VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.4% → 99.6%

Time: 20.4s

Alternatives: 18

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -155000000:\\ \;\;\;\;t_0 + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 6100000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B)))))
   (if (<= F -155000000.0)
     (+ t_0 (/ -1.0 (sin B)))
     (if (<= F 6100000000.0)
       (-
        (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5))
        (/ x (tan 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 tmp;
	if (F <= -155000000.0) {
		tmp = t_0 + (-1.0 / sin(B));
	} else if (F <= 6100000000.0) {
		tmp = ((F / sin(B)) * pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - (x / tan(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) :: tmp
    t_0 = x * ((-1.0d0) / tan(b))
    if (f <= (-155000000.0d0)) then
        tmp = t_0 + ((-1.0d0) / sin(b))
    else if (f <= 6100000000.0d0) then
        tmp = ((f / sin(b)) * ((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0))) - (x / tan(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 tmp;
	if (F <= -155000000.0) {
		tmp = t_0 + (-1.0 / Math.sin(B));
	} else if (F <= 6100000000.0) {
		tmp = ((F / Math.sin(B)) * Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - (x / Math.tan(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))
	tmp = 0
	if F <= -155000000.0:
		tmp = t_0 + (-1.0 / math.sin(B))
	elif F <= 6100000000.0:
		tmp = ((F / math.sin(B)) * math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - (x / math.tan(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)))
	tmp = 0.0
	if (F <= -155000000.0)
		tmp = Float64(t_0 + Float64(-1.0 / sin(B)));
	elseif (F <= 6100000000.0)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5)) - Float64(x / tan(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));
	tmp = 0.0;
	if (F <= -155000000.0)
		tmp = t_0 + (-1.0 / sin(B));
	elseif (F <= 6100000000.0)
		tmp = ((F / sin(B)) * ((((F * F) + 2.0) + (x * 2.0)) ^ -0.5)) - (x / tan(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]}, If[LessEqual[F, -155000000.0], N[(t$95$0 + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 6100000000.0], 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 / N[Tan[B], $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 < -1.55e8

   1. Initial program 62.1%

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

   2. Taylor expanded in F around -inf 99.8%

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

   if -1.55e8 < F < 6.1e9

   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. Step-by-step derivation

      1. div-inv 99.6%

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

      2. expm1-log1p-u 67.4%

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

      3. expm1-udef 45.6%

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

   3. Applied egg-rr 45.6%

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

      1. expm1-def 67.4%

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

      2. expm1-log1p 99.6%

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

   5. Simplified 99.6%

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

   if 6.1e9 < F

   1. Initial program 61.4%

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

   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. Recombined 3 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -155000000:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 6100000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 2: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -1.5:\\ \;\;\;\;t_0 + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;t_0 + \frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B)))))
   (if (<= F -1.5)
     (+ t_0 (/ -1.0 (sin B)))
     (if (<= F 1.4)
       (+ t_0 (* (/ F (sin B)) (sqrt (/ 1.0 (+ 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 <= -1.5) {
		tmp = t_0 + (-1.0 / sin(B));
	} else if (F <= 1.4) {
		tmp = t_0 + ((F / sin(B)) * sqrt((1.0 / (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 <= (-1.5d0)) then
        tmp = t_0 + ((-1.0d0) / sin(b))
    else if (f <= 1.4d0) then
        tmp = t_0 + ((f / sin(b)) * sqrt((1.0d0 / (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 <= -1.5) {
		tmp = t_0 + (-1.0 / Math.sin(B));
	} else if (F <= 1.4) {
		tmp = t_0 + ((F / Math.sin(B)) * Math.sqrt((1.0 / (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 <= -1.5:
		tmp = t_0 + (-1.0 / math.sin(B))
	elif F <= 1.4:
		tmp = t_0 + ((F / math.sin(B)) * math.sqrt((1.0 / (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 <= -1.5)
		tmp = Float64(t_0 + Float64(-1.0 / sin(B)));
	elseif (F <= 1.4)
		tmp = Float64(t_0 + Float64(Float64(F / sin(B)) * sqrt(Float64(1.0 / 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 <= -1.5)
		tmp = t_0 + (-1.0 / sin(B));
	elseif (F <= 1.4)
		tmp = t_0 + ((F / sin(B)) * sqrt((1.0 / (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, -1.5], N[(t$95$0 + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.4], N[(t$95$0 + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.5

   1. Initial program 64.0%

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

   2. Taylor expanded in F around -inf 97.9%

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

   if -1.5 < F < 1.3999999999999999

   1. Initial program 99.5%

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

   2. Taylor expanded in F around 0 98.1%

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

   if 1.3999999999999999 < F

   1. Initial program 61.4%

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

   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. Recombined 3 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.5:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 3: 98.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B)))))
   (if (<= F -1.4)
     (+ t_0 (/ -1.0 (sin B)))
     (if (<= F 1.4)
       (+ t_0 (/ (* F (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))) (sin 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 tmp;
	if (F <= -1.4) {
		tmp = t_0 + (-1.0 / sin(B));
	} else if (F <= 1.4) {
		tmp = t_0 + ((F * sqrt((1.0 / (2.0 + (x * 2.0))))) / sin(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) :: tmp
    t_0 = x * ((-1.0d0) / tan(b))
    if (f <= (-1.4d0)) then
        tmp = t_0 + ((-1.0d0) / sin(b))
    else if (f <= 1.4d0) then
        tmp = t_0 + ((f * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))) / sin(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 tmp;
	if (F <= -1.4) {
		tmp = t_0 + (-1.0 / Math.sin(B));
	} else if (F <= 1.4) {
		tmp = t_0 + ((F * Math.sqrt((1.0 / (2.0 + (x * 2.0))))) / Math.sin(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))
	tmp = 0
	if F <= -1.4:
		tmp = t_0 + (-1.0 / math.sin(B))
	elif F <= 1.4:
		tmp = t_0 + ((F * math.sqrt((1.0 / (2.0 + (x * 2.0))))) / math.sin(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)))
	tmp = 0.0
	if (F <= -1.4)
		tmp = Float64(t_0 + Float64(-1.0 / sin(B)));
	elseif (F <= 1.4)
		tmp = Float64(t_0 + Float64(Float64(F * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))) / sin(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));
	tmp = 0.0;
	if (F <= -1.4)
		tmp = t_0 + (-1.0 / sin(B));
	elseif (F <= 1.4)
		tmp = t_0 + ((F * sqrt((1.0 / (2.0 + (x * 2.0))))) / sin(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]}, If[LessEqual[F, -1.4], N[(t$95$0 + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.4], N[(t$95$0 + N[(N[(F * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $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 < -1.3999999999999999

   1. Initial program 64.0%

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

   2. Taylor expanded in F around -inf 97.9%

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

   if -1.3999999999999999 < F < 1.3999999999999999

   1. Initial program 99.5%

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

      1. associate-*l/ 99.5%

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

      2. +-commutative 99.5%

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

      3. *-commutative 99.5%

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

      4. fma-udef 99.5%

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

      5. fma-def 99.5%

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

      6. metadata-eval 99.5%

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

      7. metadata-eval 99.5%

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

      8. fma-def 99.5%

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

      9. fma-udef 99.5%

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

      10. *-commutative 99.5%

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

      11. fma-def 99.5%

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

      12. fma-def 99.5%

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

   3. Applied egg-rr 99.5%

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

   4. Taylor expanded in F around 0 98.1%

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

   if 1.3999999999999999 < F

   1. Initial program 61.4%

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

   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. Recombined 3 regimes into one program.

4. Final simplification 98.4%

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

Alternative 4: 89.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ t_1 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -13000000:\\ \;\;\;\;t_1 + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -6.2 \cdot 10^{-77}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 6.4 \cdot 10^{-121}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 108000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1 + \frac{1}{\sin B}\\ \end{array} \end{array} \]

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)))
        (t_1 (* x (/ -1.0 (tan B)))))
   (if (<= F -13000000.0)
     (+ t_1 (/ -1.0 (sin B)))
     (if (<= F -6.2e-77)
       t_0
       (if (<= F 6.4e-121)
         (/ (- x) (tan B))
         (if (<= F 108000.0) t_0 (+ t_1 (/ 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 t_1 = x * (-1.0 / tan(B));
	double tmp;
	if (F <= -13000000.0) {
		tmp = t_1 + (-1.0 / sin(B));
	} else if (F <= -6.2e-77) {
		tmp = t_0;
	} else if (F <= 6.4e-121) {
		tmp = -x / tan(B);
	} else if (F <= 108000.0) {
		tmp = t_0;
	} else {
		tmp = t_1 + (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 / sin(b)) * ((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0))) - (x / b)
    t_1 = x * ((-1.0d0) / tan(b))
    if (f <= (-13000000.0d0)) then
        tmp = t_1 + ((-1.0d0) / sin(b))
    else if (f <= (-6.2d-77)) then
        tmp = t_0
    else if (f <= 6.4d-121) then
        tmp = -x / tan(b)
    else if (f <= 108000.0d0) then
        tmp = t_0
    else
        tmp = t_1 + (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 t_1 = x * (-1.0 / Math.tan(B));
	double tmp;
	if (F <= -13000000.0) {
		tmp = t_1 + (-1.0 / Math.sin(B));
	} else if (F <= -6.2e-77) {
		tmp = t_0;
	} else if (F <= 6.4e-121) {
		tmp = -x / Math.tan(B);
	} else if (F <= 108000.0) {
		tmp = t_0;
	} else {
		tmp = t_1 + (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)
	t_1 = x * (-1.0 / math.tan(B))
	tmp = 0
	if F <= -13000000.0:
		tmp = t_1 + (-1.0 / math.sin(B))
	elif F <= -6.2e-77:
		tmp = t_0
	elif F <= 6.4e-121:
		tmp = -x / math.tan(B)
	elif F <= 108000.0:
		tmp = t_0
	else:
		tmp = t_1 + (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))
	t_1 = Float64(x * Float64(-1.0 / tan(B)))
	tmp = 0.0
	if (F <= -13000000.0)
		tmp = Float64(t_1 + Float64(-1.0 / sin(B)));
	elseif (F <= -6.2e-77)
		tmp = t_0;
	elseif (F <= 6.4e-121)
		tmp = Float64(Float64(-x) / tan(B));
	elseif (F <= 108000.0)
		tmp = t_0;
	else
		tmp = Float64(t_1 + 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);
	t_1 = x * (-1.0 / tan(B));
	tmp = 0.0;
	if (F <= -13000000.0)
		tmp = t_1 + (-1.0 / sin(B));
	elseif (F <= -6.2e-77)
		tmp = t_0;
	elseif (F <= 6.4e-121)
		tmp = -x / tan(B);
	elseif (F <= 108000.0)
		tmp = t_0;
	else
		tmp = t_1 + (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]}, Block[{t$95$1 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -13000000.0], N[(t$95$1 + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -6.2e-77], t$95$0, If[LessEqual[F, 6.4e-121], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 108000.0], t$95$0, N[(t$95$1 + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.3e7

   1. Initial program 62.1%

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

   2. Taylor expanded in F around -inf 99.8%

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

   if -1.3e7 < F < -6.20000000000000016e-77 or 6.40000000000000038e-121 < F < 108000

   1. Initial program 99.5%

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

   2. Taylor expanded in B around 0 84.7%

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

   if -6.20000000000000016e-77 < F < 6.40000000000000038e-121

   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. Step-by-step derivation

      1. distribute-lft-neg-in 99.5%

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

      2. +-commutative 99.5%

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

      3. fma-def 99.5%

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

      4. +-commutative 99.5%

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

      5. *-commutative 99.5%

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

      6. fma-def 99.5%

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

      7. fma-def 99.5%

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

      8. metadata-eval 99.5%

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

      9. metadata-eval 99.5%

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

      10. associate-*r/ 99.7%

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

      11. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in F around 0 81.8%

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

      1. mul-1-neg 81.8%

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

      2. associate-*l/ 81.7%

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

      3. distribute-rgt-neg-in 81.7%

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

   6. Simplified 81.7%

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

      1. distribute-rgt-neg-out 81.7%

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

      2. neg-sub0 81.7%

         \[\leadsto \color{blue}{0 - \frac{x}{\sin B} \cdot \cos B} \]

      3. clear-num 81.6%

         \[\leadsto 0 - \color{blue}{\frac{1}{\frac{\sin B}{x}}} \cdot \cos B \]

      4. associate-*l/ 81.7%

         \[\leadsto 0 - \color{blue}{\frac{1 \cdot \cos B}{\frac{\sin B}{x}}} \]

      5. *-un-lft-identity 81.7%

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

   8. Applied egg-rr 81.7%

      \[\leadsto \color{blue}{0 - \frac{\cos B}{\frac{\sin B}{x}}} \]
   9. Step-by-step derivation

      1. neg-sub0 81.7%

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

      2. distribute-neg-frac 81.7%

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

   10. Simplified 81.7%

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

       1. associate-/r/ 81.7%

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

       2. clear-num 81.6%

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

       3. add-sqr-sqrt 35.2%

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

       4. sqrt-unprod 36.4%

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

       5. sqr-neg 36.4%

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

       6. sqrt-unprod 1.2%

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

       7. add-sqr-sqrt 2.2%

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

       8. tan-quot 2.2%

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

       9. *-commutative 2.2%

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

       10. add-sqr-sqrt 1.2%

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

       11. sqrt-unprod 26.6%

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

       12. sqr-neg 26.6%

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

       13. sqrt-unprod 45.6%

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

       14. add-sqr-sqrt 81.6%

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

       15. neg-sub0 81.6%

           \[\leadsto \color{blue}{0 - x \cdot \frac{1}{\tan B}} \]

       16. un-div-inv 81.9%

           \[\leadsto 0 - \color{blue}{\frac{x}{\tan B}} \]

   12. Applied egg-rr 81.9%

       \[\leadsto \color{blue}{0 - \frac{x}{\tan B}} \]
   13. Step-by-step derivation

       1. neg-sub0 81.9%

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

       2. distribute-neg-frac 81.9%

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

   14. Simplified 81.9%

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

   if 108000 < F

   1. Initial program 61.4%

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

   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. Recombined 4 regimes into one program.

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -13000000:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -6.2 \cdot 10^{-77}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 6.4 \cdot 10^{-121}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 108000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 5: 92.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5}\\ t_1 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -15200:\\ \;\;\;\;t_1 + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{-86}:\\ \;\;\;\;t_1 + t_0 \cdot \frac{F}{B}\\ \mathbf{elif}\;F \leq 920:\\ \;\;\;\;\frac{F}{\sin B} \cdot t_0 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t_1 + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5))
        (t_1 (* x (/ -1.0 (tan B)))))
   (if (<= F -15200.0)
     (+ t_1 (/ -1.0 (sin B)))
     (if (<= F 1.8e-86)
       (+ t_1 (* t_0 (/ F B)))
       (if (<= F 920.0)
         (- (* (/ F (sin B)) t_0) (/ x B))
         (+ t_1 (/ 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 * (-1.0 / tan(B));
	double tmp;
	if (F <= -15200.0) {
		tmp = t_1 + (-1.0 / sin(B));
	} else if (F <= 1.8e-86) {
		tmp = t_1 + (t_0 * (F / B));
	} else if (F <= 920.0) {
		tmp = ((F / sin(B)) * t_0) - (x / B);
	} else {
		tmp = t_1 + (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 * ((-1.0d0) / tan(b))
    if (f <= (-15200.0d0)) then
        tmp = t_1 + ((-1.0d0) / sin(b))
    else if (f <= 1.8d-86) then
        tmp = t_1 + (t_0 * (f / b))
    else if (f <= 920.0d0) then
        tmp = ((f / sin(b)) * t_0) - (x / b)
    else
        tmp = t_1 + (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 * (-1.0 / Math.tan(B));
	double tmp;
	if (F <= -15200.0) {
		tmp = t_1 + (-1.0 / Math.sin(B));
	} else if (F <= 1.8e-86) {
		tmp = t_1 + (t_0 * (F / B));
	} else if (F <= 920.0) {
		tmp = ((F / Math.sin(B)) * t_0) - (x / B);
	} else {
		tmp = t_1 + (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 * (-1.0 / math.tan(B))
	tmp = 0
	if F <= -15200.0:
		tmp = t_1 + (-1.0 / math.sin(B))
	elif F <= 1.8e-86:
		tmp = t_1 + (t_0 * (F / B))
	elif F <= 920.0:
		tmp = ((F / math.sin(B)) * t_0) - (x / B)
	else:
		tmp = t_1 + (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(x * Float64(-1.0 / tan(B)))
	tmp = 0.0
	if (F <= -15200.0)
		tmp = Float64(t_1 + Float64(-1.0 / sin(B)));
	elseif (F <= 1.8e-86)
		tmp = Float64(t_1 + Float64(t_0 * Float64(F / B)));
	elseif (F <= 920.0)
		tmp = Float64(Float64(Float64(F / sin(B)) * t_0) - Float64(x / B));
	else
		tmp = Float64(t_1 + 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 * (-1.0 / tan(B));
	tmp = 0.0;
	if (F <= -15200.0)
		tmp = t_1 + (-1.0 / sin(B));
	elseif (F <= 1.8e-86)
		tmp = t_1 + (t_0 * (F / B));
	elseif (F <= 920.0)
		tmp = ((F / sin(B)) * t_0) - (x / B);
	else
		tmp = t_1 + (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[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -15200.0], N[(t$95$1 + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.8e-86], N[(t$95$1 + N[(t$95$0 * N[(F / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 920.0], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -15200

   1. Initial program 63.1%

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

   2. Taylor expanded in F around -inf 99.4%

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

   if -15200 < F < 1.79999999999999983e-86

   1. Initial program 99.5%

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

   2. Taylor expanded in B around 0 85.6%

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

   if 1.79999999999999983e-86 < F < 920

   1. Initial program 99.5%

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

   2. Taylor expanded in B around 0 93.2%

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

   if 920 < F

   1. Initial program 61.4%

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

   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. Recombined 4 regimes into one program.

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -15200:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{-86}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B}\\ \mathbf{elif}\;F \leq 920:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 6: 84.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -9 \cdot 10^{-18}:\\ \;\;\;\;t_0 + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 2.7 \cdot 10^{-31}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B)))))
   (if (<= F -9e-18)
     (+ t_0 (/ -1.0 (sin B)))
     (if (<= F 2.7e-31) (/ (- x) (tan 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 tmp;
	if (F <= -9e-18) {
		tmp = t_0 + (-1.0 / sin(B));
	} else if (F <= 2.7e-31) {
		tmp = -x / tan(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) :: tmp
    t_0 = x * ((-1.0d0) / tan(b))
    if (f <= (-9d-18)) then
        tmp = t_0 + ((-1.0d0) / sin(b))
    else if (f <= 2.7d-31) then
        tmp = -x / tan(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 tmp;
	if (F <= -9e-18) {
		tmp = t_0 + (-1.0 / Math.sin(B));
	} else if (F <= 2.7e-31) {
		tmp = -x / Math.tan(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))
	tmp = 0
	if F <= -9e-18:
		tmp = t_0 + (-1.0 / math.sin(B))
	elif F <= 2.7e-31:
		tmp = -x / math.tan(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)))
	tmp = 0.0
	if (F <= -9e-18)
		tmp = Float64(t_0 + Float64(-1.0 / sin(B)));
	elseif (F <= 2.7e-31)
		tmp = Float64(Float64(-x) / tan(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));
	tmp = 0.0;
	if (F <= -9e-18)
		tmp = t_0 + (-1.0 / sin(B));
	elseif (F <= 2.7e-31)
		tmp = -x / tan(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]}, If[LessEqual[F, -9e-18], N[(t$95$0 + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.7e-31], N[((-x) / N[Tan[B], $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 < -8.99999999999999987e-18

   1. Initial program 66.1%

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

   2. Taylor expanded in F around -inf 94.9%

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

   if -8.99999999999999987e-18 < F < 2.70000000000000014e-31

   1. Initial program 99.5%

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

      1. distribute-lft-neg-in 99.5%

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

      2. +-commutative 99.5%

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

      3. fma-def 99.5%

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

      4. +-commutative 99.5%

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

      5. *-commutative 99.5%

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

      6. fma-def 99.5%

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

      7. fma-def 99.5%

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

      8. metadata-eval 99.5%

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

      9. metadata-eval 99.5%

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

      10. associate-*r/ 99.7%

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

      11. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in F around 0 69.1%

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

      1. mul-1-neg 69.1%

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

      2. associate-*l/ 69.0%

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

      3. distribute-rgt-neg-in 69.0%

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

   6. Simplified 69.0%

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

      1. distribute-rgt-neg-out 69.0%

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

      2. neg-sub0 69.0%

         \[\leadsto \color{blue}{0 - \frac{x}{\sin B} \cdot \cos B} \]

      3. clear-num 68.9%

         \[\leadsto 0 - \color{blue}{\frac{1}{\frac{\sin B}{x}}} \cdot \cos B \]

      4. associate-*l/ 68.9%

         \[\leadsto 0 - \color{blue}{\frac{1 \cdot \cos B}{\frac{\sin B}{x}}} \]

      5. *-un-lft-identity 68.9%

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

   8. Applied egg-rr 68.9%

      \[\leadsto \color{blue}{0 - \frac{\cos B}{\frac{\sin B}{x}}} \]
   9. Step-by-step derivation

      1. neg-sub0 68.9%

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

      2. distribute-neg-frac 68.9%

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

   10. Simplified 68.9%

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

       1. associate-/r/ 69.0%

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

       2. clear-num 68.9%

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

       3. add-sqr-sqrt 24.6%

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

       4. sqrt-unprod 26.0%

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

       5. sqr-neg 26.0%

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

       6. sqrt-unprod 1.3%

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

       7. add-sqr-sqrt 2.3%

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

       8. tan-quot 2.3%

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

       9. *-commutative 2.3%

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

       10. add-sqr-sqrt 1.2%

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

       11. sqrt-unprod 23.0%

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

       12. sqr-neg 23.0%

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

       13. sqrt-unprod 38.4%

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

       14. add-sqr-sqrt 68.9%

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

       15. neg-sub0 68.9%

           \[\leadsto \color{blue}{0 - x \cdot \frac{1}{\tan B}} \]

       16. un-div-inv 69.1%

           \[\leadsto 0 - \color{blue}{\frac{x}{\tan B}} \]

   12. Applied egg-rr 69.1%

       \[\leadsto \color{blue}{0 - \frac{x}{\tan B}} \]
   13. Step-by-step derivation

       1. neg-sub0 69.1%

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

       2. distribute-neg-frac 69.1%

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

   14. Simplified 69.1%

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

   if 2.70000000000000014e-31 < F

   1. Initial program 65.3%

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

   2. Taylor expanded in F around inf 92.0%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -9 \cdot 10^{-18}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 2.7 \cdot 10^{-31}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 7: 77.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -1.2 \cdot 10^{-17}:\\ \;\;\;\;t_0 + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 2.8 \cdot 10^{-11}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B)))))
   (if (<= F -1.2e-17)
     (+ t_0 (/ -1.0 (sin B)))
     (if (<= F 2.8e-11) (/ (- x) (tan B)) (+ t_0 (/ 1.0 B))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double tmp;
	if (F <= -1.2e-17) {
		tmp = t_0 + (-1.0 / sin(B));
	} else if (F <= 2.8e-11) {
		tmp = -x / tan(B);
	} else {
		tmp = t_0 + (1.0 / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * ((-1.0d0) / tan(b))
    if (f <= (-1.2d-17)) then
        tmp = t_0 + ((-1.0d0) / sin(b))
    else if (f <= 2.8d-11) then
        tmp = -x / tan(b)
    else
        tmp = t_0 + (1.0d0 / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / Math.tan(B));
	double tmp;
	if (F <= -1.2e-17) {
		tmp = t_0 + (-1.0 / Math.sin(B));
	} else if (F <= 2.8e-11) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = t_0 + (1.0 / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * (-1.0 / math.tan(B))
	tmp = 0
	if F <= -1.2e-17:
		tmp = t_0 + (-1.0 / math.sin(B))
	elif F <= 2.8e-11:
		tmp = -x / math.tan(B)
	else:
		tmp = t_0 + (1.0 / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	tmp = 0.0
	if (F <= -1.2e-17)
		tmp = Float64(t_0 + Float64(-1.0 / sin(B)));
	elseif (F <= 2.8e-11)
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(t_0 + Float64(1.0 / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x * (-1.0 / tan(B));
	tmp = 0.0;
	if (F <= -1.2e-17)
		tmp = t_0 + (-1.0 / sin(B));
	elseif (F <= 2.8e-11)
		tmp = -x / tan(B);
	else
		tmp = t_0 + (1.0 / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.2e-17], N[(t$95$0 + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.8e-11], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.19999999999999993e-17

   1. Initial program 66.1%

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

   2. Taylor expanded in F around -inf 94.9%

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

   if -1.19999999999999993e-17 < F < 2.8e-11

   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. Step-by-step derivation

      1. distribute-lft-neg-in 99.5%

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

      2. +-commutative 99.5%

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

      3. fma-def 99.5%

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

      4. +-commutative 99.5%

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

      5. *-commutative 99.5%

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

      6. fma-def 99.5%

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

      7. fma-def 99.5%

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

      8. metadata-eval 99.5%

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

      9. metadata-eval 99.5%

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

      10. associate-*r/ 99.7%

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

      11. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in F around 0 67.5%

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

      1. mul-1-neg 67.5%

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

      2. associate-*l/ 67.4%

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

      3. distribute-rgt-neg-in 67.4%

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

   6. Simplified 67.4%

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

      1. distribute-rgt-neg-out 67.4%

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

      2. neg-sub0 67.4%

         \[\leadsto \color{blue}{0 - \frac{x}{\sin B} \cdot \cos B} \]

      3. clear-num 67.3%

         \[\leadsto 0 - \color{blue}{\frac{1}{\frac{\sin B}{x}}} \cdot \cos B \]

      4. associate-*l/ 67.4%

         \[\leadsto 0 - \color{blue}{\frac{1 \cdot \cos B}{\frac{\sin B}{x}}} \]

      5. *-un-lft-identity 67.4%

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

   8. Applied egg-rr 67.4%

      \[\leadsto \color{blue}{0 - \frac{\cos B}{\frac{\sin B}{x}}} \]
   9. Step-by-step derivation

      1. neg-sub0 67.4%

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

      2. distribute-neg-frac 67.4%

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

   10. Simplified 67.4%

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

       1. associate-/r/ 67.4%

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

       2. clear-num 67.3%

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

       3. add-sqr-sqrt 23.7%

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

       4. sqrt-unprod 25.2%

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

       5. sqr-neg 25.2%

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

       6. sqrt-unprod 1.4%

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

       7. add-sqr-sqrt 2.4%

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

       8. tan-quot 2.4%

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

       9. *-commutative 2.4%

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

       10. add-sqr-sqrt 1.4%

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

       11. sqrt-unprod 23.3%

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

       12. sqr-neg 23.3%

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

       13. sqrt-unprod 37.9%

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

       14. add-sqr-sqrt 67.3%

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

       15. neg-sub0 67.3%

           \[\leadsto \color{blue}{0 - x \cdot \frac{1}{\tan B}} \]

       16. un-div-inv 67.5%

           \[\leadsto 0 - \color{blue}{\frac{x}{\tan B}} \]

   12. Applied egg-rr 67.5%

       \[\leadsto \color{blue}{0 - \frac{x}{\tan B}} \]
   13. Step-by-step derivation

       1. neg-sub0 67.5%

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

       2. distribute-neg-frac 67.5%

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

   14. Simplified 67.5%

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

   if 2.8e-11 < F

   1. Initial program 63.2%

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

   2. Taylor expanded in B around 0 44.1%

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

   3. Taylor expanded in F around inf 69.0%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.2 \cdot 10^{-17}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 2.8 \cdot 10^{-11}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \end{array} \]

Alternative 8: 64.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.25 \cdot 10^{-15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 5600:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 3 \cdot 10^{+82} \lor \neg \left(F \leq 1.9 \cdot 10^{+153}\right):\\ \;\;\;\;0.3333333333333333 \cdot \left(x \cdot B\right) + \frac{1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ -1.0 B) (/ x (tan B)))))
   (if (<= F -1.25e-15)
     t_0
     (if (<= F 5600.0)
       (/ (- x) (tan B))
       (if (or (<= F 3e+82) (not (<= F 1.9e+153)))
         (+ (* 0.3333333333333333 (* x B)) (/ (- 1.0 x) B))
         t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 / B) - (x / tan(B));
	double tmp;
	if (F <= -1.25e-15) {
		tmp = t_0;
	} else if (F <= 5600.0) {
		tmp = -x / tan(B);
	} else if ((F <= 3e+82) || !(F <= 1.9e+153)) {
		tmp = (0.3333333333333333 * (x * B)) + ((1.0 - x) / B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-1.0d0) / b) - (x / tan(b))
    if (f <= (-1.25d-15)) then
        tmp = t_0
    else if (f <= 5600.0d0) then
        tmp = -x / tan(b)
    else if ((f <= 3d+82) .or. (.not. (f <= 1.9d+153))) then
        tmp = (0.3333333333333333d0 * (x * b)) + ((1.0d0 - x) / b)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (-1.0 / B) - (x / Math.tan(B));
	double tmp;
	if (F <= -1.25e-15) {
		tmp = t_0;
	} else if (F <= 5600.0) {
		tmp = -x / Math.tan(B);
	} else if ((F <= 3e+82) || !(F <= 1.9e+153)) {
		tmp = (0.3333333333333333 * (x * B)) + ((1.0 - x) / B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (-1.0 / B) - (x / math.tan(B))
	tmp = 0
	if F <= -1.25e-15:
		tmp = t_0
	elif F <= 5600.0:
		tmp = -x / math.tan(B)
	elif (F <= 3e+82) or not (F <= 1.9e+153):
		tmp = (0.3333333333333333 * (x * B)) + ((1.0 - x) / B)
	else:
		tmp = t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 / B) - Float64(x / tan(B)))
	tmp = 0.0
	if (F <= -1.25e-15)
		tmp = t_0;
	elseif (F <= 5600.0)
		tmp = Float64(Float64(-x) / tan(B));
	elseif ((F <= 3e+82) || !(F <= 1.9e+153))
		tmp = Float64(Float64(0.3333333333333333 * Float64(x * B)) + Float64(Float64(1.0 - x) / B));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (-1.0 / B) - (x / tan(B));
	tmp = 0.0;
	if (F <= -1.25e-15)
		tmp = t_0;
	elseif (F <= 5600.0)
		tmp = -x / tan(B);
	elseif ((F <= 3e+82) || ~((F <= 1.9e+153)))
		tmp = (0.3333333333333333 * (x * B)) + ((1.0 - x) / B);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.25e-15], t$95$0, If[LessEqual[F, 5600.0], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[F, 3e+82], N[Not[LessEqual[F, 1.9e+153]], $MachinePrecision]], N[(N[(0.3333333333333333 * N[(x * B), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.25e-15 or 2.99999999999999989e82 < F < 1.89999999999999983e153

   1. Initial program 69.8%

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

   2. Taylor expanded in B around 0 46.7%

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

   3. Taylor expanded in F around inf 34.7%

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

      1. neg-sub0 34.7%

         \[\leadsto \color{blue}{\left(0 - x \cdot \frac{1}{\tan B}\right)} + \frac{F}{B} \cdot \frac{1}{F} \]

      2. associate-+l- 34.7%

         \[\leadsto \color{blue}{0 - \left(x \cdot \frac{1}{\tan B} - \frac{F}{B} \cdot \frac{1}{F}\right)} \]

      3. add-sqr-sqrt 15.8%

         \[\leadsto 0 - \left(\color{blue}{\sqrt{x \cdot \frac{1}{\tan B}} \cdot \sqrt{x \cdot \frac{1}{\tan B}}} - \frac{F}{B} \cdot \frac{1}{F}\right) \]

      4. sqrt-unprod 10.3%

         \[\leadsto 0 - \left(\color{blue}{\sqrt{\left(x \cdot \frac{1}{\tan B}\right) \cdot \left(x \cdot \frac{1}{\tan B}\right)}} - \frac{F}{B} \cdot \frac{1}{F}\right) \]

      5. sqr-neg 10.3%

         \[\leadsto 0 - \left(\sqrt{\color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(-x \cdot \frac{1}{\tan B}\right)}} - \frac{F}{B} \cdot \frac{1}{F}\right) \]

      6. sqrt-unprod 0.9%

         \[\leadsto 0 - \left(\color{blue}{\sqrt{-x \cdot \frac{1}{\tan B}} \cdot \sqrt{-x \cdot \frac{1}{\tan B}}} - \frac{F}{B} \cdot \frac{1}{F}\right) \]

      7. add-sqr-sqrt 1.8%

         \[\leadsto 0 - \left(\color{blue}{\left(-x \cdot \frac{1}{\tan B}\right)} - \frac{F}{B} \cdot \frac{1}{F}\right) \]

      8. *-commutative 1.8%

         \[\leadsto 0 - \left(\left(-x \cdot \frac{1}{\tan B}\right) - \color{blue}{\frac{1}{F} \cdot \frac{F}{B}}\right) \]

      9. cancel-sign-sub-inv 1.8%

         \[\leadsto 0 - \color{blue}{\left(\left(-x \cdot \frac{1}{\tan B}\right) + \left(-\frac{1}{F}\right) \cdot \frac{F}{B}\right)} \]

   5. Applied egg-rr 65.5%

      \[\leadsto \color{blue}{0 - \left(\frac{1}{B} + \frac{x}{\tan B}\right)} \]
   6. Step-by-step derivation

      1. associate--r+ 65.5%

         \[\leadsto \color{blue}{\left(0 - \frac{1}{B}\right) - \frac{x}{\tan B}} \]

      2. neg-sub0 65.5%

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

      3. distribute-neg-frac 65.5%

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

      4. metadata-eval 65.5%

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

   7. Simplified 65.5%

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

   if -1.25e-15 < F < 5600

   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. Step-by-step derivation

      1. distribute-lft-neg-in 99.5%

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

      2. +-commutative 99.5%

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

      3. fma-def 99.5%

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

      4. +-commutative 99.5%

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

      5. *-commutative 99.5%

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

      6. fma-def 99.5%

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

      7. fma-def 99.5%

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

      8. metadata-eval 99.5%

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

      9. metadata-eval 99.5%

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

      10. associate-*r/ 99.7%

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

      11. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in F around 0 65.2%

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

      1. mul-1-neg 65.2%

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

      2. associate-*l/ 65.2%

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

      3. distribute-rgt-neg-in 65.2%

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

   6. Simplified 65.2%

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

      1. distribute-rgt-neg-out 65.2%

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

      2. neg-sub0 65.2%

         \[\leadsto \color{blue}{0 - \frac{x}{\sin B} \cdot \cos B} \]

      3. clear-num 65.1%

         \[\leadsto 0 - \color{blue}{\frac{1}{\frac{\sin B}{x}}} \cdot \cos B \]

      4. associate-*l/ 65.2%

         \[\leadsto 0 - \color{blue}{\frac{1 \cdot \cos B}{\frac{\sin B}{x}}} \]

      5. *-un-lft-identity 65.2%

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

   8. Applied egg-rr 65.2%

      \[\leadsto \color{blue}{0 - \frac{\cos B}{\frac{\sin B}{x}}} \]
   9. Step-by-step derivation

      1. neg-sub0 65.2%

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

      2. distribute-neg-frac 65.2%

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

   10. Simplified 65.2%

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

       1. associate-/r/ 65.2%

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

       2. clear-num 65.1%

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

       3. add-sqr-sqrt 22.8%

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

       4. sqrt-unprod 24.4%

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

       5. sqr-neg 24.4%

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

       6. sqrt-unprod 1.5%

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

       7. add-sqr-sqrt 2.4%

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

       8. tan-quot 2.4%

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

       9. *-commutative 2.4%

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

       10. add-sqr-sqrt 1.3%

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

       11. sqrt-unprod 22.6%

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

       12. sqr-neg 22.6%

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

       13. sqrt-unprod 36.7%

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

       14. add-sqr-sqrt 65.1%

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

       15. neg-sub0 65.1%

           \[\leadsto \color{blue}{0 - x \cdot \frac{1}{\tan B}} \]

       16. un-div-inv 65.3%

           \[\leadsto 0 - \color{blue}{\frac{x}{\tan B}} \]

   12. Applied egg-rr 65.3%

       \[\leadsto \color{blue}{0 - \frac{x}{\tan B}} \]
   13. Step-by-step derivation

       1. neg-sub0 65.3%

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

       2. distribute-neg-frac 65.3%

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

   14. Simplified 65.3%

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

   if 5600 < F < 2.99999999999999989e82 or 1.89999999999999983e153 < F

   1. Initial program 51.4%

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

   2. Taylor expanded in B around 0 43.5%

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

   3. Taylor expanded in F around inf 47.1%

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

   4. Taylor expanded in B around 0 61.7%

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

      1. associate--l+ 61.7%

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

      2. div-sub 61.7%

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

   6. Simplified 61.7%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.25 \cdot 10^{-15}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 5600:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 3 \cdot 10^{+82} \lor \neg \left(F \leq 1.9 \cdot 10^{+153}\right):\\ \;\;\;\;0.3333333333333333 \cdot \left(x \cdot B\right) + \frac{1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 9: 70.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.35 \cdot 10^{-17}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.35e-17)
   (- (/ -1.0 B) (/ x (tan B)))
   (if (<= F 4.5e-12) (/ (- x) (tan B)) (+ (* x (/ -1.0 (tan B))) (/ 1.0 B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.35e-17) {
		tmp = (-1.0 / B) - (x / tan(B));
	} else if (F <= 4.5e-12) {
		tmp = -x / tan(B);
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-2.35d-17)) then
        tmp = ((-1.0d0) / b) - (x / tan(b))
    else if (f <= 4.5d-12) then
        tmp = -x / tan(b)
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.35e-17) {
		tmp = (-1.0 / B) - (x / Math.tan(B));
	} else if (F <= 4.5e-12) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.35e-17:
		tmp = (-1.0 / B) - (x / math.tan(B))
	elif F <= 4.5e-12:
		tmp = -x / math.tan(B)
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.35e-17)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / tan(B)));
	elseif (F <= 4.5e-12)
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.35e-17)
		tmp = (-1.0 / B) - (x / tan(B));
	elseif (F <= 4.5e-12)
		tmp = -x / tan(B);
	else
		tmp = (x * (-1.0 / tan(B))) + (1.0 / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.35e-17], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.5e-12], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.35e-17

   1. Initial program 65.7%

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

   2. Taylor expanded in B around 0 46.6%

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

   3. Taylor expanded in F around inf 32.4%

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

      1. neg-sub0 32.4%

         \[\leadsto \color{blue}{\left(0 - x \cdot \frac{1}{\tan B}\right)} + \frac{F}{B} \cdot \frac{1}{F} \]

      2. associate-+l- 32.4%

         \[\leadsto \color{blue}{0 - \left(x \cdot \frac{1}{\tan B} - \frac{F}{B} \cdot \frac{1}{F}\right)} \]

      3. add-sqr-sqrt 14.9%

         \[\leadsto 0 - \left(\color{blue}{\sqrt{x \cdot \frac{1}{\tan B}} \cdot \sqrt{x \cdot \frac{1}{\tan B}}} - \frac{F}{B} \cdot \frac{1}{F}\right) \]

      4. sqrt-unprod 9.4%

         \[\leadsto 0 - \left(\color{blue}{\sqrt{\left(x \cdot \frac{1}{\tan B}\right) \cdot \left(x \cdot \frac{1}{\tan B}\right)}} - \frac{F}{B} \cdot \frac{1}{F}\right) \]

      5. sqr-neg 9.4%

         \[\leadsto 0 - \left(\sqrt{\color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(-x \cdot \frac{1}{\tan B}\right)}} - \frac{F}{B} \cdot \frac{1}{F}\right) \]

      6. sqrt-unprod 0.9%

         \[\leadsto 0 - \left(\color{blue}{\sqrt{-x \cdot \frac{1}{\tan B}} \cdot \sqrt{-x \cdot \frac{1}{\tan B}}} - \frac{F}{B} \cdot \frac{1}{F}\right) \]

      7. add-sqr-sqrt 1.6%

         \[\leadsto 0 - \left(\color{blue}{\left(-x \cdot \frac{1}{\tan B}\right)} - \frac{F}{B} \cdot \frac{1}{F}\right) \]

      8. *-commutative 1.6%

         \[\leadsto 0 - \left(\left(-x \cdot \frac{1}{\tan B}\right) - \color{blue}{\frac{1}{F} \cdot \frac{F}{B}}\right) \]

      9. cancel-sign-sub-inv 1.6%

         \[\leadsto 0 - \color{blue}{\left(\left(-x \cdot \frac{1}{\tan B}\right) + \left(-\frac{1}{F}\right) \cdot \frac{F}{B}\right)} \]

   5. Applied egg-rr 67.4%

      \[\leadsto \color{blue}{0 - \left(\frac{1}{B} + \frac{x}{\tan B}\right)} \]
   6. Step-by-step derivation

      1. associate--r+ 67.4%

         \[\leadsto \color{blue}{\left(0 - \frac{1}{B}\right) - \frac{x}{\tan B}} \]

      2. neg-sub0 67.4%

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

      3. distribute-neg-frac 67.4%

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

      4. metadata-eval 67.4%

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

   7. Simplified 67.4%

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

   if -2.35e-17 < F < 4.49999999999999981e-12

   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. Step-by-step derivation

      1. distribute-lft-neg-in 99.5%

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

      2. +-commutative 99.5%

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

      3. fma-def 99.5%

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

      4. +-commutative 99.5%

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

      5. *-commutative 99.5%

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

      6. fma-def 99.5%

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

      7. fma-def 99.5%

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

      8. metadata-eval 99.5%

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

      9. metadata-eval 99.5%

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

      10. associate-*r/ 99.7%

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

      11. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in F around 0 66.9%

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

      1. mul-1-neg 66.9%

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

      2. associate-*l/ 66.9%

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

      3. distribute-rgt-neg-in 66.9%

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

   6. Simplified 66.9%

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

      1. distribute-rgt-neg-out 66.9%

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

      2. neg-sub0 66.9%

         \[\leadsto \color{blue}{0 - \frac{x}{\sin B} \cdot \cos B} \]

      3. clear-num 66.8%

         \[\leadsto 0 - \color{blue}{\frac{1}{\frac{\sin B}{x}}} \cdot \cos B \]

      4. associate-*l/ 66.8%

         \[\leadsto 0 - \color{blue}{\frac{1 \cdot \cos B}{\frac{\sin B}{x}}} \]

      5. *-un-lft-identity 66.8%

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

   8. Applied egg-rr 66.8%

      \[\leadsto \color{blue}{0 - \frac{\cos B}{\frac{\sin B}{x}}} \]
   9. Step-by-step derivation

      1. neg-sub0 66.8%

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

      2. distribute-neg-frac 66.8%

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

   10. Simplified 66.8%

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

       1. associate-/r/ 66.8%

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

       2. clear-num 66.8%

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

       3. add-sqr-sqrt 23.5%

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

       4. sqrt-unprod 24.9%

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

       5. sqr-neg 24.9%

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

       6. sqrt-unprod 1.4%

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

       7. add-sqr-sqrt 2.4%

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

       8. tan-quot 2.4%

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

       9. *-commutative 2.4%

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

       10. add-sqr-sqrt 1.4%

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

       11. sqrt-unprod 23.1%

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

       12. sqr-neg 23.1%

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

       13. sqrt-unprod 37.5%

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

       14. add-sqr-sqrt 66.8%

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

       15. neg-sub0 66.8%

           \[\leadsto \color{blue}{0 - x \cdot \frac{1}{\tan B}} \]

       16. un-div-inv 67.0%

           \[\leadsto 0 - \color{blue}{\frac{x}{\tan B}} \]

   12. Applied egg-rr 67.0%

       \[\leadsto \color{blue}{0 - \frac{x}{\tan B}} \]
   13. Step-by-step derivation

       1. neg-sub0 67.0%

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

       2. distribute-neg-frac 67.0%

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

   14. Simplified 67.0%

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

   if 4.49999999999999981e-12 < F

   1. Initial program 63.2%

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

   2. Taylor expanded in B around 0 44.1%

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

   3. Taylor expanded in F around inf 69.0%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.35 \cdot 10^{-17}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \end{array} \]

Alternative 10: 70.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.9 \cdot 10^{-5}:\\ \;\;\;\;\frac{-F}{F \cdot \sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 6.4 \cdot 10^{-13}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.9e-5)
   (- (/ (- F) (* F (sin B))) (/ x B))
   (if (<= F 6.4e-13) (/ (- x) (tan B)) (+ (* x (/ -1.0 (tan B))) (/ 1.0 B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.9e-5) {
		tmp = (-F / (F * sin(B))) - (x / B);
	} else if (F <= 6.4e-13) {
		tmp = -x / tan(B);
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-2.9d-5)) then
        tmp = (-f / (f * sin(b))) - (x / b)
    else if (f <= 6.4d-13) then
        tmp = -x / tan(b)
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.9e-5) {
		tmp = (-F / (F * Math.sin(B))) - (x / B);
	} else if (F <= 6.4e-13) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.9e-5:
		tmp = (-F / (F * math.sin(B))) - (x / B)
	elif F <= 6.4e-13:
		tmp = -x / math.tan(B)
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.9e-5)
		tmp = Float64(Float64(Float64(-F) / Float64(F * sin(B))) - Float64(x / B));
	elseif (F <= 6.4e-13)
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.9e-5)
		tmp = (-F / (F * sin(B))) - (x / B);
	elseif (F <= 6.4e-13)
		tmp = -x / tan(B);
	else
		tmp = (x * (-1.0 / tan(B))) + (1.0 / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.9e-5], N[(N[((-F) / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 6.4e-13], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.9e-5

   1. Initial program 64.4%

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

   2. Taylor expanded in F around -inf 82.8%

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

      1. *-commutative 82.8%

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

      2. frac-times 96.7%

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

      3. neg-mul-1 96.7%

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

   4. Applied egg-rr 96.7%

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

   5. Taylor expanded in B around 0 78.5%

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

   if -2.9e-5 < F < 6.39999999999999999e-13

   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. Step-by-step derivation

      1. distribute-lft-neg-in 99.5%

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

      2. +-commutative 99.5%

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

      3. fma-def 99.5%

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

      4. +-commutative 99.5%

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

      5. *-commutative 99.5%

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

      6. fma-def 99.5%

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

      7. fma-def 99.5%

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

      8. metadata-eval 99.5%

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

      9. metadata-eval 99.5%

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

      10. associate-*r/ 99.7%

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

      11. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in F around 0 66.9%

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

      1. mul-1-neg 66.9%

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

      2. associate-*l/ 66.9%

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

      3. distribute-rgt-neg-in 66.9%

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

   6. Simplified 66.9%

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

      1. distribute-rgt-neg-out 66.9%

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

      2. neg-sub0 66.9%

         \[\leadsto \color{blue}{0 - \frac{x}{\sin B} \cdot \cos B} \]

      3. clear-num 66.8%

         \[\leadsto 0 - \color{blue}{\frac{1}{\frac{\sin B}{x}}} \cdot \cos B \]

      4. associate-*l/ 66.8%

         \[\leadsto 0 - \color{blue}{\frac{1 \cdot \cos B}{\frac{\sin B}{x}}} \]

      5. *-un-lft-identity 66.8%

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

   8. Applied egg-rr 66.8%

      \[\leadsto \color{blue}{0 - \frac{\cos B}{\frac{\sin B}{x}}} \]
   9. Step-by-step derivation

      1. neg-sub0 66.8%

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

      2. distribute-neg-frac 66.8%

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

   10. Simplified 66.8%

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

       1. associate-/r/ 66.8%

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

       2. clear-num 66.8%

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

       3. add-sqr-sqrt 23.8%

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

       4. sqrt-unprod 25.2%

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

       5. sqr-neg 25.2%

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

       6. sqrt-unprod 1.4%

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

       7. add-sqr-sqrt 2.4%

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

       8. tan-quot 2.4%

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

       9. *-commutative 2.4%

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

       10. add-sqr-sqrt 1.4%

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

       11. sqrt-unprod 23.5%

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

       12. sqr-neg 23.5%

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

       13. sqrt-unprod 38.3%

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

       14. add-sqr-sqrt 66.8%

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

       15. neg-sub0 66.8%

           \[\leadsto \color{blue}{0 - x \cdot \frac{1}{\tan B}} \]

       16. un-div-inv 67.0%

           \[\leadsto 0 - \color{blue}{\frac{x}{\tan B}} \]

   12. Applied egg-rr 67.0%

       \[\leadsto \color{blue}{0 - \frac{x}{\tan B}} \]
   13. Step-by-step derivation

       1. neg-sub0 67.0%

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

       2. distribute-neg-frac 67.0%

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

   14. Simplified 67.0%

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

   if 6.39999999999999999e-13 < F

   1. Initial program 63.2%

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

   2. Taylor expanded in B around 0 44.1%

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

   3. Taylor expanded in F around inf 69.0%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.9 \cdot 10^{-5}:\\ \;\;\;\;\frac{-F}{F \cdot \sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 6.4 \cdot 10^{-13}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \end{array} \]

Alternative 11: 57.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -6.1 \cdot 10^{-5}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333\right) + \frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 44 \lor \neg \left(F \leq 8 \cdot 10^{+81}\right) \land F \leq 1.45 \cdot 10^{+153}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(x \cdot B\right) + \frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -6.1e-5)
   (+ (* B (* x 0.3333333333333333)) (/ (- -1.0 x) B))
   (if (or (<= F 44.0) (and (not (<= F 8e+81)) (<= F 1.45e+153)))
     (/ (- x) (tan B))
     (+ (* 0.3333333333333333 (* x B)) (/ (- 1.0 x) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -6.1e-5) {
		tmp = (B * (x * 0.3333333333333333)) + ((-1.0 - x) / B);
	} else if ((F <= 44.0) || (!(F <= 8e+81) && (F <= 1.45e+153))) {
		tmp = -x / tan(B);
	} else {
		tmp = (0.3333333333333333 * (x * B)) + ((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 <= (-6.1d-5)) then
        tmp = (b * (x * 0.3333333333333333d0)) + (((-1.0d0) - x) / b)
    else if ((f <= 44.0d0) .or. (.not. (f <= 8d+81)) .and. (f <= 1.45d+153)) then
        tmp = -x / tan(b)
    else
        tmp = (0.3333333333333333d0 * (x * b)) + ((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 <= -6.1e-5) {
		tmp = (B * (x * 0.3333333333333333)) + ((-1.0 - x) / B);
	} else if ((F <= 44.0) || (!(F <= 8e+81) && (F <= 1.45e+153))) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (0.3333333333333333 * (x * B)) + ((1.0 - x) / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -6.1e-5:
		tmp = (B * (x * 0.3333333333333333)) + ((-1.0 - x) / B)
	elif (F <= 44.0) or (not (F <= 8e+81) and (F <= 1.45e+153)):
		tmp = -x / math.tan(B)
	else:
		tmp = (0.3333333333333333 * (x * B)) + ((1.0 - x) / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -6.1e-5)
		tmp = Float64(Float64(B * Float64(x * 0.3333333333333333)) + Float64(Float64(-1.0 - x) / B));
	elseif ((F <= 44.0) || (!(F <= 8e+81) && (F <= 1.45e+153)))
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(0.3333333333333333 * Float64(x * B)) + Float64(Float64(1.0 - x) / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -6.1e-5)
		tmp = (B * (x * 0.3333333333333333)) + ((-1.0 - x) / B);
	elseif ((F <= 44.0) || (~((F <= 8e+81)) && (F <= 1.45e+153)))
		tmp = -x / tan(B);
	else
		tmp = (0.3333333333333333 * (x * B)) + ((1.0 - x) / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -6.1e-5], N[(N[(B * N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[F, 44.0], And[N[Not[LessEqual[F, 8e+81]], $MachinePrecision], LessEqual[F, 1.45e+153]]], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 * N[(x * B), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -6.09999999999999987e-5

   1. Initial program 64.4%

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

   2. Taylor expanded in B around 0 44.6%

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

   3. Taylor expanded in F around inf 31.1%

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

      1. neg-sub0 31.1%

         \[\leadsto \color{blue}{\left(0 - x \cdot \frac{1}{\tan B}\right)} + \frac{F}{B} \cdot \frac{1}{F} \]

      2. associate-+l- 31.1%

         \[\leadsto \color{blue}{0 - \left(x \cdot \frac{1}{\tan B} - \frac{F}{B} \cdot \frac{1}{F}\right)} \]

      3. add-sqr-sqrt 15.5%

         \[\leadsto 0 - \left(\color{blue}{\sqrt{x \cdot \frac{1}{\tan B}} \cdot \sqrt{x \cdot \frac{1}{\tan B}}} - \frac{F}{B} \cdot \frac{1}{F}\right) \]

      4. sqrt-unprod 9.7%

         \[\leadsto 0 - \left(\color{blue}{\sqrt{\left(x \cdot \frac{1}{\tan B}\right) \cdot \left(x \cdot \frac{1}{\tan B}\right)}} - \frac{F}{B} \cdot \frac{1}{F}\right) \]

      5. sqr-neg 9.7%

         \[\leadsto 0 - \left(\sqrt{\color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(-x \cdot \frac{1}{\tan B}\right)}} - \frac{F}{B} \cdot \frac{1}{F}\right) \]

      6. sqrt-unprod 0.9%

         \[\leadsto 0 - \left(\color{blue}{\sqrt{-x \cdot \frac{1}{\tan B}} \cdot \sqrt{-x \cdot \frac{1}{\tan B}}} - \frac{F}{B} \cdot \frac{1}{F}\right) \]

      7. add-sqr-sqrt 1.6%

         \[\leadsto 0 - \left(\color{blue}{\left(-x \cdot \frac{1}{\tan B}\right)} - \frac{F}{B} \cdot \frac{1}{F}\right) \]

      8. *-commutative 1.6%

         \[\leadsto 0 - \left(\left(-x \cdot \frac{1}{\tan B}\right) - \color{blue}{\frac{1}{F} \cdot \frac{F}{B}}\right) \]

      9. cancel-sign-sub-inv 1.6%

         \[\leadsto 0 - \color{blue}{\left(\left(-x \cdot \frac{1}{\tan B}\right) + \left(-\frac{1}{F}\right) \cdot \frac{F}{B}\right)} \]

   5. Applied egg-rr 67.3%

      \[\leadsto \color{blue}{0 - \left(\frac{1}{B} + \frac{x}{\tan B}\right)} \]
   6. Step-by-step derivation

      1. associate--r+ 67.3%

         \[\leadsto \color{blue}{\left(0 - \frac{1}{B}\right) - \frac{x}{\tan B}} \]

      2. neg-sub0 67.3%

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

      3. distribute-neg-frac 67.3%

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

      4. metadata-eval 67.3%

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

   7. Simplified 67.3%

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

   8. Taylor expanded in B around 0 51.1%

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

      1. +-commutative 51.1%

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

      2. mul-1-neg 51.1%

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

      3. unsub-neg 51.1%

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

      4. *-commutative 51.1%

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

      5. associate-*l* 51.1%

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

   10. Simplified 51.1%

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

   if -6.09999999999999987e-5 < F < 44 or 7.99999999999999937e81 < F < 1.45000000000000001e153

   1. Initial program 98.7%

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

      1. distribute-lft-neg-in 98.7%

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

      2. +-commutative 98.7%

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

      3. fma-def 98.7%

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

      4. +-commutative 98.7%

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

      5. *-commutative 98.7%

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

      6. fma-def 98.7%

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

      7. fma-def 98.7%

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

      8. metadata-eval 98.7%

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

      9. metadata-eval 98.7%

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

      10. associate-*r/ 98.9%

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

      11. *-rgt-identity 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in F around 0 64.0%

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

      1. mul-1-neg 64.0%

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

      2. associate-*l/ 64.0%

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

      3. distribute-rgt-neg-in 64.0%

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

   6. Simplified 64.0%

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

      1. distribute-rgt-neg-out 64.0%

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

      2. neg-sub0 64.0%

         \[\leadsto \color{blue}{0 - \frac{x}{\sin B} \cdot \cos B} \]

      3. clear-num 63.9%

         \[\leadsto 0 - \color{blue}{\frac{1}{\frac{\sin B}{x}}} \cdot \cos B \]

      4. associate-*l/ 63.9%

         \[\leadsto 0 - \color{blue}{\frac{1 \cdot \cos B}{\frac{\sin B}{x}}} \]

      5. *-un-lft-identity 63.9%

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

   8. Applied egg-rr 63.9%

      \[\leadsto \color{blue}{0 - \frac{\cos B}{\frac{\sin B}{x}}} \]
   9. Step-by-step derivation

      1. neg-sub0 63.9%

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

      2. distribute-neg-frac 63.9%

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

   10. Simplified 63.9%

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

       1. associate-/r/ 63.9%

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

       2. clear-num 63.9%

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

       3. add-sqr-sqrt 22.8%

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

       4. sqrt-unprod 24.4%

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

       5. sqr-neg 24.4%

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

       6. sqrt-unprod 1.5%

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

       7. add-sqr-sqrt 2.5%

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

       8. tan-quot 2.5%

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

       9. *-commutative 2.5%

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

       10. add-sqr-sqrt 1.5%

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

       11. sqrt-unprod 22.3%

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

       12. sqr-neg 22.3%

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

       13. sqrt-unprod 36.2%

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

       14. add-sqr-sqrt 63.9%

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

       15. neg-sub0 63.9%

           \[\leadsto \color{blue}{0 - x \cdot \frac{1}{\tan B}} \]

       16. un-div-inv 64.1%

           \[\leadsto 0 - \color{blue}{\frac{x}{\tan B}} \]

   12. Applied egg-rr 64.1%

       \[\leadsto \color{blue}{0 - \frac{x}{\tan B}} \]
   13. Step-by-step derivation

       1. neg-sub0 64.1%

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

       2. distribute-neg-frac 64.1%

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

   14. Simplified 64.1%

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

   if 44 < F < 7.99999999999999937e81 or 1.45000000000000001e153 < F

   1. Initial program 51.4%

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

   2. Taylor expanded in B around 0 43.5%

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

   3. Taylor expanded in F around inf 47.1%

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

   4. Taylor expanded in B around 0 61.7%

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

      1. associate--l+ 61.7%

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

      2. div-sub 61.7%

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

   6. Simplified 61.7%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6.1 \cdot 10^{-5}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333\right) + \frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 44 \lor \neg \left(F \leq 8 \cdot 10^{+81}\right) \land F \leq 1.45 \cdot 10^{+153}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(x \cdot B\right) + \frac{1 - x}{B}\\ \end{array} \]

Alternative 12: 43.2% accurate, 21.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.4 \cdot 10^{-116}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(x \cdot B\right) + \frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.4e-116)
   (- (/ -1.0 B) (/ x B))
   (if (<= F 1.4e-9)
     (/ (- x) B)
     (+ (* 0.3333333333333333 (* x B)) (/ (- 1.0 x) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.4e-116) {
		tmp = (-1.0 / B) - (x / B);
	} else if (F <= 1.4e-9) {
		tmp = -x / B;
	} else {
		tmp = (0.3333333333333333 * (x * B)) + ((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 <= (-2.4d-116)) then
        tmp = ((-1.0d0) / b) - (x / b)
    else if (f <= 1.4d-9) then
        tmp = -x / b
    else
        tmp = (0.3333333333333333d0 * (x * b)) + ((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 <= -2.4e-116) {
		tmp = (-1.0 / B) - (x / B);
	} else if (F <= 1.4e-9) {
		tmp = -x / B;
	} else {
		tmp = (0.3333333333333333 * (x * B)) + ((1.0 - x) / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.4e-116:
		tmp = (-1.0 / B) - (x / B)
	elif F <= 1.4e-9:
		tmp = -x / B
	else:
		tmp = (0.3333333333333333 * (x * B)) + ((1.0 - x) / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.4e-116)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / B));
	elseif (F <= 1.4e-9)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(0.3333333333333333 * Float64(x * B)) + Float64(Float64(1.0 - x) / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.4e-116)
		tmp = (-1.0 / B) - (x / B);
	elseif (F <= 1.4e-9)
		tmp = -x / B;
	else
		tmp = (0.3333333333333333 * (x * B)) + ((1.0 - x) / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.4e-116], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.4e-9], N[((-x) / B), $MachinePrecision], N[(N[(0.3333333333333333 * N[(x * B), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.39999999999999993e-116

   1. Initial program 72.5%

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

   2. Taylor expanded in B around 0 49.5%

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

   3. Taylor expanded in B around 0 29.7%

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

   4. Taylor expanded in F around -inf 43.8%

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

      1. distribute-lft-in 43.8%

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

      2. neg-mul-1 43.8%

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

      3. unsub-neg 43.8%

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

      4. associate-*r/ 43.8%

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

      5. metadata-eval 43.8%

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

   6. Simplified 43.8%

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

   if -2.39999999999999993e-116 < F < 1.39999999999999992e-9

   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. Step-by-step derivation

      1. distribute-lft-neg-in 99.5%

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

      2. +-commutative 99.5%

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

      3. fma-def 99.5%

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

      4. +-commutative 99.5%

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

      5. *-commutative 99.5%

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

      6. fma-def 99.5%

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

      7. fma-def 99.5%

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

      8. metadata-eval 99.5%

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

      9. metadata-eval 99.5%

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

      10. associate-*r/ 99.7%

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

      11. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in F around 0 70.8%

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

      1. mul-1-neg 70.8%

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

      2. associate-*l/ 70.8%

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

      3. distribute-rgt-neg-in 70.8%

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

   6. Simplified 70.8%

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

   7. Taylor expanded in B around 0 35.6%

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

      1. associate-*r/ 35.6%

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

      2. neg-mul-1 35.6%

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

   9. Simplified 35.6%

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

   if 1.39999999999999992e-9 < F

   1. Initial program 62.6%

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

   2. Taylor expanded in B around 0 44.8%

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

   3. Taylor expanded in F around inf 46.1%

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

   4. Taylor expanded in B around 0 51.0%

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

      1. associate--l+ 51.0%

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

      2. div-sub 51.0%

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

   6. Simplified 51.0%

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

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.4 \cdot 10^{-116}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(x \cdot B\right) + \frac{1 - x}{B}\\ \end{array} \]

Alternative 13: 43.2% accurate, 21.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -4.4 \cdot 10^{-126}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333\right) + \frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(x \cdot B\right) + \frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4.4e-126)
   (+ (* B (* x 0.3333333333333333)) (/ (- -1.0 x) B))
   (if (<= F 1.4e-9)
     (/ (- x) B)
     (+ (* 0.3333333333333333 (* x B)) (/ (- 1.0 x) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.4e-126) {
		tmp = (B * (x * 0.3333333333333333)) + ((-1.0 - x) / B);
	} else if (F <= 1.4e-9) {
		tmp = -x / B;
	} else {
		tmp = (0.3333333333333333 * (x * B)) + ((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 <= (-4.4d-126)) then
        tmp = (b * (x * 0.3333333333333333d0)) + (((-1.0d0) - x) / b)
    else if (f <= 1.4d-9) then
        tmp = -x / b
    else
        tmp = (0.3333333333333333d0 * (x * b)) + ((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 <= -4.4e-126) {
		tmp = (B * (x * 0.3333333333333333)) + ((-1.0 - x) / B);
	} else if (F <= 1.4e-9) {
		tmp = -x / B;
	} else {
		tmp = (0.3333333333333333 * (x * B)) + ((1.0 - x) / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -4.4e-126:
		tmp = (B * (x * 0.3333333333333333)) + ((-1.0 - x) / B)
	elif F <= 1.4e-9:
		tmp = -x / B
	else:
		tmp = (0.3333333333333333 * (x * B)) + ((1.0 - x) / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -4.4e-126)
		tmp = Float64(Float64(B * Float64(x * 0.3333333333333333)) + Float64(Float64(-1.0 - x) / B));
	elseif (F <= 1.4e-9)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(0.3333333333333333 * Float64(x * B)) + Float64(Float64(1.0 - x) / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -4.4e-126)
		tmp = (B * (x * 0.3333333333333333)) + ((-1.0 - x) / B);
	elseif (F <= 1.4e-9)
		tmp = -x / B;
	else
		tmp = (0.3333333333333333 * (x * B)) + ((1.0 - x) / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -4.4e-126], N[(N[(B * N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.4e-9], N[((-x) / B), $MachinePrecision], N[(N[(0.3333333333333333 * N[(x * B), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.40000000000000029e-126

   1. Initial program 72.7%

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

   2. Taylor expanded in B around 0 50.0%

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

   3. Taylor expanded in F around inf 34.3%

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

      1. neg-sub0 34.3%

         \[\leadsto \color{blue}{\left(0 - x \cdot \frac{1}{\tan B}\right)} + \frac{F}{B} \cdot \frac{1}{F} \]

      2. associate-+l- 34.3%

         \[\leadsto \color{blue}{0 - \left(x \cdot \frac{1}{\tan B} - \frac{F}{B} \cdot \frac{1}{F}\right)} \]

      3. add-sqr-sqrt 12.7%

         \[\leadsto 0 - \left(\color{blue}{\sqrt{x \cdot \frac{1}{\tan B}} \cdot \sqrt{x \cdot \frac{1}{\tan B}}} - \frac{F}{B} \cdot \frac{1}{F}\right) \]

      4. sqrt-unprod 8.6%

         \[\leadsto 0 - \left(\color{blue}{\sqrt{\left(x \cdot \frac{1}{\tan B}\right) \cdot \left(x \cdot \frac{1}{\tan B}\right)}} - \frac{F}{B} \cdot \frac{1}{F}\right) \]

      5. sqr-neg 8.6%

         \[\leadsto 0 - \left(\sqrt{\color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(-x \cdot \frac{1}{\tan B}\right)}} - \frac{F}{B} \cdot \frac{1}{F}\right) \]

      6. sqrt-unprod 1.0%

         \[\leadsto 0 - \left(\color{blue}{\sqrt{-x \cdot \frac{1}{\tan B}} \cdot \sqrt{-x \cdot \frac{1}{\tan B}}} - \frac{F}{B} \cdot \frac{1}{F}\right) \]

      7. add-sqr-sqrt 1.9%

         \[\leadsto 0 - \left(\color{blue}{\left(-x \cdot \frac{1}{\tan B}\right)} - \frac{F}{B} \cdot \frac{1}{F}\right) \]

      8. *-commutative 1.9%

         \[\leadsto 0 - \left(\left(-x \cdot \frac{1}{\tan B}\right) - \color{blue}{\frac{1}{F} \cdot \frac{F}{B}}\right) \]

      9. cancel-sign-sub-inv 1.9%

         \[\leadsto 0 - \color{blue}{\left(\left(-x \cdot \frac{1}{\tan B}\right) + \left(-\frac{1}{F}\right) \cdot \frac{F}{B}\right)} \]

   5. Applied egg-rr 62.3%

      \[\leadsto \color{blue}{0 - \left(\frac{1}{B} + \frac{x}{\tan B}\right)} \]
   6. Step-by-step derivation

      1. associate--r+ 62.3%

         \[\leadsto \color{blue}{\left(0 - \frac{1}{B}\right) - \frac{x}{\tan B}} \]

      2. neg-sub0 62.3%

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

      3. distribute-neg-frac 62.3%

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

      4. metadata-eval 62.3%

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

   7. Simplified 62.3%

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

   8. Taylor expanded in B around 0 44.0%

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

      1. +-commutative 44.0%

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

      2. mul-1-neg 44.0%

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

      3. unsub-neg 44.0%

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

      4. *-commutative 44.0%

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

      5. associate-*l* 44.0%

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

   10. Simplified 44.0%

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

   if -4.40000000000000029e-126 < F < 1.39999999999999992e-9

   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. Step-by-step derivation

      1. distribute-lft-neg-in 99.5%

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

      2. +-commutative 99.5%

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

      3. fma-def 99.5%

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

      4. +-commutative 99.5%

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

      5. *-commutative 99.5%

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

      6. fma-def 99.5%

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

      7. fma-def 99.5%

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

      8. metadata-eval 99.5%

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

      9. metadata-eval 99.5%

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

      10. associate-*r/ 99.7%

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

      11. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in F around 0 70.5%

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

      1. mul-1-neg 70.5%

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

      2. associate-*l/ 70.5%

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

      3. distribute-rgt-neg-in 70.5%

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

   6. Simplified 70.5%

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

   7. Taylor expanded in B around 0 36.0%

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

      1. associate-*r/ 36.0%

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

      2. neg-mul-1 36.0%

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

   9. Simplified 36.0%

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

   if 1.39999999999999992e-9 < F

   1. Initial program 62.6%

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

   2. Taylor expanded in B around 0 44.8%

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

   3. Taylor expanded in F around inf 46.1%

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

   4. Taylor expanded in B around 0 51.0%

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

      1. associate--l+ 51.0%

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

      2. div-sub 51.0%

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

   6. Simplified 51.0%

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

4. Final simplification 43.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.4 \cdot 10^{-126}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333\right) + \frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(x \cdot B\right) + \frac{1 - x}{B}\\ \end{array} \]

Alternative 14: 43.1% accurate, 35.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.3 \cdot 10^{-116}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.3e-116)
   (/ (- -1.0 x) B)
   (if (<= F 1.4e-9) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.3e-116) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.4e-9) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-2.3d-116)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 1.4d-9) then
        tmp = -x / b
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.3e-116) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.4e-9) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.3e-116:
		tmp = (-1.0 - x) / B
	elif F <= 1.4e-9:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.3e-116)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 1.4e-9)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.3e-116)
		tmp = (-1.0 - x) / B;
	elseif (F <= 1.4e-9)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.3e-116], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.4e-9], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.30000000000000002e-116

   1. Initial program 72.5%

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

   2. Taylor expanded in B around 0 49.5%

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

   3. Taylor expanded in F around inf 33.6%

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

      1. neg-sub0 33.6%

         \[\leadsto \color{blue}{\left(0 - x \cdot \frac{1}{\tan B}\right)} + \frac{F}{B} \cdot \frac{1}{F} \]

      2. associate-+l- 33.6%

         \[\leadsto \color{blue}{0 - \left(x \cdot \frac{1}{\tan B} - \frac{F}{B} \cdot \frac{1}{F}\right)} \]

      3. add-sqr-sqrt 12.8%

         \[\leadsto 0 - \left(\color{blue}{\sqrt{x \cdot \frac{1}{\tan B}} \cdot \sqrt{x \cdot \frac{1}{\tan B}}} - \frac{F}{B} \cdot \frac{1}{F}\right) \]

      4. sqrt-unprod 8.7%

         \[\leadsto 0 - \left(\color{blue}{\sqrt{\left(x \cdot \frac{1}{\tan B}\right) \cdot \left(x \cdot \frac{1}{\tan B}\right)}} - \frac{F}{B} \cdot \frac{1}{F}\right) \]

      5. sqr-neg 8.7%

         \[\leadsto 0 - \left(\sqrt{\color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(-x \cdot \frac{1}{\tan B}\right)}} - \frac{F}{B} \cdot \frac{1}{F}\right) \]

      6. sqrt-unprod 1.0%

         \[\leadsto 0 - \left(\color{blue}{\sqrt{-x \cdot \frac{1}{\tan B}} \cdot \sqrt{-x \cdot \frac{1}{\tan B}}} - \frac{F}{B} \cdot \frac{1}{F}\right) \]

      7. add-sqr-sqrt 1.9%

         \[\leadsto 0 - \left(\color{blue}{\left(-x \cdot \frac{1}{\tan B}\right)} - \frac{F}{B} \cdot \frac{1}{F}\right) \]

      8. *-commutative 1.9%

         \[\leadsto 0 - \left(\left(-x \cdot \frac{1}{\tan B}\right) - \color{blue}{\frac{1}{F} \cdot \frac{F}{B}}\right) \]

      9. cancel-sign-sub-inv 1.9%

         \[\leadsto 0 - \color{blue}{\left(\left(-x \cdot \frac{1}{\tan B}\right) + \left(-\frac{1}{F}\right) \cdot \frac{F}{B}\right)} \]

   5. Applied egg-rr 61.9%

      \[\leadsto \color{blue}{0 - \left(\frac{1}{B} + \frac{x}{\tan B}\right)} \]
   6. Step-by-step derivation

      1. associate--r+ 61.9%

         \[\leadsto \color{blue}{\left(0 - \frac{1}{B}\right) - \frac{x}{\tan B}} \]

      2. neg-sub0 61.9%

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

      3. distribute-neg-frac 61.9%

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

      4. metadata-eval 61.9%

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

   7. Simplified 61.9%

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

   8. Taylor expanded in B around 0 43.8%

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

      1. associate-*r/ 43.8%

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

      2. distribute-lft-in 43.8%

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

      3. metadata-eval 43.8%

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

      4. neg-mul-1 43.8%

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

   10. Simplified 43.8%

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

   if -2.30000000000000002e-116 < F < 1.39999999999999992e-9

   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. Step-by-step derivation

      1. distribute-lft-neg-in 99.5%

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

      2. +-commutative 99.5%

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

      3. fma-def 99.5%

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

      4. +-commutative 99.5%

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

      5. *-commutative 99.5%

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

      6. fma-def 99.5%

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

      7. fma-def 99.5%

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

      8. metadata-eval 99.5%

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

      9. metadata-eval 99.5%

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

      10. associate-*r/ 99.7%

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

      11. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in F around 0 70.8%

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

      1. mul-1-neg 70.8%

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

      2. associate-*l/ 70.8%

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

      3. distribute-rgt-neg-in 70.8%

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

   6. Simplified 70.8%

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

   7. Taylor expanded in B around 0 35.6%

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

      1. associate-*r/ 35.6%

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

      2. neg-mul-1 35.6%

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

   9. Simplified 35.6%

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

   if 1.39999999999999992e-9 < F

   1. Initial program 62.6%

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

   2. Taylor expanded in B around 0 44.8%

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

   3. Taylor expanded in F around inf 46.1%

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

   4. Taylor expanded in B around 0 50.5%

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

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.3 \cdot 10^{-116}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 15: 43.1% accurate, 35.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.4 \cdot 10^{-116}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.4e-116)
   (- (/ -1.0 B) (/ x B))
   (if (<= F 1.4e-9) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.4e-116) {
		tmp = (-1.0 / B) - (x / B);
	} else if (F <= 1.4e-9) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-2.4d-116)) then
        tmp = ((-1.0d0) / b) - (x / b)
    else if (f <= 1.4d-9) then
        tmp = -x / b
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.4e-116) {
		tmp = (-1.0 / B) - (x / B);
	} else if (F <= 1.4e-9) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.4e-116:
		tmp = (-1.0 / B) - (x / B)
	elif F <= 1.4e-9:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.4e-116)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / B));
	elseif (F <= 1.4e-9)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.4e-116)
		tmp = (-1.0 / B) - (x / B);
	elseif (F <= 1.4e-9)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.4e-116], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.4e-9], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.39999999999999993e-116

   1. Initial program 72.5%

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

   2. Taylor expanded in B around 0 49.5%

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

   3. Taylor expanded in B around 0 29.7%

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

   4. Taylor expanded in F around -inf 43.8%

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

      1. distribute-lft-in 43.8%

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

      2. neg-mul-1 43.8%

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

      3. unsub-neg 43.8%

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

      4. associate-*r/ 43.8%

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

      5. metadata-eval 43.8%

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

   6. Simplified 43.8%

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

   if -2.39999999999999993e-116 < F < 1.39999999999999992e-9

   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. Step-by-step derivation

      1. distribute-lft-neg-in 99.5%

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

      2. +-commutative 99.5%

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

      3. fma-def 99.5%

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

      4. +-commutative 99.5%

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

      5. *-commutative 99.5%

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

      6. fma-def 99.5%

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

      7. fma-def 99.5%

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

      8. metadata-eval 99.5%

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

      9. metadata-eval 99.5%

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

      10. associate-*r/ 99.7%

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

      11. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in F around 0 70.8%

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

      1. mul-1-neg 70.8%

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

      2. associate-*l/ 70.8%

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

      3. distribute-rgt-neg-in 70.8%

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

   6. Simplified 70.8%

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

   7. Taylor expanded in B around 0 35.6%

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

      1. associate-*r/ 35.6%

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

      2. neg-mul-1 35.6%

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

   9. Simplified 35.6%

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

   if 1.39999999999999992e-9 < F

   1. Initial program 62.6%

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

   2. Taylor expanded in B around 0 44.8%

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

   3. Taylor expanded in F around inf 46.1%

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

   4. Taylor expanded in B around 0 50.5%

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

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.4 \cdot 10^{-116}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 16: 30.1% accurate, 39.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-122} \lor \neg \left(x \leq 0.00166\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (or (<= x -2.3e-122) (not (<= x 0.00166))) (/ (- x) B) (/ -1.0 B)))

C

double code(double F, double B, double x) {
	double tmp;
	if ((x <= -2.3e-122) || !(x <= 0.00166)) {
		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 <= (-2.3d-122)) .or. (.not. (x <= 0.00166d0))) 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 <= -2.3e-122) || !(x <= 0.00166)) {
		tmp = -x / B;
	} else {
		tmp = -1.0 / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if (x <= -2.3e-122) or not (x <= 0.00166):
		tmp = -x / B
	else:
		tmp = -1.0 / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if ((x <= -2.3e-122) || !(x <= 0.00166))
		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 <= -2.3e-122) || ~((x <= 0.00166)))
		tmp = -x / B;
	else
		tmp = -1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[Or[LessEqual[x, -2.3e-122], N[Not[LessEqual[x, 0.00166]], $MachinePrecision]], N[((-x) / B), $MachinePrecision], N[(-1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.30000000000000007e-122 or 0.00166 < x

   1. Initial program 85.3%

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

      1. distribute-lft-neg-in 85.3%

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

      2. +-commutative 85.3%

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

      3. fma-def 85.3%

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

      4. +-commutative 85.3%

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

      5. *-commutative 85.3%

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

      6. fma-def 85.3%

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

      7. fma-def 85.3%

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

      8. metadata-eval 85.3%

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

      9. metadata-eval 85.3%

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

      10. associate-*r/ 85.4%

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

      11. *-rgt-identity 85.4%

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

   3. Simplified 85.4%

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

   4. Taylor expanded in F around 0 83.4%

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

      1. mul-1-neg 83.4%

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

      2. associate-*l/ 83.4%

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

      3. distribute-rgt-neg-in 83.4%

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

   6. Simplified 83.4%

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

   7. Taylor expanded in B around 0 45.7%

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

      1. associate-*r/ 45.7%

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

      2. neg-mul-1 45.7%

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

   9. Simplified 45.7%

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

   if -2.30000000000000007e-122 < x < 0.00166

   1. Initial program 72.1%

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

   2. Taylor expanded in B around 0 39.6%

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

   3. Taylor expanded in F around inf 17.1%

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

      1. neg-sub0 17.1%

         \[\leadsto \color{blue}{\left(0 - x \cdot \frac{1}{\tan B}\right)} + \frac{F}{B} \cdot \frac{1}{F} \]

      2. associate-+l- 17.1%

         \[\leadsto \color{blue}{0 - \left(x \cdot \frac{1}{\tan B} - \frac{F}{B} \cdot \frac{1}{F}\right)} \]

      3. add-sqr-sqrt 9.2%

         \[\leadsto 0 - \left(\color{blue}{\sqrt{x \cdot \frac{1}{\tan B}} \cdot \sqrt{x \cdot \frac{1}{\tan B}}} - \frac{F}{B} \cdot \frac{1}{F}\right) \]

      4. sqrt-unprod 12.8%

         \[\leadsto 0 - \left(\color{blue}{\sqrt{\left(x \cdot \frac{1}{\tan B}\right) \cdot \left(x \cdot \frac{1}{\tan B}\right)}} - \frac{F}{B} \cdot \frac{1}{F}\right) \]

      5. sqr-neg 12.8%

         \[\leadsto 0 - \left(\sqrt{\color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(-x \cdot \frac{1}{\tan B}\right)}} - \frac{F}{B} \cdot \frac{1}{F}\right) \]

      6. sqrt-unprod 4.3%

         \[\leadsto 0 - \left(\color{blue}{\sqrt{-x \cdot \frac{1}{\tan B}} \cdot \sqrt{-x \cdot \frac{1}{\tan B}}} - \frac{F}{B} \cdot \frac{1}{F}\right) \]

      7. add-sqr-sqrt 8.6%

         \[\leadsto 0 - \left(\color{blue}{\left(-x \cdot \frac{1}{\tan B}\right)} - \frac{F}{B} \cdot \frac{1}{F}\right) \]

      8. *-commutative 8.6%

         \[\leadsto 0 - \left(\left(-x \cdot \frac{1}{\tan B}\right) - \color{blue}{\frac{1}{F} \cdot \frac{F}{B}}\right) \]

      9. cancel-sign-sub-inv 8.6%

         \[\leadsto 0 - \color{blue}{\left(\left(-x \cdot \frac{1}{\tan B}\right) + \left(-\frac{1}{F}\right) \cdot \frac{F}{B}\right)} \]

   5. Applied egg-rr 21.8%

      \[\leadsto \color{blue}{0 - \left(\frac{1}{B} + \frac{x}{\tan B}\right)} \]
   6. Step-by-step derivation

      1. associate--r+ 21.8%

         \[\leadsto \color{blue}{\left(0 - \frac{1}{B}\right) - \frac{x}{\tan B}} \]

      2. neg-sub0 21.8%

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

      3. distribute-neg-frac 21.8%

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

      4. metadata-eval 21.8%

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

   7. Simplified 21.8%

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

   8. Taylor expanded in x around 0 15.3%

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

4. Final simplification 31.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-122} \lor \neg \left(x \leq 0.00166\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B}\\ \end{array} \]

Alternative 17: 36.8% accurate, 45.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 1.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F 1.4e-9) (/ (- x) B) (/ (- 1.0 x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= 1.4e-9) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= 1.4d-9) then
        tmp = -x / b
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= 1.4e-9) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= 1.4e-9:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= 1.4e-9)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= 1.4e-9)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, 1.4e-9], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 1.39999999999999992e-9

   1. Initial program 84.7%

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

      1. distribute-lft-neg-in 84.7%

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

      2. +-commutative 84.7%

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

      3. fma-def 84.7%

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

      4. +-commutative 84.7%

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

      5. *-commutative 84.7%

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

      6. fma-def 84.7%

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

      7. fma-def 84.7%

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

      8. metadata-eval 84.7%

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

      9. metadata-eval 84.7%

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

      10. associate-*r/ 84.8%

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

      11. *-rgt-identity 84.8%

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

   3. Simplified 84.8%

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

   4. Taylor expanded in F around 0 56.9%

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

      1. mul-1-neg 56.9%

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

      2. associate-*l/ 56.8%

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

      3. distribute-rgt-neg-in 56.8%

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

   6. Simplified 56.8%

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

   7. Taylor expanded in B around 0 30.1%

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

      1. associate-*r/ 30.1%

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

      2. neg-mul-1 30.1%

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

   9. Simplified 30.1%

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

   if 1.39999999999999992e-9 < F

   1. Initial program 62.6%

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

   2. Taylor expanded in B around 0 44.8%

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

   3. Taylor expanded in F around inf 46.1%

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

   4. Taylor expanded in B around 0 50.5%

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

4. Final simplification 35.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 18: 10.0% accurate, 108.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

2. Taylor expanded in B around 0 59.9%

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

3. Taylor expanded in F around inf 43.0%

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

   1. neg-sub0 43.0%

      \[\leadsto \color{blue}{\left(0 - x \cdot \frac{1}{\tan B}\right)} + \frac{F}{B} \cdot \frac{1}{F} \]

   2. associate-+l- 43.0%

      \[\leadsto \color{blue}{0 - \left(x \cdot \frac{1}{\tan B} - \frac{F}{B} \cdot \frac{1}{F}\right)} \]

   3. add-sqr-sqrt 19.2%

      \[\leadsto 0 - \left(\color{blue}{\sqrt{x \cdot \frac{1}{\tan B}} \cdot \sqrt{x \cdot \frac{1}{\tan B}}} - \frac{F}{B} \cdot \frac{1}{F}\right) \]

   4. sqrt-unprod 15.8%

      \[\leadsto 0 - \left(\color{blue}{\sqrt{\left(x \cdot \frac{1}{\tan B}\right) \cdot \left(x \cdot \frac{1}{\tan B}\right)}} - \frac{F}{B} \cdot \frac{1}{F}\right) \]

   5. sqr-neg 15.8%

      \[\leadsto 0 - \left(\sqrt{\color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) \cdot \left(-x \cdot \frac{1}{\tan B}\right)}} - \frac{F}{B} \cdot \frac{1}{F}\right) \]

   6. sqrt-unprod 2.4%

      \[\leadsto 0 - \left(\color{blue}{\sqrt{-x \cdot \frac{1}{\tan B}} \cdot \sqrt{-x \cdot \frac{1}{\tan B}}} - \frac{F}{B} \cdot \frac{1}{F}\right) \]

   7. add-sqr-sqrt 5.2%

      \[\leadsto 0 - \left(\color{blue}{\left(-x \cdot \frac{1}{\tan B}\right)} - \frac{F}{B} \cdot \frac{1}{F}\right) \]

   8. *-commutative 5.2%

      \[\leadsto 0 - \left(\left(-x \cdot \frac{1}{\tan B}\right) - \color{blue}{\frac{1}{F} \cdot \frac{F}{B}}\right) \]

   9. cancel-sign-sub-inv 5.2%

      \[\leadsto 0 - \color{blue}{\left(\left(-x \cdot \frac{1}{\tan B}\right) + \left(-\frac{1}{F}\right) \cdot \frac{F}{B}\right)} \]

5. Applied egg-rr 52.4%

   \[\leadsto \color{blue}{0 - \left(\frac{1}{B} + \frac{x}{\tan B}\right)} \]
6. Step-by-step derivation

   1. associate--r+ 52.4%

      \[\leadsto \color{blue}{\left(0 - \frac{1}{B}\right) - \frac{x}{\tan B}} \]

   2. neg-sub0 52.4%

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

   3. distribute-neg-frac 52.4%

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

   4. metadata-eval 52.4%

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

7. Simplified 52.4%

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

8. Taylor expanded in x around 0 11.0%

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

9. Final simplification 11.0%

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

Reproduce

herbie shell --seed 2023299 
(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))))))