   3. Applied egg-rr 99.4%

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

   4. Taylor expanded in F around 0 99.5%

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

   if 1.70000000000000007e-10 < F

   1. Initial program 53.6%

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

   2. Taylor expanded in F around inf 98.9%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.95:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.7 \cdot 10^{-10}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\frac{\sin B}{F} \cdot \sqrt{2 + x \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 5: 87.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{\frac{1}{2 + x \cdot 2}}}{\frac{\sin B}{F}} - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \mathbf{if}\;F \leq -0.96:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1 \cdot 10^{-93}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 2.75 \cdot 10^{-132}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{elif}\;F \leq 1.7 \cdot 10^{-10}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0
         (-
          (/ (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))) (/ (sin B) F))
          (+ (/ x B) (* -0.3333333333333333 (* B x))))))
   (if (<= F -0.96)
     (- (/ -1.0 (sin B)) (/ x (tan B)))
     (if (<= F -1e-93)
       t_0
       (if (<= F 2.75e-132)
         (/ (* x (cos B)) (- (sin B)))
         (if (<= F 1.7e-10)
           t_0
           (+ (* x (/ -1.0 (tan B))) (/ 1.0 (sin B)))))))))

C

double code(double F, double B, double x) {
	double t_0 = (sqrt((1.0 / (2.0 + (x * 2.0)))) / (sin(B) / F)) - ((x / B) + (-0.3333333333333333 * (B * x)));
	double tmp;
	if (F <= -0.96) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= -1e-93) {
		tmp = t_0;
	} else if (F <= 2.75e-132) {
		tmp = (x * cos(B)) / -sin(B);
	} else if (F <= 1.7e-10) {
		tmp = t_0;
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))) / (sin(b) / f)) - ((x / b) + ((-0.3333333333333333d0) * (b * x)))
    if (f <= (-0.96d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= (-1d-93)) then
        tmp = t_0
    else if (f <= 2.75d-132) then
        tmp = (x * cos(b)) / -sin(b)
    else if (f <= 1.7d-10) then
        tmp = t_0
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / sin(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)))) / (Math.sin(B) / F)) - ((x / B) + (-0.3333333333333333 * (B * x)));
	double tmp;
	if (F <= -0.96) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= -1e-93) {
		tmp = t_0;
	} else if (F <= 2.75e-132) {
		tmp = (x * Math.cos(B)) / -Math.sin(B);
	} else if (F <= 1.7e-10) {
		tmp = t_0;
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (math.sqrt((1.0 / (2.0 + (x * 2.0)))) / (math.sin(B) / F)) - ((x / B) + (-0.3333333333333333 * (B * x)))
	tmp = 0
	if F <= -0.96:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= -1e-93:
		tmp = t_0
	elif F <= 2.75e-132:
		tmp = (x * math.cos(B)) / -math.sin(B)
	elif F <= 1.7e-10:
		tmp = t_0
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / math.sin(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(sin(B) / F)) - Float64(Float64(x / B) + Float64(-0.3333333333333333 * Float64(B * x))))
	tmp = 0.0
	if (F <= -0.96)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= -1e-93)
		tmp = t_0;
	elseif (F <= 2.75e-132)
		tmp = Float64(Float64(x * cos(B)) / Float64(-sin(B)));
	elseif (F <= 1.7e-10)
		tmp = t_0;
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (sqrt((1.0 / (2.0 + (x * 2.0)))) / (sin(B) / F)) - ((x / B) + (-0.3333333333333333 * (B * x)));
	tmp = 0.0;
	if (F <= -0.96)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= -1e-93)
		tmp = t_0;
	elseif (F <= 2.75e-132)
		tmp = (x * cos(B)) / -sin(B);
	elseif (F <= 1.7e-10)
		tmp = t_0;
	else
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(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[(N[Sin[B], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision] - N[(N[(x / B), $MachinePrecision] + N[(-0.3333333333333333 * N[(B * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.96], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1e-93], t$95$0, If[LessEqual[F, 2.75e-132], N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 1.7e-10], t$95$0, N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.95999999999999996

   1. Initial program 60.4%

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

   2. Taylor expanded in F around -inf 96.4%

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

      1. +-commutative 96.4%

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

      2. unsub-neg 96.4%

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

      3. un-div-inv 96.5%

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

   4. Applied egg-rr 96.5%

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

   if -0.95999999999999996 < F < -9.999999999999999e-94 or 2.75e-132 < F < 1.70000000000000007e-10

   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 85.4%

      \[\leadsto \left(-\color{blue}{\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. *-commutative 85.4%

         \[\leadsto \left(-\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)\right) + \color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \cdot \frac{F}{\sin B}} \]

      2. clear-num 85.3%

         \[\leadsto \left(-\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)\right) + {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \cdot \color{blue}{\frac{1}{\frac{\sin B}{F}}} \]

      3. un-div-inv 85.4%

         \[\leadsto \left(-\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)\right) + \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\frac{\sin B}{F}}} \]

      4. +-commutative 85.4%

         \[\leadsto \left(-\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)\right) + \frac{{\color{blue}{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)}}{\frac{\sin B}{F}} \]

      5. fma-def 85.4%

         \[\leadsto \left(-\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)\right) + \frac{{\color{blue}{\left(\mathsf{fma}\left(2, x, F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)}}{\frac{\sin B}{F}} \]

      6. fma-def 85.4%

         \[\leadsto \left(-\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)\right) + \frac{{\left(\mathsf{fma}\left(2, x, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)\right)}^{\left(-\frac{1}{2}\right)}}{\frac{\sin B}{F}} \]

      7. metadata-eval 85.4%

         \[\leadsto \left(-\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)\right) + \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(-\color{blue}{0.5}\right)}}{\frac{\sin B}{F}} \]

      8. metadata-eval 85.4%

         \[\leadsto \left(-\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)\right) + \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{-0.5}}}{\frac{\sin B}{F}} \]

   4. Applied egg-rr 85.4%

      \[\leadsto \left(-\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)\right) + \color{blue}{\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\frac{\sin B}{F}}} \]

   5. Taylor expanded in F around 0 85.4%

      \[\leadsto \left(-\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)\right) + \frac{\color{blue}{\sqrt{\frac{1}{2 + 2 \cdot x}}}}{\frac{\sin B}{F}} \]

   if -9.999999999999999e-94 < F < 2.75e-132

   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 F around -inf 37.7%

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

   3. Taylor expanded in x around inf 76.4%

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

      1. associate-/l* 76.4%

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

      2. associate-*r/ 76.4%

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

      3. neg-mul-1 76.4%

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

      4. associate-/l* 76.4%

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

      5. associate-*l/ 76.3%

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

      6. *-commutative 76.3%

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

   5. Simplified 76.3%

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

      1. *-commutative 76.3%

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

      2. frac-2neg 76.3%

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

      3. associate-*l/ 76.4%

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

      4. remove-double-neg 76.4%

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

   7. Applied egg-rr 76.4%

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

   if 1.70000000000000007e-10 < F

   1. Initial program 53.6%

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

   2. Taylor expanded in F around inf 98.9%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.96:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1 \cdot 10^{-93}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2 + x \cdot 2}}}{\frac{\sin B}{F}} - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \mathbf{elif}\;F \leq 2.75 \cdot 10^{-132}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{elif}\;F \leq 1.7 \cdot 10^{-10}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{2 + x \cdot 2}}}{\frac{\sin B}{F}} - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 6: 90.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{if}\;F \leq -3.6 \cdot 10^{+14}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -2.4 \cdot 10^{-134}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 8.5 \cdot 10^{-136}:\\ \;\;\;\;\frac{x}{\frac{-\sin B}{\cos B}}\\ \mathbf{elif}\;F \leq 1.55 \cdot 10^{+25}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0
         (-
          (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5))
          (/ x B))))
   (if (<= F -3.6e+14)
     (- (/ -1.0 (sin B)) (/ x (tan B)))
     (if (<= F -2.4e-134)
       t_0
       (if (<= F 8.5e-136)
         (/ x (/ (- (sin B)) (cos B)))
         (if (<= F 1.55e+25)
           t_0
           (+ (* x (/ -1.0 (tan B))) (/ 1.0 (sin B)))))))))

C

double code(double F, double B, double x) {
	double t_0 = ((F / sin(B)) * pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - (x / B);
	double tmp;
	if (F <= -3.6e+14) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= -2.4e-134) {
		tmp = t_0;
	} else if (F <= 8.5e-136) {
		tmp = x / (-sin(B) / cos(B));
	} else if (F <= 1.55e+25) {
		tmp = t_0;
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((f / sin(b)) * ((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0))) - (x / b)
    if (f <= (-3.6d+14)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= (-2.4d-134)) then
        tmp = t_0
    else if (f <= 8.5d-136) then
        tmp = x / (-sin(b) / cos(b))
    else if (f <= 1.55d+25) then
        tmp = t_0
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = ((F / Math.sin(B)) * Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - (x / B);
	double tmp;
	if (F <= -3.6e+14) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= -2.4e-134) {
		tmp = t_0;
	} else if (F <= 8.5e-136) {
		tmp = x / (-Math.sin(B) / Math.cos(B));
	} else if (F <= 1.55e+25) {
		tmp = t_0;
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = ((F / math.sin(B)) * math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - (x / B)
	tmp = 0
	if F <= -3.6e+14:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= -2.4e-134:
		tmp = t_0
	elif F <= 8.5e-136:
		tmp = x / (-math.sin(B) / math.cos(B))
	elif F <= 1.55e+25:
		tmp = t_0
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5)) - Float64(x / B))
	tmp = 0.0
	if (F <= -3.6e+14)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= -2.4e-134)
		tmp = t_0;
	elseif (F <= 8.5e-136)
		tmp = Float64(x / Float64(Float64(-sin(B)) / cos(B)));
	elseif (F <= 1.55e+25)
		tmp = t_0;
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = ((F / sin(B)) * ((((F * F) + 2.0) + (x * 2.0)) ^ -0.5)) - (x / B);
	tmp = 0.0;
	if (F <= -3.6e+14)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= -2.4e-134)
		tmp = t_0;
	elseif (F <= 8.5e-136)
		tmp = x / (-sin(B) / cos(B));
	elseif (F <= 1.55e+25)
		tmp = t_0;
	else
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -3.6e+14], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -2.4e-134], t$95$0, If[LessEqual[F, 8.5e-136], N[(x / N[((-N[Sin[B], $MachinePrecision]) / N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.55e+25], t$95$0, N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.6e14

   1. Initial program 57.3%

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

   2. Taylor expanded in F around -inf 99.6%

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

      1. +-commutative 99.6%

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

      2. unsub-neg 99.6%

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

      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 -3.6e14 < F < -2.4000000000000001e-134 or 8.49999999999999973e-136 < F < 1.5499999999999999e25

   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 91.6%

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

   if -2.4000000000000001e-134 < F < 8.49999999999999973e-136

   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 F around -inf 37.3%

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

   3. Taylor expanded in x around inf 78.2%

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

      1. associate-/l* 78.2%

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

      2. associate-*r/ 78.2%

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

      3. neg-mul-1 78.2%

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

      4. associate-/l* 78.2%

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

      5. associate-*l/ 78.1%

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

      6. *-commutative 78.1%

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

   5. Simplified 78.1%

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

      1. *-commutative 78.1%

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

      2. frac-2neg 78.1%

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

      3. associate-*l/ 78.2%

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

      4. remove-double-neg 78.2%

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

   7. Applied egg-rr 78.2%

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

      1. associate-/l* 78.2%

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

   9. Simplified 78.2%

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

   if 1.5499999999999999e25 < F

   1. Initial program 50.1%

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

   2. Taylor expanded in F around inf 99.7%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.6 \cdot 10^{+14}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -2.4 \cdot 10^{-134}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 8.5 \cdot 10^{-136}:\\ \;\;\;\;\frac{x}{\frac{-\sin B}{\cos B}}\\ \mathbf{elif}\;F \leq 1.55 \cdot 10^{+25}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 7: 92.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ t_1 := {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{if}\;F \leq -3.6 \cdot 10^{+14}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.3 \cdot 10^{-71}:\\ \;\;\;\;\frac{F}{\sin B} \cdot t_1 - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.7 \cdot 10^{-10}:\\ \;\;\;\;t_0 + t_1 \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B))))
        (t_1 (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5)))
   (if (<= F -3.6e+14)
     (- (/ -1.0 (sin B)) (/ x (tan B)))
     (if (<= F -1.3e-71)
       (- (* (/ F (sin B)) t_1) (/ x B))
       (if (<= F 1.7e-10) (+ t_0 (* t_1 (/ F B))) (+ t_0 (/ 1.0 (sin B))))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double t_1 = pow((((F * F) + 2.0) + (x * 2.0)), -0.5);
	double tmp;
	if (F <= -3.6e+14) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= -1.3e-71) {
		tmp = ((F / sin(B)) * t_1) - (x / B);
	} else if (F <= 1.7e-10) {
		tmp = t_0 + (t_1 * (F / B));
	} else {
		tmp = t_0 + (1.0 / sin(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x * ((-1.0d0) / tan(b))
    t_1 = (((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0)
    if (f <= (-3.6d+14)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= (-1.3d-71)) then
        tmp = ((f / sin(b)) * t_1) - (x / b)
    else if (f <= 1.7d-10) then
        tmp = t_0 + (t_1 * (f / b))
    else
        tmp = t_0 + (1.0d0 / sin(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 t_1 = Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5);
	double tmp;
	if (F <= -3.6e+14) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= -1.3e-71) {
		tmp = ((F / Math.sin(B)) * t_1) - (x / B);
	} else if (F <= 1.7e-10) {
		tmp = t_0 + (t_1 * (F / B));
	} else {
		tmp = t_0 + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * (-1.0 / math.tan(B))
	t_1 = math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)
	tmp = 0
	if F <= -3.6e+14:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= -1.3e-71:
		tmp = ((F / math.sin(B)) * t_1) - (x / B)
	elif F <= 1.7e-10:
		tmp = t_0 + (t_1 * (F / B))
	else:
		tmp = t_0 + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	t_1 = Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5
	tmp = 0.0
	if (F <= -3.6e+14)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= -1.3e-71)
		tmp = Float64(Float64(Float64(F / sin(B)) * t_1) - Float64(x / B));
	elseif (F <= 1.7e-10)
		tmp = Float64(t_0 + Float64(t_1 * Float64(F / B)));
	else
		tmp = Float64(t_0 + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x * (-1.0 / tan(B));
	t_1 = (((F * F) + 2.0) + (x * 2.0)) ^ -0.5;
	tmp = 0.0;
	if (F <= -3.6e+14)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= -1.3e-71)
		tmp = ((F / sin(B)) * t_1) - (x / B);
	elseif (F <= 1.7e-10)
		tmp = t_0 + (t_1 * (F / B));
	else
		tmp = t_0 + (1.0 / sin(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]}, Block[{t$95$1 = N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]}, If[LessEqual[F, -3.6e+14], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.3e-71], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.7e-10], N[(t$95$0 + N[(t$95$1 * N[(F / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.6e14

   1. Initial program 57.3%

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

   2. Taylor expanded in F around -inf 99.6%

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

      1. +-commutative 99.6%

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

      2. unsub-neg 99.6%

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

      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 -3.6e14 < F < -1.2999999999999999e-71

   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 99.2%

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

   if -1.2999999999999999e-71 < F < 1.70000000000000007e-10

   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 82.9%

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

   if 1.70000000000000007e-10 < F

   1. Initial program 53.6%

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

   2. Taylor expanded in F around inf 98.9%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.6 \cdot 10^{+14}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.3 \cdot 10^{-71}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.7 \cdot 10^{-10}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 8: 85.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5}\\ t_1 := \frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\\ \mathbf{if}\;F \leq -1350:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -9.5 \cdot 10^{-94}:\\ \;\;\;\;t_0 \cdot \left(F \cdot \frac{1}{B}\right) - t_1\\ \mathbf{elif}\;F \leq 1.06 \cdot 10^{-135}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{elif}\;F \leq 1.7 \cdot 10^{-10}:\\ \;\;\;\;t_0 \cdot \left(\frac{F}{B} + 0.16666666666666666 \cdot \left(F \cdot B\right)\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5))
        (t_1 (+ (/ x B) (* -0.3333333333333333 (* B x)))))
   (if (<= F -1350.0)
     (- (/ -1.0 (sin B)) (/ x (tan B)))
     (if (<= F -9.5e-94)
       (- (* t_0 (* F (/ 1.0 B))) t_1)
       (if (<= F 1.06e-135)
         (/ (* x (cos B)) (- (sin B)))
         (if (<= F 1.7e-10)
           (- (* t_0 (+ (/ F B) (* 0.16666666666666666 (* F B)))) t_1)
           (+ (* x (/ -1.0 (tan B))) (/ 1.0 (sin B)))))))))

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 / B) + (-0.3333333333333333 * (B * x));
	double tmp;
	if (F <= -1350.0) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= -9.5e-94) {
		tmp = (t_0 * (F * (1.0 / B))) - t_1;
	} else if (F <= 1.06e-135) {
		tmp = (x * cos(B)) / -sin(B);
	} else if (F <= 1.7e-10) {
		tmp = (t_0 * ((F / B) + (0.16666666666666666 * (F * B)))) - t_1;
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0)
    t_1 = (x / b) + ((-0.3333333333333333d0) * (b * x))
    if (f <= (-1350.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= (-9.5d-94)) then
        tmp = (t_0 * (f * (1.0d0 / b))) - t_1
    else if (f <= 1.06d-135) then
        tmp = (x * cos(b)) / -sin(b)
    else if (f <= 1.7d-10) then
        tmp = (t_0 * ((f / b) + (0.16666666666666666d0 * (f * b)))) - t_1
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / sin(b))
    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 / B) + (-0.3333333333333333 * (B * x));
	double tmp;
	if (F <= -1350.0) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= -9.5e-94) {
		tmp = (t_0 * (F * (1.0 / B))) - t_1;
	} else if (F <= 1.06e-135) {
		tmp = (x * Math.cos(B)) / -Math.sin(B);
	} else if (F <= 1.7e-10) {
		tmp = (t_0 * ((F / B) + (0.16666666666666666 * (F * B)))) - t_1;
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)
	t_1 = (x / B) + (-0.3333333333333333 * (B * x))
	tmp = 0
	if F <= -1350.0:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= -9.5e-94:
		tmp = (t_0 * (F * (1.0 / B))) - t_1
	elif F <= 1.06e-135:
		tmp = (x * math.cos(B)) / -math.sin(B)
	elif F <= 1.7e-10:
		tmp = (t_0 * ((F / B) + (0.16666666666666666 * (F * B)))) - t_1
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / math.sin(B))
	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(Float64(x / B) + Float64(-0.3333333333333333 * Float64(B * x)))
	tmp = 0.0
	if (F <= -1350.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= -9.5e-94)
		tmp = Float64(Float64(t_0 * Float64(F * Float64(1.0 / B))) - t_1);
	elseif (F <= 1.06e-135)
		tmp = Float64(Float64(x * cos(B)) / Float64(-sin(B)));
	elseif (F <= 1.7e-10)
		tmp = Float64(Float64(t_0 * Float64(Float64(F / B) + Float64(0.16666666666666666 * Float64(F * B)))) - t_1);
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / sin(B)));
	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 / B) + (-0.3333333333333333 * (B * x));
	tmp = 0.0;
	if (F <= -1350.0)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= -9.5e-94)
		tmp = (t_0 * (F * (1.0 / B))) - t_1;
	elseif (F <= 1.06e-135)
		tmp = (x * cos(B)) / -sin(B);
	elseif (F <= 1.7e-10)
		tmp = (t_0 * ((F / B) + (0.16666666666666666 * (F * B)))) - t_1;
	else
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	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[(N[(x / B), $MachinePrecision] + N[(-0.3333333333333333 * N[(B * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1350.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -9.5e-94], N[(N[(t$95$0 * N[(F * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, 1.06e-135], N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 1.7e-10], N[(N[(t$95$0 * N[(N[(F / B), $MachinePrecision] + N[(0.16666666666666666 * N[(F * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -1350

   1. Initial program 58.6%

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

   2. Taylor expanded in F around -inf 99.1%

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

      1. +-commutative 99.1%

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

      2. unsub-neg 99.1%

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

      3. un-div-inv 99.3%

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

   4. Applied egg-rr 99.3%

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

   if -1350 < F < -9.4999999999999997e-94

   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 91.5%

      \[\leadsto \left(-\color{blue}{\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. div-inv 91.5%

         \[\leadsto \left(-\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)\right) + \color{blue}{\left(F \cdot \frac{1}{\sin B}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. *-commutative 91.5%

         \[\leadsto \left(-\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)\right) + \color{blue}{\left(\frac{1}{\sin B} \cdot F\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied egg-rr 91.5%

      \[\leadsto \left(-\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)\right) + \color{blue}{\left(\frac{1}{\sin B} \cdot F\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   5. Taylor expanded in B around 0 76.0%

      \[\leadsto \left(-\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)\right) + \left(\color{blue}{\frac{1}{B}} \cdot F\right) \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   if -9.4999999999999997e-94 < F < 1.06000000000000004e-135

   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 F around -inf 37.7%

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

   3. Taylor expanded in x around inf 76.4%

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

      1. associate-/l* 76.4%

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

      2. associate-*r/ 76.4%

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

      3. neg-mul-1 76.4%

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

      4. associate-/l* 76.4%

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

      5. associate-*l/ 76.3%

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

      6. *-commutative 76.3%

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

   5. Simplified 76.3%

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

      1. *-commutative 76.3%

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

      2. frac-2neg 76.3%

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

      3. associate-*l/ 76.4%

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

      4. remove-double-neg 76.4%

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

   7. Applied egg-rr 76.4%

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

   if 1.06000000000000004e-135 < F < 1.70000000000000007e-10

   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 80.1%

      \[\leadsto \left(-\color{blue}{\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in B around 0 75.3%

      \[\leadsto \left(-\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)\right) + \color{blue}{\left(0.16666666666666666 \cdot \left(B \cdot F\right) + \frac{F}{B}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   if 1.70000000000000007e-10 < F

   1. Initial program 53.6%

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

   2. Taylor expanded in F around inf 98.9%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1350:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -9.5 \cdot 10^{-94}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \left(F \cdot \frac{1}{B}\right) - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \mathbf{elif}\;F \leq 1.06 \cdot 10^{-135}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{elif}\;F \leq 1.7 \cdot 10^{-10}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \left(\frac{F}{B} + 0.16666666666666666 \cdot \left(F \cdot B\right)\right) - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 9: 66.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos B \cdot \frac{-x}{\sin B}\\ t_1 := \frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\\ t_2 := {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \left(F \cdot \frac{1}{B}\right) - t_1\\ \mathbf{if}\;F \leq -5200:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -8.6 \cdot 10^{-94}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;F \leq 8 \cdot 10^{-135}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 1.06 \cdot 10^{+71}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;F \leq 6.8 \cdot 10^{+212}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (cos B) (/ (- x) (sin B))))
        (t_1 (+ (/ x B) (* -0.3333333333333333 (* B x))))
        (t_2
         (- (* (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5) (* F (/ 1.0 B))) t_1)))
   (if (<= F -5200.0)
     (- (/ -1.0 (sin B)) (/ x B))
     (if (<= F -8.6e-94)
       t_2
       (if (<= F 8e-135)
         t_0
         (if (<= F 1.06e+71)
           t_2
           (if (<= F 6.8e+212) t_0 (- (/ 1.0 (sin B)) t_1))))))))

C

double code(double F, double B, double x) {
	double t_0 = cos(B) * (-x / sin(B));
	double t_1 = (x / B) + (-0.3333333333333333 * (B * x));
	double t_2 = (pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F * (1.0 / B))) - t_1;
	double tmp;
	if (F <= -5200.0) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -8.6e-94) {
		tmp = t_2;
	} else if (F <= 8e-135) {
		tmp = t_0;
	} else if (F <= 1.06e+71) {
		tmp = t_2;
	} else if (F <= 6.8e+212) {
		tmp = t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_1;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = cos(b) * (-x / sin(b))
    t_1 = (x / b) + ((-0.3333333333333333d0) * (b * x))
    t_2 = (((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0)) * (f * (1.0d0 / b))) - t_1
    if (f <= (-5200.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-8.6d-94)) then
        tmp = t_2
    else if (f <= 8d-135) then
        tmp = t_0
    else if (f <= 1.06d+71) then
        tmp = t_2
    else if (f <= 6.8d+212) then
        tmp = t_0
    else
        tmp = (1.0d0 / sin(b)) - t_1
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = Math.cos(B) * (-x / Math.sin(B));
	double t_1 = (x / B) + (-0.3333333333333333 * (B * x));
	double t_2 = (Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F * (1.0 / B))) - t_1;
	double tmp;
	if (F <= -5200.0) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -8.6e-94) {
		tmp = t_2;
	} else if (F <= 8e-135) {
		tmp = t_0;
	} else if (F <= 1.06e+71) {
		tmp = t_2;
	} else if (F <= 6.8e+212) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = math.cos(B) * (-x / math.sin(B))
	t_1 = (x / B) + (-0.3333333333333333 * (B * x))
	t_2 = (math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F * (1.0 / B))) - t_1
	tmp = 0
	if F <= -5200.0:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -8.6e-94:
		tmp = t_2
	elif F <= 8e-135:
		tmp = t_0
	elif F <= 1.06e+71:
		tmp = t_2
	elif F <= 6.8e+212:
		tmp = t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(cos(B) * Float64(Float64(-x) / sin(B)))
	t_1 = Float64(Float64(x / B) + Float64(-0.3333333333333333 * Float64(B * x)))
	t_2 = Float64(Float64((Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5) * Float64(F * Float64(1.0 / B))) - t_1)
	tmp = 0.0
	if (F <= -5200.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -8.6e-94)
		tmp = t_2;
	elseif (F <= 8e-135)
		tmp = t_0;
	elseif (F <= 1.06e+71)
		tmp = t_2;
	elseif (F <= 6.8e+212)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = cos(B) * (-x / sin(B));
	t_1 = (x / B) + (-0.3333333333333333 * (B * x));
	t_2 = (((((F * F) + 2.0) + (x * 2.0)) ^ -0.5) * (F * (1.0 / B))) - t_1;
	tmp = 0.0;
	if (F <= -5200.0)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -8.6e-94)
		tmp = t_2;
	elseif (F <= 8e-135)
		tmp = t_0;
	elseif (F <= 1.06e+71)
		tmp = t_2;
	elseif (F <= 6.8e+212)
		tmp = t_0;
	else
		tmp = (1.0 / sin(B)) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[Cos[B], $MachinePrecision] * N[((-x) / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x / B), $MachinePrecision] + N[(-0.3333333333333333 * N[(B * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]}, If[LessEqual[F, -5200.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -8.6e-94], t$95$2, If[LessEqual[F, 8e-135], t$95$0, If[LessEqual[F, 1.06e+71], t$95$2, If[LessEqual[F, 6.8e+212], t$95$0, N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -5200

   1. Initial program 58.6%

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

   2. Taylor expanded in F around -inf 99.1%

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

   3. Taylor expanded in B around 0 84.9%

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

   if -5200 < F < -8.5999999999999997e-94 or 8.0000000000000003e-135 < F < 1.06e71

   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 83.8%

      \[\leadsto \left(-\color{blue}{\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. div-inv 83.9%

         \[\leadsto \left(-\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)\right) + \color{blue}{\left(F \cdot \frac{1}{\sin B}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. *-commutative 83.9%

         \[\leadsto \left(-\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)\right) + \color{blue}{\left(\frac{1}{\sin B} \cdot F\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied egg-rr 83.9%

      \[\leadsto \left(-\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)\right) + \color{blue}{\left(\frac{1}{\sin B} \cdot F\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   5. Taylor expanded in B around 0 75.3%

      \[\leadsto \left(-\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)\right) + \left(\color{blue}{\frac{1}{B}} \cdot F\right) \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   if -8.5999999999999997e-94 < F < 8.0000000000000003e-135 or 1.06e71 < F < 6.80000000000000073e212

   1. Initial program 91.3%

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

   2. Taylor expanded in F around -inf 45.3%

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

   3. Taylor expanded in x around inf 73.7%

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

      1. associate-/l* 73.7%

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

      2. associate-*r/ 73.7%

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

      3. neg-mul-1 73.7%

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

      4. associate-/l* 73.7%

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

      5. associate-*l/ 73.7%

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

      6. *-commutative 73.7%

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

   5. Simplified 73.7%

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

   if 6.80000000000000073e212 < F

   1. Initial program 21.4%

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

   2. Taylor expanded in B around 0 5.0%

      \[\leadsto \left(-\color{blue}{\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in F around inf 69.3%

      \[\leadsto \left(-\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)\right) + \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5200:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -8.6 \cdot 10^{-94}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \left(F \cdot \frac{1}{B}\right) - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \mathbf{elif}\;F \leq 8 \cdot 10^{-135}:\\ \;\;\;\;\cos B \cdot \frac{-x}{\sin B}\\ \mathbf{elif}\;F \leq 1.06 \cdot 10^{+71}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \left(F \cdot \frac{1}{B}\right) - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \mathbf{elif}\;F \leq 6.8 \cdot 10^{+212}:\\ \;\;\;\;\cos B \cdot \frac{-x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \end{array} \]

Alternative 10: 66.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\\ t_1 := {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \left(F \cdot \frac{1}{B}\right) - t_0\\ \mathbf{if}\;F \leq -1350:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -9.2 \cdot 10^{-94}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 10^{-132}:\\ \;\;\;\;\frac{x}{\frac{-\sin B}{\cos B}}\\ \mathbf{elif}\;F \leq 1.22 \cdot 10^{+67}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 7.2 \cdot 10^{+213}:\\ \;\;\;\;\cos B \cdot \frac{-x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (+ (/ x B) (* -0.3333333333333333 (* B x))))
        (t_1
         (- (* (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5) (* F (/ 1.0 B))) t_0)))
   (if (<= F -1350.0)
     (- (/ -1.0 (sin B)) (/ x B))
     (if (<= F -9.2e-94)
       t_1
       (if (<= F 1e-132)
         (/ x (/ (- (sin B)) (cos B)))
         (if (<= F 1.22e+67)
           t_1
           (if (<= F 7.2e+213)
             (* (cos B) (/ (- x) (sin B)))
             (- (/ 1.0 (sin B)) t_0))))))))

C

double code(double F, double B, double x) {
	double t_0 = (x / B) + (-0.3333333333333333 * (B * x));
	double t_1 = (pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F * (1.0 / B))) - t_0;
	double tmp;
	if (F <= -1350.0) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -9.2e-94) {
		tmp = t_1;
	} else if (F <= 1e-132) {
		tmp = x / (-sin(B) / cos(B));
	} else if (F <= 1.22e+67) {
		tmp = t_1;
	} else if (F <= 7.2e+213) {
		tmp = cos(B) * (-x / sin(B));
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x / b) + ((-0.3333333333333333d0) * (b * x))
    t_1 = (((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0)) * (f * (1.0d0 / b))) - t_0
    if (f <= (-1350.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-9.2d-94)) then
        tmp = t_1
    else if (f <= 1d-132) then
        tmp = x / (-sin(b) / cos(b))
    else if (f <= 1.22d+67) then
        tmp = t_1
    else if (f <= 7.2d+213) then
        tmp = cos(b) * (-x / sin(b))
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (x / B) + (-0.3333333333333333 * (B * x));
	double t_1 = (Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F * (1.0 / B))) - t_0;
	double tmp;
	if (F <= -1350.0) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -9.2e-94) {
		tmp = t_1;
	} else if (F <= 1e-132) {
		tmp = x / (-Math.sin(B) / Math.cos(B));
	} else if (F <= 1.22e+67) {
		tmp = t_1;
	} else if (F <= 7.2e+213) {
		tmp = Math.cos(B) * (-x / Math.sin(B));
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (x / B) + (-0.3333333333333333 * (B * x))
	t_1 = (math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F * (1.0 / B))) - t_0
	tmp = 0
	if F <= -1350.0:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -9.2e-94:
		tmp = t_1
	elif F <= 1e-132:
		tmp = x / (-math.sin(B) / math.cos(B))
	elif F <= 1.22e+67:
		tmp = t_1
	elif F <= 7.2e+213:
		tmp = math.cos(B) * (-x / math.sin(B))
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(x / B) + Float64(-0.3333333333333333 * Float64(B * x)))
	t_1 = Float64(Float64((Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5) * Float64(F * Float64(1.0 / B))) - t_0)
	tmp = 0.0
	if (F <= -1350.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -9.2e-94)
		tmp = t_1;
	elseif (F <= 1e-132)
		tmp = Float64(x / Float64(Float64(-sin(B)) / cos(B)));
	elseif (F <= 1.22e+67)
		tmp = t_1;
	elseif (F <= 7.2e+213)
		tmp = Float64(cos(B) * Float64(Float64(-x) / sin(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (x / B) + (-0.3333333333333333 * (B * x));
	t_1 = (((((F * F) + 2.0) + (x * 2.0)) ^ -0.5) * (F * (1.0 / B))) - t_0;
	tmp = 0.0;
	if (F <= -1350.0)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -9.2e-94)
		tmp = t_1;
	elseif (F <= 1e-132)
		tmp = x / (-sin(B) / cos(B));
	elseif (F <= 1.22e+67)
		tmp = t_1;
	elseif (F <= 7.2e+213)
		tmp = cos(B) * (-x / sin(B));
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(x / B), $MachinePrecision] + N[(-0.3333333333333333 * N[(B * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[F, -1350.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -9.2e-94], t$95$1, If[LessEqual[F, 1e-132], N[(x / N[((-N[Sin[B], $MachinePrecision]) / N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.22e+67], t$95$1, If[LessEqual[F, 7.2e+213], N[(N[Cos[B], $MachinePrecision] * N[((-x) / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -1350

   1. Initial program 58.6%

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

   2. Taylor expanded in F around -inf 99.1%

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

   3. Taylor expanded in B around 0 84.9%

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

   if -1350 < F < -9.1999999999999997e-94 or 9.9999999999999999e-133 < F < 1.22000000000000004e67

   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 83.8%

      \[\leadsto \left(-\color{blue}{\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. div-inv 83.9%

         \[\leadsto \left(-\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)\right) + \color{blue}{\left(F \cdot \frac{1}{\sin B}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. *-commutative 83.9%

         \[\leadsto \left(-\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)\right) + \color{blue}{\left(\frac{1}{\sin B} \cdot F\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied egg-rr 83.9%

      \[\leadsto \left(-\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)\right) + \color{blue}{\left(\frac{1}{\sin B} \cdot F\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   5. Taylor expanded in B around 0 75.3%

      \[\leadsto \left(-\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)\right) + \left(\color{blue}{\frac{1}{B}} \cdot F\right) \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   if -9.1999999999999997e-94 < F < 9.9999999999999999e-133

   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 F around -inf 37.7%

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

   3. Taylor expanded in x around inf 76.4%

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

      1. associate-/l* 76.4%

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

      2. associate-*r/ 76.4%

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

      3. neg-mul-1 76.4%

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

      4. associate-/l* 76.4%

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

      5. associate-*l/ 76.3%

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

      6. *-commutative 76.3%

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

   5. Simplified 76.3%

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

      1. *-commutative 76.3%

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

      2. frac-2neg 76.3%

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

      3. associate-*l/ 76.4%

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

      4. remove-double-neg 76.4%

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

   7. Applied egg-rr 76.4%

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

      1. associate-/l* 76.4%

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

   9. Simplified 76.4%

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

   if 1.22000000000000004e67 < F < 7.2000000000000002e213

   1. Initial program 69.5%

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

   2. Taylor expanded in F around -inf 65.6%

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

   3. Taylor expanded in x around inf 66.4%

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

      1. associate-/l* 66.6%

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

      2. associate-*r/ 66.6%

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

      3. neg-mul-1 66.6%

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

      4. associate-/l* 66.4%

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

      5. associate-*l/ 66.6%

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

      6. *-commutative 66.6%

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

   5. Simplified 66.6%

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

   if 7.2000000000000002e213 < F

   1. Initial program 21.4%

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

   2. Taylor expanded in B around 0 5.0%

      \[\leadsto \left(-\color{blue}{\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in F around inf 69.3%

      \[\leadsto \left(-\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)\right) + \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1350:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -9.2 \cdot 10^{-94}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \left(F \cdot \frac{1}{B}\right) - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \mathbf{elif}\;F \leq 10^{-132}:\\ \;\;\;\;\frac{x}{\frac{-\sin B}{\cos B}}\\ \mathbf{elif}\;F \leq 1.22 \cdot 10^{+67}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \left(F \cdot \frac{1}{B}\right) - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \mathbf{elif}\;F \leq 7.2 \cdot 10^{+213}:\\ \;\;\;\;\cos B \cdot \frac{-x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \end{array} \]

Alternative 11: 66.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\\ t_1 := {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \left(F \cdot \frac{1}{B}\right) - t_0\\ \mathbf{if}\;F \leq -3300:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -9.5 \cdot 10^{-94}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{-131}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{elif}\;F \leq 9 \cdot 10^{+70}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 6.8 \cdot 10^{+212}:\\ \;\;\;\;\cos B \cdot \frac{-x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (+ (/ x B) (* -0.3333333333333333 (* B x))))
        (t_1
         (- (* (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5) (* F (/ 1.0 B))) t_0)))
   (if (<= F -3300.0)
     (- (/ -1.0 (sin B)) (/ x B))
     (if (<= F -9.5e-94)
       t_1
       (if (<= F 1.4e-131)
         (/ (* x (cos B)) (- (sin B)))
         (if (<= F 9e+70)
           t_1
           (if (<= F 6.8e+212)
             (* (cos B) (/ (- x) (sin B)))
             (- (/ 1.0 (sin B)) t_0))))))))

C

double code(double F, double B, double x) {
	double t_0 = (x / B) + (-0.3333333333333333 * (B * x));
	double t_1 = (pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F * (1.0 / B))) - t_0;
	double tmp;
	if (F <= -3300.0) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -9.5e-94) {
		tmp = t_1;
	} else if (F <= 1.4e-131) {
		tmp = (x * cos(B)) / -sin(B);
	} else if (F <= 9e+70) {
		tmp = t_1;
	} else if (F <= 6.8e+212) {
		tmp = cos(B) * (-x / sin(B));
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x / b) + ((-0.3333333333333333d0) * (b * x))
    t_1 = (((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0)) * (f * (1.0d0 / b))) - t_0
    if (f <= (-3300.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-9.5d-94)) then
        tmp = t_1
    else if (f <= 1.4d-131) then
        tmp = (x * cos(b)) / -sin(b)
    else if (f <= 9d+70) then
        tmp = t_1
    else if (f <= 6.8d+212) then
        tmp = cos(b) * (-x / sin(b))
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (x / B) + (-0.3333333333333333 * (B * x));
	double t_1 = (Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F * (1.0 / B))) - t_0;
	double tmp;
	if (F <= -3300.0) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -9.5e-94) {
		tmp = t_1;
	} else if (F <= 1.4e-131) {
		tmp = (x * Math.cos(B)) / -Math.sin(B);
	} else if (F <= 9e+70) {
		tmp = t_1;
	} else if (F <= 6.8e+212) {
		tmp = Math.cos(B) * (-x / Math.sin(B));
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (x / B) + (-0.3333333333333333 * (B * x))
	t_1 = (math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F * (1.0 / B))) - t_0
	tmp = 0
	if F <= -3300.0:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -9.5e-94:
		tmp = t_1
	elif F <= 1.4e-131:
		tmp = (x * math.cos(B)) / -math.sin(B)
	elif F <= 9e+70:
		tmp = t_1
	elif F <= 6.8e+212:
		tmp = math.cos(B) * (-x / math.sin(B))
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(x / B) + Float64(-0.3333333333333333 * Float64(B * x)))
	t_1 = Float64(Float64((Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5) * Float64(F * Float64(1.0 / B))) - t_0)
	tmp = 0.0
	if (F <= -3300.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -9.5e-94)
		tmp = t_1;
	elseif (F <= 1.4e-131)
		tmp = Float64(Float64(x * cos(B)) / Float64(-sin(B)));
	elseif (F <= 9e+70)
		tmp = t_1;
	elseif (F <= 6.8e+212)
		tmp = Float64(cos(B) * Float64(Float64(-x) / sin(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (x / B) + (-0.3333333333333333 * (B * x));
	t_1 = (((((F * F) + 2.0) + (x * 2.0)) ^ -0.5) * (F * (1.0 / B))) - t_0;
	tmp = 0.0;
	if (F <= -3300.0)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -9.5e-94)
		tmp = t_1;
	elseif (F <= 1.4e-131)
		tmp = (x * cos(B)) / -sin(B);
	elseif (F <= 9e+70)
		tmp = t_1;
	elseif (F <= 6.8e+212)
		tmp = cos(B) * (-x / sin(B));
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(x / B), $MachinePrecision] + N[(-0.3333333333333333 * N[(B * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[F, -3300.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -9.5e-94], t$95$1, If[LessEqual[F, 1.4e-131], N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 9e+70], t$95$1, If[LessEqual[F, 6.8e+212], N[(N[Cos[B], $MachinePrecision] * N[((-x) / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -3300

   1. Initial program 58.6%

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

   2. Taylor expanded in F around -inf 99.1%

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

   3. Taylor expanded in B around 0 84.9%

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

   if -3300 < F < -9.4999999999999997e-94 or 1.4e-131 < F < 8.9999999999999999e70

   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 83.8%

      \[\leadsto \left(-\color{blue}{\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. div-inv 83.9%

         \[\leadsto \left(-\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)\right) + \color{blue}{\left(F \cdot \frac{1}{\sin B}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. *-commutative 83.9%

         \[\leadsto \left(-\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)\right) + \color{blue}{\left(\frac{1}{\sin B} \cdot F\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied egg-rr 83.9%

      \[\leadsto \left(-\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)\right) + \color{blue}{\left(\frac{1}{\sin B} \cdot F\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   5. Taylor expanded in B around 0 75.3%

      \[\leadsto \left(-\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)\right) + \left(\color{blue}{\frac{1}{B}} \cdot F\right) \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   if -9.4999999999999997e-94 < F < 1.4e-131

   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 F around -inf 37.7%

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

   3. Taylor expanded in x around inf 76.4%

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

      1. associate-/l* 76.4%

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

      2. associate-*r/ 76.4%

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

      3. neg-mul-1 76.4%

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

      4. associate-/l* 76.4%

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

      5. associate-*l/ 76.3%

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

      6. *-commutative 76.3%

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

   5. Simplified 76.3%

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

      1. *-commutative 76.3%

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

      2. frac-2neg 76.3%

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

      3. associate-*l/ 76.4%

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

      4. remove-double-neg 76.4%

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

   7. Applied egg-rr 76.4%

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

   if 8.9999999999999999e70 < F < 6.80000000000000073e212

   1. Initial program 69.5%

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

   2. Taylor expanded in F around -inf 65.6%

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

   3. Taylor expanded in x around inf 66.4%

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

      1. associate-/l* 66.6%

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

      2. associate-*r/ 66.6%

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

      3. neg-mul-1 66.6%

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

      4. associate-/l* 66.4%

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

      5. associate-*l/ 66.6%

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

      6. *-commutative 66.6%

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

   5. Simplified 66.6%

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

   if 6.80000000000000073e212 < F

   1. Initial program 21.4%

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

   2. Taylor expanded in B around 0 5.0%

      \[\leadsto \left(-\color{blue}{\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in F around inf 69.3%

      \[\leadsto \left(-\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)\right) + \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3300:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -9.5 \cdot 10^{-94}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \left(F \cdot \frac{1}{B}\right) - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{-131}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{elif}\;F \leq 9 \cdot 10^{+70}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \left(F \cdot \frac{1}{B}\right) - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \mathbf{elif}\;F \leq 6.8 \cdot 10^{+212}:\\ \;\;\;\;\cos B \cdot \frac{-x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \end{array} \]

Alternative 12: 72.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\\ t_1 := {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \left(F \cdot \frac{1}{B}\right) - t_0\\ \mathbf{if}\;F \leq -1000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -7.5 \cdot 10^{-94}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{-135}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{elif}\;F \leq 2.15 \cdot 10^{+67}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 7.2 \cdot 10^{+213}:\\ \;\;\;\;\cos B \cdot \frac{-x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (+ (/ x B) (* -0.3333333333333333 (* B x))))
        (t_1
         (- (* (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5) (* F (/ 1.0 B))) t_0)))
   (if (<= F -1000.0)
     (- (/ -1.0 (sin B)) (/ x (tan B)))
     (if (<= F -7.5e-94)
       t_1
       (if (<= F 1.8e-135)
         (/ (* x (cos B)) (- (sin B)))
         (if (<= F 2.15e+67)
           t_1
           (if (<= F 7.2e+213)
             (* (cos B) (/ (- x) (sin B)))
             (- (/ 1.0 (sin B)) t_0))))))))

C

double code(double F, double B, double x) {
	double t_0 = (x / B) + (-0.3333333333333333 * (B * x));
	double t_1 = (pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F * (1.0 / B))) - t_0;
	double tmp;
	if (F <= -1000.0) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= -7.5e-94) {
		tmp = t_1;
	} else if (F <= 1.8e-135) {
		tmp = (x * cos(B)) / -sin(B);
	} else if (F <= 2.15e+67) {
		tmp = t_1;
	} else if (F <= 7.2e+213) {
		tmp = cos(B) * (-x / sin(B));
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x / b) + ((-0.3333333333333333d0) * (b * x))
    t_1 = (((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0)) * (f * (1.0d0 / b))) - t_0
    if (f <= (-1000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= (-7.5d-94)) then
        tmp = t_1
    else if (f <= 1.8d-135) then
        tmp = (x * cos(b)) / -sin(b)
    else if (f <= 2.15d+67) then
        tmp = t_1
    else if (f <= 7.2d+213) then
        tmp = cos(b) * (-x / sin(b))
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (x / B) + (-0.3333333333333333 * (B * x));
	double t_1 = (Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F * (1.0 / B))) - t_0;
	double tmp;
	if (F <= -1000.0) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= -7.5e-94) {
		tmp = t_1;
	} else if (F <= 1.8e-135) {
		tmp = (x * Math.cos(B)) / -Math.sin(B);
	} else if (F <= 2.15e+67) {
		tmp = t_1;
	} else if (F <= 7.2e+213) {
		tmp = Math.cos(B) * (-x / Math.sin(B));
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (x / B) + (-0.3333333333333333 * (B * x))
	t_1 = (math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F * (1.0 / B))) - t_0
	tmp = 0
	if F <= -1000.0:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= -7.5e-94:
		tmp = t_1
	elif F <= 1.8e-135:
		tmp = (x * math.cos(B)) / -math.sin(B)
	elif F <= 2.15e+67:
		tmp = t_1
	elif F <= 7.2e+213:
		tmp = math.cos(B) * (-x / math.sin(B))
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(x / B) + Float64(-0.3333333333333333 * Float64(B * x)))
	t_1 = Float64(Float64((Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5) * Float64(F * Float64(1.0 / B))) - t_0)
	tmp = 0.0
	if (F <= -1000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= -7.5e-94)
		tmp = t_1;
	elseif (F <= 1.8e-135)
		tmp = Float64(Float64(x * cos(B)) / Float64(-sin(B)));
	elseif (F <= 2.15e+67)
		tmp = t_1;
	elseif (F <= 7.2e+213)
		tmp = Float64(cos(B) * Float64(Float64(-x) / sin(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (x / B) + (-0.3333333333333333 * (B * x));
	t_1 = (((((F * F) + 2.0) + (x * 2.0)) ^ -0.5) * (F * (1.0 / B))) - t_0;
	tmp = 0.0;
	if (F <= -1000.0)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= -7.5e-94)
		tmp = t_1;
	elseif (F <= 1.8e-135)
		tmp = (x * cos(B)) / -sin(B);
	elseif (F <= 2.15e+67)
		tmp = t_1;
	elseif (F <= 7.2e+213)
		tmp = cos(B) * (-x / sin(B));
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(x / B), $MachinePrecision] + N[(-0.3333333333333333 * N[(B * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[F, -1000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -7.5e-94], t$95$1, If[LessEqual[F, 1.8e-135], N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 2.15e+67], t$95$1, If[LessEqual[F, 7.2e+213], N[(N[Cos[B], $MachinePrecision] * N[((-x) / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -1e3

   1. Initial program 58.6%

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

   2. Taylor expanded in F around -inf 99.1%

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

      1. +-commutative 99.1%

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

      2. unsub-neg 99.1%

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

      3. un-div-inv 99.3%

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

   4. Applied egg-rr 99.3%

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

   if -1e3 < F < -7.5000000000000003e-94 or 1.79999999999999989e-135 < F < 2.1500000000000001e67

   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 83.8%

      \[\leadsto \left(-\color{blue}{\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. div-inv 83.9%

         \[\leadsto \left(-\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)\right) + \color{blue}{\left(F \cdot \frac{1}{\sin B}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. *-commutative 83.9%

         \[\leadsto \left(-\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)\right) + \color{blue}{\left(\frac{1}{\sin B} \cdot F\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied egg-rr 83.9%

      \[\leadsto \left(-\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)\right) + \color{blue}{\left(\frac{1}{\sin B} \cdot F\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   5. Taylor expanded in B around 0 75.3%

      \[\leadsto \left(-\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)\right) + \left(\color{blue}{\frac{1}{B}} \cdot F\right) \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   if -7.5000000000000003e-94 < F < 1.79999999999999989e-135

   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 F around -inf 37.7%

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

   3. Taylor expanded in x around inf 76.4%

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

      1. associate-/l* 76.4%

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

      2. associate-*r/ 76.4%

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

      3. neg-mul-1 76.4%

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

      4. associate-/l* 76.4%

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

      5. associate-*l/ 76.3%

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

      6. *-commutative 76.3%

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

   5. Simplified 76.3%

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

      1. *-commutative 76.3%

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

      2. frac-2neg 76.3%

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

      3. associate-*l/ 76.4%

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

      4. remove-double-neg 76.4%

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

   7. Applied egg-rr 76.4%

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

   if 2.1500000000000001e67 < F < 7.2000000000000002e213

   1. Initial program 69.5%

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

   2. Taylor expanded in F around -inf 65.6%

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

   3. Taylor expanded in x around inf 66.4%

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

      1. associate-/l* 66.6%

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

      2. associate-*r/ 66.6%

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

      3. neg-mul-1 66.6%

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

      4. associate-/l* 66.4%

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

      5. associate-*l/ 66.6%

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

      6. *-commutative 66.6%

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

   5. Simplified 66.6%

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

   if 7.2000000000000002e213 < F

   1. Initial program 21.4%

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

   2. Taylor expanded in B around 0 5.0%

      \[\leadsto \left(-\color{blue}{\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in F around inf 69.3%

      \[\leadsto \left(-\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)\right) + \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -7.5 \cdot 10^{-94}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \left(F \cdot \frac{1}{B}\right) - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{-135}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{elif}\;F \leq 2.15 \cdot 10^{+67}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \left(F \cdot \frac{1}{B}\right) - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \mathbf{elif}\;F \leq 7.2 \cdot 10^{+213}:\\ \;\;\;\;\cos B \cdot \frac{-x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \end{array} \]

Alternative 13: 58.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ t_1 := \frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\\ t_2 := {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \left(F \cdot \frac{1}{B}\right) - t_1\\ \mathbf{if}\;F \leq -1650:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.5 \cdot 10^{-138}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;F \leq 5.6 \cdot 10^{-141}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{+68}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;F \leq 6.8 \cdot 10^{+212}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (+ (* x (/ -1.0 (tan B))) (/ -1.0 B)))
        (t_1 (+ (/ x B) (* -0.3333333333333333 (* B x))))
        (t_2
         (- (* (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5) (* F (/ 1.0 B))) t_1)))
   (if (<= F -1650.0)
     (- (/ -1.0 (sin B)) (/ x B))
     (if (<= F -1.5e-138)
       t_2
       (if (<= F 5.6e-141)
         t_0
         (if (<= F 2.3e+68)
           t_2
           (if (<= F 6.8e+212) t_0 (- (/ 1.0 (sin B)) t_1))))))))

C

double code(double F, double B, double x) {
	double t_0 = (x * (-1.0 / tan(B))) + (-1.0 / B);
	double t_1 = (x / B) + (-0.3333333333333333 * (B * x));
	double t_2 = (pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F * (1.0 / B))) - t_1;
	double tmp;
	if (F <= -1650.0) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -1.5e-138) {
		tmp = t_2;
	} else if (F <= 5.6e-141) {
		tmp = t_0;
	} else if (F <= 2.3e+68) {
		tmp = t_2;
	} else if (F <= 6.8e+212) {
		tmp = t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_1;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (x * ((-1.0d0) / tan(b))) + ((-1.0d0) / b)
    t_1 = (x / b) + ((-0.3333333333333333d0) * (b * x))
    t_2 = (((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0)) * (f * (1.0d0 / b))) - t_1
    if (f <= (-1650.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-1.5d-138)) then
        tmp = t_2
    else if (f <= 5.6d-141) then
        tmp = t_0
    else if (f <= 2.3d+68) then
        tmp = t_2
    else if (f <= 6.8d+212) then
        tmp = t_0
    else
        tmp = (1.0d0 / sin(b)) - t_1
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (x * (-1.0 / Math.tan(B))) + (-1.0 / B);
	double t_1 = (x / B) + (-0.3333333333333333 * (B * x));
	double t_2 = (Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F * (1.0 / B))) - t_1;
	double tmp;
	if (F <= -1650.0) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -1.5e-138) {
		tmp = t_2;
	} else if (F <= 5.6e-141) {
		tmp = t_0;
	} else if (F <= 2.3e+68) {
		tmp = t_2;
	} else if (F <= 6.8e+212) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (x * (-1.0 / math.tan(B))) + (-1.0 / B)
	t_1 = (x / B) + (-0.3333333333333333 * (B * x))
	t_2 = (math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F * (1.0 / B))) - t_1
	tmp = 0
	if F <= -1650.0:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -1.5e-138:
		tmp = t_2
	elif F <= 5.6e-141:
		tmp = t_0
	elif F <= 2.3e+68:
		tmp = t_2
	elif F <= 6.8e+212:
		tmp = t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(-1.0 / B))
	t_1 = Float64(Float64(x / B) + Float64(-0.3333333333333333 * Float64(B * x)))
	t_2 = Float64(Float64((Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5) * Float64(F * Float64(1.0 / B))) - t_1)
	tmp = 0.0
	if (F <= -1650.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -1.5e-138)
		tmp = t_2;
	elseif (F <= 5.6e-141)
		tmp = t_0;
	elseif (F <= 2.3e+68)
		tmp = t_2;
	elseif (F <= 6.8e+212)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (x * (-1.0 / tan(B))) + (-1.0 / B);
	t_1 = (x / B) + (-0.3333333333333333 * (B * x));
	t_2 = (((((F * F) + 2.0) + (x * 2.0)) ^ -0.5) * (F * (1.0 / B))) - t_1;
	tmp = 0.0;
	if (F <= -1650.0)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -1.5e-138)
		tmp = t_2;
	elseif (F <= 5.6e-141)
		tmp = t_0;
	elseif (F <= 2.3e+68)
		tmp = t_2;
	elseif (F <= 6.8e+212)
		tmp = t_0;
	else
		tmp = (1.0 / sin(B)) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x / B), $MachinePrecision] + N[(-0.3333333333333333 * N[(B * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]}, If[LessEqual[F, -1650.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.5e-138], t$95$2, If[LessEqual[F, 5.6e-141], t$95$0, If[LessEqual[F, 2.3e+68], t$95$2, If[LessEqual[F, 6.8e+212], t$95$0, N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1650

   1. Initial program 58.6%

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

   2. Taylor expanded in F around -inf 99.1%

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

   3. Taylor expanded in B around 0 84.9%

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

   if -1650 < F < -1.5e-138 or 5.60000000000000023e-141 < F < 2.3e68

   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 78.1%

      \[\leadsto \left(-\color{blue}{\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. div-inv 78.1%

         \[\leadsto \left(-\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)\right) + \color{blue}{\left(F \cdot \frac{1}{\sin B}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. *-commutative 78.1%

         \[\leadsto \left(-\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)\right) + \color{blue}{\left(\frac{1}{\sin B} \cdot F\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied egg-rr 78.1%

      \[\leadsto \left(-\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)\right) + \color{blue}{\left(\frac{1}{\sin B} \cdot F\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   5. Taylor expanded in B around 0 70.9%

      \[\leadsto \left(-\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)\right) + \left(\color{blue}{\frac{1}{B}} \cdot F\right) \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   if -1.5e-138 < F < 5.60000000000000023e-141 or 2.3e68 < F < 6.80000000000000073e212

   1. Initial program 90.4%

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

   2. Taylor expanded in F around -inf 46.5%

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

   3. Taylor expanded in B around 0 58.3%

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

   if 6.80000000000000073e212 < F

   1. Initial program 21.4%

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

   2. Taylor expanded in B around 0 5.0%

      \[\leadsto \left(-\color{blue}{\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in F around inf 69.3%

      \[\leadsto \left(-\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)\right) + \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1650:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.5 \cdot 10^{-138}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \left(F \cdot \frac{1}{B}\right) - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \mathbf{elif}\;F \leq 5.6 \cdot 10^{-141}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{+68}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \left(F \cdot \frac{1}{B}\right) - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \mathbf{elif}\;F \leq 6.8 \cdot 10^{+212}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \end{array} \]

Alternative 14: 58.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ t_1 := \frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\\ t_2 := {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - t_1\\ \mathbf{if}\;F \leq -3900:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -3.6 \cdot 10^{-140}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;F \leq 4.6 \cdot 10^{-141}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 3.8 \cdot 10^{+71}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;F \leq 7.2 \cdot 10^{+213}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (+ (* x (/ -1.0 (tan B))) (/ -1.0 B)))
        (t_1 (+ (/ x B) (* -0.3333333333333333 (* B x))))
        (t_2 (- (* (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5) (/ F B)) t_1)))
   (if (<= F -3900.0)
     (- (/ -1.0 (sin B)) (/ x B))
     (if (<= F -3.6e-140)
       t_2
       (if (<= F 4.6e-141)
         t_0
         (if (<= F 3.8e+71)
           t_2
           (if (<= F 7.2e+213) t_0 (- (/ 1.0 (sin B)) t_1))))))))

C

double code(double F, double B, double x) {
	double t_0 = (x * (-1.0 / tan(B))) + (-1.0 / B);
	double t_1 = (x / B) + (-0.3333333333333333 * (B * x));
	double t_2 = (pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B)) - t_1;
	double tmp;
	if (F <= -3900.0) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -3.6e-140) {
		tmp = t_2;
	} else if (F <= 4.6e-141) {
		tmp = t_0;
	} else if (F <= 3.8e+71) {
		tmp = t_2;
	} else if (F <= 7.2e+213) {
		tmp = t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_1;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (x * ((-1.0d0) / tan(b))) + ((-1.0d0) / b)
    t_1 = (x / b) + ((-0.3333333333333333d0) * (b * x))
    t_2 = (((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0)) * (f / b)) - t_1
    if (f <= (-3900.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-3.6d-140)) then
        tmp = t_2
    else if (f <= 4.6d-141) then
        tmp = t_0
    else if (f <= 3.8d+71) then
        tmp = t_2
    else if (f <= 7.2d+213) then
        tmp = t_0
    else
        tmp = (1.0d0 / sin(b)) - t_1
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (x * (-1.0 / Math.tan(B))) + (-1.0 / B);
	double t_1 = (x / B) + (-0.3333333333333333 * (B * x));
	double t_2 = (Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B)) - t_1;
	double tmp;
	if (F <= -3900.0) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -3.6e-140) {
		tmp = t_2;
	} else if (F <= 4.6e-141) {
		tmp = t_0;
	} else if (F <= 3.8e+71) {
		tmp = t_2;
	} else if (F <= 7.2e+213) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (x * (-1.0 / math.tan(B))) + (-1.0 / B)
	t_1 = (x / B) + (-0.3333333333333333 * (B * x))
	t_2 = (math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5) * (F / B)) - t_1
	tmp = 0
	if F <= -3900.0:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -3.6e-140:
		tmp = t_2
	elif F <= 4.6e-141:
		tmp = t_0
	elif F <= 3.8e+71:
		tmp = t_2
	elif F <= 7.2e+213:
		tmp = t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(-1.0 / B))
	t_1 = Float64(Float64(x / B) + Float64(-0.3333333333333333 * Float64(B * x)))
	t_2 = Float64(Float64((Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5) * Float64(F / B)) - t_1)
	tmp = 0.0
	if (F <= -3900.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -3.6e-140)
		tmp = t_2;
	elseif (F <= 4.6e-141)
		tmp = t_0;
	elseif (F <= 3.8e+71)
		tmp = t_2;
	elseif (F <= 7.2e+213)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (x * (-1.0 / tan(B))) + (-1.0 / B);
	t_1 = (x / B) + (-0.3333333333333333 * (B * x));
	t_2 = (((((F * F) + 2.0) + (x * 2.0)) ^ -0.5) * (F / B)) - t_1;
	tmp = 0.0;
	if (F <= -3900.0)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -3.6e-140)
		tmp = t_2;
	elseif (F <= 4.6e-141)
		tmp = t_0;
	elseif (F <= 3.8e+71)
		tmp = t_2;
	elseif (F <= 7.2e+213)
		tmp = t_0;
	else
		tmp = (1.0 / sin(B)) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x / B), $MachinePrecision] + N[(-0.3333333333333333 * N[(B * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]}, If[LessEqual[F, -3900.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -3.6e-140], t$95$2, If[LessEqual[F, 4.6e-141], t$95$0, If[LessEqual[F, 3.8e+71], t$95$2, If[LessEqual[F, 7.2e+213], t$95$0, N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3900

   1. Initial program 58.6%

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

   2. Taylor expanded in F around -inf 99.1%

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

   3. Taylor expanded in B around 0 84.9%

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

   if -3900 < F < -3.6e-140 or 4.5999999999999999e-141 < F < 3.8000000000000001e71

   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 78.1%

      \[\leadsto \left(-\color{blue}{\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in B around 0 70.9%

      \[\leadsto \left(-\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   if -3.6e-140 < F < 4.5999999999999999e-141 or 3.8000000000000001e71 < F < 7.2000000000000002e213

   1. Initial program 90.4%

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

   2. Taylor expanded in F around -inf 46.5%

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

   3. Taylor expanded in B around 0 58.3%

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

   if 7.2000000000000002e213 < F

   1. Initial program 21.4%

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

   2. Taylor expanded in B around 0 5.0%

      \[\leadsto \left(-\color{blue}{\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in F around inf 69.3%

      \[\leadsto \left(-\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)\right) + \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3900:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -3.6 \cdot 10^{-140}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \mathbf{elif}\;F \leq 4.6 \cdot 10^{-141}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq 3.8 \cdot 10^{+71}:\\ \;\;\;\;{\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \mathbf{elif}\;F \leq 7.2 \cdot 10^{+213}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \end{array} \]

Alternative 15: 55.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{if}\;F \leq -5.8 \cdot 10^{+19}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.05 \cdot 10^{+212}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{+265}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{elif}\;F \leq 8.5 \cdot 10^{+290}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (+ (* x (/ -1.0 (tan B))) (/ -1.0 B))))
   (if (<= F -5.8e+19)
     (- (/ -1.0 (sin B)) (/ x B))
     (if (<= F 3.05e+212)
       t_0
       (if (<= F 7.5e+265)
         (/ 1.0 (sin B))
         (if (<= F 8.5e+290)
           t_0
           (+
            (* B (+ 0.16666666666666666 (* x 0.3333333333333333)))
            (/ (- 1.0 x) B))))))))

C

double code(double F, double B, double x) {
	double t_0 = (x * (-1.0 / tan(B))) + (-1.0 / B);
	double tmp;
	if (F <= -5.8e+19) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 3.05e+212) {
		tmp = t_0;
	} else if (F <= 7.5e+265) {
		tmp = 1.0 / sin(B);
	} else if (F <= 8.5e+290) {
		tmp = t_0;
	} else {
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((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) :: t_0
    real(8) :: tmp
    t_0 = (x * ((-1.0d0) / tan(b))) + ((-1.0d0) / b)
    if (f <= (-5.8d+19)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 3.05d+212) then
        tmp = t_0
    else if (f <= 7.5d+265) then
        tmp = 1.0d0 / sin(b)
    else if (f <= 8.5d+290) then
        tmp = t_0
    else
        tmp = (b * (0.16666666666666666d0 + (x * 0.3333333333333333d0))) + ((1.0d0 - 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))) + (-1.0 / B);
	double tmp;
	if (F <= -5.8e+19) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 3.05e+212) {
		tmp = t_0;
	} else if (F <= 7.5e+265) {
		tmp = 1.0 / Math.sin(B);
	} else if (F <= 8.5e+290) {
		tmp = t_0;
	} else {
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (x * (-1.0 / math.tan(B))) + (-1.0 / B)
	tmp = 0
	if F <= -5.8e+19:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 3.05e+212:
		tmp = t_0
	elif F <= 7.5e+265:
		tmp = 1.0 / math.sin(B)
	elif F <= 8.5e+290:
		tmp = t_0
	else:
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(-1.0 / B))
	tmp = 0.0
	if (F <= -5.8e+19)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 3.05e+212)
		tmp = t_0;
	elseif (F <= 7.5e+265)
		tmp = Float64(1.0 / sin(B));
	elseif (F <= 8.5e+290)
		tmp = t_0;
	else
		tmp = Float64(Float64(B * Float64(0.16666666666666666 + Float64(x * 0.3333333333333333))) + Float64(Float64(1.0 - x) / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (x * (-1.0 / tan(B))) + (-1.0 / B);
	tmp = 0.0;
	if (F <= -5.8e+19)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 3.05e+212)
		tmp = t_0;
	elseif (F <= 7.5e+265)
		tmp = 1.0 / sin(B);
	elseif (F <= 8.5e+290)
		tmp = t_0;
	else
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -5.8e+19], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.05e+212], t$95$0, If[LessEqual[F, 7.5e+265], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 8.5e+290], t$95$0, N[(N[(B * N[(0.16666666666666666 + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -5.8e19

   1. Initial program 56.6%

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

   2. Taylor expanded in F around -inf 99.6%

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

   3. Taylor expanded in B around 0 86.1%

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

   if -5.8e19 < F < 3.0499999999999999e212 or 7.49999999999999951e265 < F < 8.50000000000000061e290

   1. Initial program 91.5%

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

   2. Taylor expanded in F around -inf 46.2%

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

   3. Taylor expanded in B around 0 52.6%

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

   if 3.0499999999999999e212 < F < 7.49999999999999951e265

   1. Initial program 21.1%

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

   2. Taylor expanded in B around 0 8.9%

      \[\leadsto \left(-\color{blue}{\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in F around inf 61.2%

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

   4. Taylor expanded in x around 0 66.8%

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

   if 8.50000000000000061e290 < F

   1. Initial program 0.4%

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

   2. Taylor expanded in B around 0 0.2%

      \[\leadsto \left(-\color{blue}{\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in F around inf 83.5%

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

   4. Taylor expanded in B around 0 83.9%

      \[\leadsto \color{blue}{\left(B \cdot \left(0.16666666666666666 - -0.3333333333333333 \cdot x\right) + \frac{1}{B}\right) - \frac{x}{B}} \]
   5. Step-by-step derivation

      1. associate--l+ 83.9%

         \[\leadsto \color{blue}{B \cdot \left(0.16666666666666666 - -0.3333333333333333 \cdot x\right) + \left(\frac{1}{B} - \frac{x}{B}\right)} \]

      2. cancel-sign-sub-inv 83.9%

         \[\leadsto B \cdot \color{blue}{\left(0.16666666666666666 + \left(--0.3333333333333333\right) \cdot x\right)} + \left(\frac{1}{B} - \frac{x}{B}\right) \]

      3. metadata-eval 83.9%

         \[\leadsto B \cdot \left(0.16666666666666666 + \color{blue}{0.3333333333333333} \cdot x\right) + \left(\frac{1}{B} - \frac{x}{B}\right) \]

      4. *-commutative 83.9%

         \[\leadsto B \cdot \left(0.16666666666666666 + \color{blue}{x \cdot 0.3333333333333333}\right) + \left(\frac{1}{B} - \frac{x}{B}\right) \]

      5. div-sub 83.9%

         \[\leadsto B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \color{blue}{\frac{1 - x}{B}} \]

   6. Simplified 83.9%

      \[\leadsto \color{blue}{B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5.8 \cdot 10^{+19}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.05 \cdot 10^{+212}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{+265}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{elif}\;F \leq 8.5 \cdot 10^{+290}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \end{array} \]

Alternative 16: 55.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -6.2 \cdot 10^{+19}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 7.2 \cdot 10^{+214}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -6.2e+19)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 7.2e+214)
     (+ (* x (/ -1.0 (tan B))) (/ -1.0 B))
     (- (/ 1.0 (sin B)) (+ (/ x B) (* -0.3333333333333333 (* B x)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -6.2e+19) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 7.2e+214) {
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / B);
	} else {
		tmp = (1.0 / sin(B)) - ((x / B) + (-0.3333333333333333 * (B * x)));
	}
	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 <= (-6.2d+19)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 7.2d+214) then
        tmp = (x * ((-1.0d0) / tan(b))) + ((-1.0d0) / b)
    else
        tmp = (1.0d0 / sin(b)) - ((x / b) + ((-0.3333333333333333d0) * (b * x)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -6.2e+19) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 7.2e+214) {
		tmp = (x * (-1.0 / Math.tan(B))) + (-1.0 / B);
	} else {
		tmp = (1.0 / Math.sin(B)) - ((x / B) + (-0.3333333333333333 * (B * x)));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -6.2e+19:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 7.2e+214:
		tmp = (x * (-1.0 / math.tan(B))) + (-1.0 / B)
	else:
		tmp = (1.0 / math.sin(B)) - ((x / B) + (-0.3333333333333333 * (B * x)))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -6.2e+19)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 7.2e+214)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(-1.0 / B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(Float64(x / B) + Float64(-0.3333333333333333 * Float64(B * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -6.2e+19)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 7.2e+214)
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / B);
	else
		tmp = (1.0 / sin(B)) - ((x / B) + (-0.3333333333333333 * (B * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -6.2e+19], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7.2e+214], 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[(N[(x / B), $MachinePrecision] + N[(-0.3333333333333333 * N[(B * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -6.2e19

   1. Initial program 56.6%

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

   2. Taylor expanded in F around -inf 99.6%

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

   3. Taylor expanded in B around 0 86.1%

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

   if -6.2e19 < F < 7.2000000000000002e214

   1. Initial program 94.1%

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

   2. Taylor expanded in F around -inf 44.7%

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

   3. Taylor expanded in B around 0 51.4%

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

   if 7.2000000000000002e214 < F

   1. Initial program 21.4%

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

   2. Taylor expanded in B around 0 5.0%

      \[\leadsto \left(-\color{blue}{\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in F around inf 69.3%

      \[\leadsto \left(-\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)\right) + \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6.2 \cdot 10^{+19}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 7.2 \cdot 10^{+214}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \end{array} \]

Alternative 17: 43.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.3 \cdot 10^{-63}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 3.9 \cdot 10^{-72}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 2.5 \cdot 10^{+136}:\\ \;\;\;\;\frac{1}{B} - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \mathbf{elif}\;F \leq 3.2 \cdot 10^{+270}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.3e-63)
   (/ (- -1.0 x) B)
   (if (<= F 3.9e-72)
     (/ (- x) B)
     (if (<= F 2.5e+136)
       (- (/ 1.0 B) (+ (/ x B) (* -0.3333333333333333 (* B x))))
       (if (<= F 3.2e+270) (/ 1.0 (sin B)) (/ (- 1.0 x) B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.3e-63) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 3.9e-72) {
		tmp = -x / B;
	} else if (F <= 2.5e+136) {
		tmp = (1.0 / B) - ((x / B) + (-0.3333333333333333 * (B * x)));
	} else if (F <= 3.2e+270) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-2.3d-63)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 3.9d-72) then
        tmp = -x / b
    else if (f <= 2.5d+136) then
        tmp = (1.0d0 / b) - ((x / b) + ((-0.3333333333333333d0) * (b * x)))
    else if (f <= 3.2d+270) then
        tmp = 1.0d0 / sin(b)
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.3e-63) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 3.9e-72) {
		tmp = -x / B;
	} else if (F <= 2.5e+136) {
		tmp = (1.0 / B) - ((x / B) + (-0.3333333333333333 * (B * x)));
	} else if (F <= 3.2e+270) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.3e-63:
		tmp = (-1.0 - x) / B
	elif F <= 3.9e-72:
		tmp = -x / B
	elif F <= 2.5e+136:
		tmp = (1.0 / B) - ((x / B) + (-0.3333333333333333 * (B * x)))
	elif F <= 3.2e+270:
		tmp = 1.0 / math.sin(B)
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.3e-63)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 3.9e-72)
		tmp = Float64(Float64(-x) / B);
	elseif (F <= 2.5e+136)
		tmp = Float64(Float64(1.0 / B) - Float64(Float64(x / B) + Float64(-0.3333333333333333 * Float64(B * x))));
	elseif (F <= 3.2e+270)
		tmp = Float64(1.0 / sin(B));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.3e-63)
		tmp = (-1.0 - x) / B;
	elseif (F <= 3.9e-72)
		tmp = -x / B;
	elseif (F <= 2.5e+136)
		tmp = (1.0 / B) - ((x / B) + (-0.3333333333333333 * (B * x)));
	elseif (F <= 3.2e+270)
		tmp = 1.0 / sin(B);
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.3e-63], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 3.9e-72], N[((-x) / B), $MachinePrecision], If[LessEqual[F, 2.5e+136], N[(N[(1.0 / B), $MachinePrecision] - N[(N[(x / B), $MachinePrecision] + N[(-0.3333333333333333 * N[(B * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.2e+270], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -2.3e-63

   1. Initial program 67.4%

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

   2. Taylor expanded in F around -inf 85.9%

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

   3. Taylor expanded in B around 0 54.8%

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

      1. mul-1-neg 54.8%

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

   5. Simplified 54.8%

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

   if -2.3e-63 < F < 3.9e-72

   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 37.7%

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

   3. Taylor expanded in B around 0 22.0%

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

      1. mul-1-neg 22.0%

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

   5. Simplified 22.0%

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

   6. Taylor expanded in x around inf 37.4%

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

   if 3.9e-72 < F < 2.5000000000000001e136

   1. Initial program 95.7%

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

   2. Taylor expanded in B around 0 66.5%

      \[\leadsto \left(-\color{blue}{\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in F around inf 63.6%

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

   4. Taylor expanded in B around 0 63.6%

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

   if 2.5000000000000001e136 < F < 3.2000000000000001e270

   1. Initial program 36.8%

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

   2. Taylor expanded in B around 0 15.0%

      \[\leadsto \left(-\color{blue}{\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in F around inf 60.9%

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

   4. Taylor expanded in x around 0 54.1%

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

   if 3.2000000000000001e270 < F

   1. Initial program 27.3%

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

   2. Taylor expanded in B around 0 0.2%

      \[\leadsto \left(-\color{blue}{\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in F around inf 63.8%

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

   4. Taylor expanded in B around 0 65.5%

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

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.3 \cdot 10^{-63}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 3.9 \cdot 10^{-72}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 2.5 \cdot 10^{+136}:\\ \;\;\;\;\frac{1}{B} - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \mathbf{elif}\;F \leq 3.2 \cdot 10^{+270}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 18: 50.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.85 \cdot 10^{-90}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.9 \cdot 10^{-72}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{+136}:\\ \;\;\;\;\frac{1}{B} - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \mathbf{elif}\;F \leq 1.45 \cdot 10^{+270}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.85e-90)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 3.9e-72)
     (/ (- x) B)
     (if (<= F 5e+136)
       (- (/ 1.0 B) (+ (/ x B) (* -0.3333333333333333 (* B x))))
       (if (<= F 1.45e+270) (/ 1.0 (sin B)) (/ (- 1.0 x) B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.85e-90) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 3.9e-72) {
		tmp = -x / B;
	} else if (F <= 5e+136) {
		tmp = (1.0 / B) - ((x / B) + (-0.3333333333333333 * (B * x)));
	} else if (F <= 1.45e+270) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-3.85d-90)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 3.9d-72) then
        tmp = -x / b
    else if (f <= 5d+136) then
        tmp = (1.0d0 / b) - ((x / b) + ((-0.3333333333333333d0) * (b * x)))
    else if (f <= 1.45d+270) then
        tmp = 1.0d0 / sin(b)
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.85e-90) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 3.9e-72) {
		tmp = -x / B;
	} else if (F <= 5e+136) {
		tmp = (1.0 / B) - ((x / B) + (-0.3333333333333333 * (B * x)));
	} else if (F <= 1.45e+270) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.85e-90:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 3.9e-72:
		tmp = -x / B
	elif F <= 5e+136:
		tmp = (1.0 / B) - ((x / B) + (-0.3333333333333333 * (B * x)))
	elif F <= 1.45e+270:
		tmp = 1.0 / math.sin(B)
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.85e-90)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 3.9e-72)
		tmp = Float64(Float64(-x) / B);
	elseif (F <= 5e+136)
		tmp = Float64(Float64(1.0 / B) - Float64(Float64(x / B) + Float64(-0.3333333333333333 * Float64(B * x))));
	elseif (F <= 1.45e+270)
		tmp = Float64(1.0 / sin(B));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -3.85e-90)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 3.9e-72)
		tmp = -x / B;
	elseif (F <= 5e+136)
		tmp = (1.0 / B) - ((x / B) + (-0.3333333333333333 * (B * x)));
	elseif (F <= 1.45e+270)
		tmp = 1.0 / sin(B);
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.85e-90], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.9e-72], N[((-x) / B), $MachinePrecision], If[LessEqual[F, 5e+136], N[(N[(1.0 / B), $MachinePrecision] - N[(N[(x / B), $MachinePrecision] + N[(-0.3333333333333333 * N[(B * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.45e+270], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -3.84999999999999986e-90

   1. Initial program 68.6%

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

   2. Taylor expanded in F around -inf 84.2%

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

   3. Taylor expanded in B around 0 72.4%

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

   if -3.84999999999999986e-90 < F < 3.9e-72

   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 37.7%

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

   3. Taylor expanded in B around 0 22.6%

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

      1. mul-1-neg 22.6%

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

   5. Simplified 22.6%

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

   6. Taylor expanded in x around inf 38.5%

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

   if 3.9e-72 < F < 5.0000000000000002e136

   1. Initial program 95.7%

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

   2. Taylor expanded in B around 0 66.5%

      \[\leadsto \left(-\color{blue}{\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in F around inf 63.6%

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

   4. Taylor expanded in B around 0 63.6%

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

   if 5.0000000000000002e136 < F < 1.44999999999999995e270

   1. Initial program 36.8%

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

   2. Taylor expanded in B around 0 15.0%

      \[\leadsto \left(-\color{blue}{\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in F around inf 60.9%

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

   4. Taylor expanded in x around 0 54.1%

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

   if 1.44999999999999995e270 < F

   1. Initial program 27.3%

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

   2. Taylor expanded in B around 0 0.2%

      \[\leadsto \left(-\color{blue}{\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in F around inf 63.8%

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

   4. Taylor expanded in B around 0 65.5%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.85 \cdot 10^{-90}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.9 \cdot 10^{-72}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{+136}:\\ \;\;\;\;\frac{1}{B} - \left(\frac{x}{B} + -0.3333333333333333 \cdot \left(B \cdot x\right)\right)\\ \mathbf{elif}\;F \leq 1.45 \cdot 10^{+270}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 19: 37.5% accurate, 24.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{B}\\ \mathbf{if}\;F \leq -4 \cdot 10^{+222}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq -2.8 \cdot 10^{+70}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -7.6:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq 2.4 \cdot 10^{-109}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) B)))
   (if (<= F -4e+222)
     (/ -1.0 B)
     (if (<= F -2.8e+70)
       t_0
       (if (<= F -7.6) (/ -1.0 B) (if (<= F 2.4e-109) t_0 (/ (- 1.0 x) B)))))))

C

double code(double F, double B, double x) {
	double t_0 = -x / B;
	double tmp;
	if (F <= -4e+222) {
		tmp = -1.0 / B;
	} else if (F <= -2.8e+70) {
		tmp = t_0;
	} else if (F <= -7.6) {
		tmp = -1.0 / B;
	} else if (F <= 2.4e-109) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = -x / b
    if (f <= (-4d+222)) then
        tmp = (-1.0d0) / b
    else if (f <= (-2.8d+70)) then
        tmp = t_0
    else if (f <= (-7.6d0)) then
        tmp = (-1.0d0) / b
    else if (f <= 2.4d-109) then
        tmp = t_0
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = -x / B;
	double tmp;
	if (F <= -4e+222) {
		tmp = -1.0 / B;
	} else if (F <= -2.8e+70) {
		tmp = t_0;
	} else if (F <= -7.6) {
		tmp = -1.0 / B;
	} else if (F <= 2.4e-109) {
		tmp = t_0;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -x / B
	tmp = 0
	if F <= -4e+222:
		tmp = -1.0 / B
	elif F <= -2.8e+70:
		tmp = t_0
	elif F <= -7.6:
		tmp = -1.0 / B
	elif F <= 2.4e-109:
		tmp = t_0
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / B)
	tmp = 0.0
	if (F <= -4e+222)
		tmp = Float64(-1.0 / B);
	elseif (F <= -2.8e+70)
		tmp = t_0;
	elseif (F <= -7.6)
		tmp = Float64(-1.0 / B);
	elseif (F <= 2.4e-109)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = -x / B;
	tmp = 0.0;
	if (F <= -4e+222)
		tmp = -1.0 / B;
	elseif (F <= -2.8e+70)
		tmp = t_0;
	elseif (F <= -7.6)
		tmp = -1.0 / B;
	elseif (F <= 2.4e-109)
		tmp = t_0;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-x) / B), $MachinePrecision]}, If[LessEqual[F, -4e+222], N[(-1.0 / B), $MachinePrecision], If[LessEqual[F, -2.8e+70], t$95$0, If[LessEqual[F, -7.6], N[(-1.0 / B), $MachinePrecision], If[LessEqual[F, 2.4e-109], t$95$0, N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.0000000000000002e222 or -2.7999999999999999e70 < F < -7.5999999999999996

   1. Initial program 63.9%

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

   2. Taylor expanded in F around -inf 93.5%

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

   3. Taylor expanded in B around 0 54.6%

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

      1. mul-1-neg 54.6%

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

   5. Simplified 54.6%

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

   6. Taylor expanded in x around 0 44.9%

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

   if -4.0000000000000002e222 < F < -2.7999999999999999e70 or -7.5999999999999996 < F < 2.39999999999999989e-109

   1. Initial program 89.6%

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

   2. Taylor expanded in F around -inf 50.6%

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

   3. Taylor expanded in B around 0 31.1%

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

      1. mul-1-neg 31.1%

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

   5. Simplified 31.1%

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

   6. Taylor expanded in x around inf 37.6%

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

   if 2.39999999999999989e-109 < 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 0 33.6%

      \[\leadsto \left(-\color{blue}{\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in F around inf 62.7%

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

   4. Taylor expanded in B around 0 53.1%

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

4. Final simplification 43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4 \cdot 10^{+222}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq -2.8 \cdot 10^{+70}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq -7.6:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq 2.4 \cdot 10^{-109}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 20: 43.9% accurate, 35.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -4.2 \cdot 10^{-73}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 3 \cdot 10^{-109}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4.2e-73)
   (/ (- -1.0 x) B)
   (if (<= F 3e-109) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.2e-73) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 3e-109) {
		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 <= (-4.2d-73)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 3d-109) 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 <= -4.2e-73) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 3e-109) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -4.2e-73:
		tmp = (-1.0 - x) / B
	elif F <= 3e-109:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -4.2e-73)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 3e-109)
		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 <= -4.2e-73)
		tmp = (-1.0 - x) / B;
	elseif (F <= 3e-109)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -4.2e-73], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 3e-109], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.1999999999999997e-73

   1. Initial program 67.4%

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

   2. Taylor expanded in F around -inf 85.9%

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

   3. Taylor expanded in B around 0 54.8%

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

      1. mul-1-neg 54.8%

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

   5. Simplified 54.8%

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

   if -4.1999999999999997e-73 < F < 3.00000000000000021e-109

   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 35.8%

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

   3. Taylor expanded in B around 0 19.0%

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

      1. mul-1-neg 19.0%

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

   5. Simplified 19.0%

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

   6. Taylor expanded in x around inf 35.5%

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

   if 3.00000000000000021e-109 < 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 0 33.6%

      \[\leadsto \left(-\color{blue}{\left(-0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{x}{B}\right)}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in F around inf 62.7%

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

   4. Taylor expanded in B around 0 53.1%

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

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.2 \cdot 10^{-73}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 3 \cdot 10^{-109}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 21: 31.5% accurate, 39.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-58} \lor \neg \left(x \leq 1.15 \cdot 10^{-120}\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (or (<= x -3.6e-58) (not (<= x 1.15e-120))) (/ (- x) B) (/ -1.0 B)))

C

double code(double F, double B, double x) {
	double tmp;
	if ((x <= -3.6e-58) || !(x <= 1.15e-120)) {
		tmp = -x / B;
	} else {
		tmp = -1.0 / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-3.6d-58)) .or. (.not. (x <= 1.15d-120))) then
        tmp = -x / b
    else
        tmp = (-1.0d0) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if ((x <= -3.6e-58) || !(x <= 1.15e-120)) {
		tmp = -x / B;
	} else {
		tmp = -1.0 / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if (x <= -3.6e-58) or not (x <= 1.15e-120):
		tmp = -x / B
	else:
		tmp = -1.0 / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if ((x <= -3.6e-58) || !(x <= 1.15e-120))
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(-1.0 / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if ((x <= -3.6e-58) || ~((x <= 1.15e-120)))
		tmp = -x / B;
	else
		tmp = -1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[Or[LessEqual[x, -3.6e-58], N[Not[LessEqual[x, 1.15e-120]], $MachinePrecision]], N[((-x) / B), $MachinePrecision], N[(-1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.60000000000000009e-58 or 1.14999999999999993e-120 < x

   1. Initial program 80.4%

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

   2. Taylor expanded in F around -inf 78.2%

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

   3. Taylor expanded in B around 0 47.1%

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

      1. mul-1-neg 47.1%

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

   5. Simplified 47.1%

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

   6. Taylor expanded in x around inf 52.3%

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

   if -3.60000000000000009e-58 < x < 1.14999999999999993e-120

   1. Initial program 71.5%

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

   2. Taylor expanded in F around -inf 31.8%

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

   3. Taylor expanded in B around 0 21.8%

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

      1. mul-1-neg 21.8%

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

   5. Simplified 21.8%

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

   6. Taylor expanded in x around 0 21.8%

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

4. Final simplification 38.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-58} \lor \neg \left(x \leq 1.15 \cdot 10^{-120}\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B}\\ \end{array} \]

Alternative 22: 10.6% 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 76.5%

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

2. Taylor expanded in F around -inf 57.9%

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

3. Taylor expanded in B around 0 36.0%

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

   1. mul-1-neg 36.0%

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

5. Simplified 36.0%

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

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 2023301 
(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))))))