VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.9% → 99.6%

Time: 24.4s

Alternatives: 25

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{\sin B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -100000000:\\ \;\;\;\;t_0 - t_1\\ \mathbf{elif}\;F \leq 14600000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} + x \cdot \left(t_0 \cdot \cos B\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ -1.0 (sin B))) (t_1 (/ x (tan B))))
   (if (<= F -100000000.0)
     (- t_0 t_1)
     (if (<= F 14600000000.0)
       (- (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5)) t_1)
       (+ (/ 1.0 (sin B)) (* x (* t_0 (cos B))))))))

C

double code(double F, double B, double x) {
	double t_0 = -1.0 / sin(B);
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -100000000.0) {
		tmp = t_0 - t_1;
	} else if (F <= 14600000000.0) {
		tmp = ((F / sin(B)) * pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - t_1;
	} else {
		tmp = (1.0 / sin(B)) + (x * (t_0 * cos(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 = (-1.0d0) / sin(b)
    t_1 = x / tan(b)
    if (f <= (-100000000.0d0)) then
        tmp = t_0 - t_1
    else if (f <= 14600000000.0d0) then
        tmp = ((f / sin(b)) * ((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0))) - t_1
    else
        tmp = (1.0d0 / sin(b)) + (x * (t_0 * cos(b)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = -1.0 / Math.sin(B);
	double t_1 = x / Math.tan(B);
	double tmp;
	if (F <= -100000000.0) {
		tmp = t_0 - t_1;
	} else if (F <= 14600000000.0) {
		tmp = ((F / Math.sin(B)) * Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - t_1;
	} else {
		tmp = (1.0 / Math.sin(B)) + (x * (t_0 * Math.cos(B)));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -1.0 / math.sin(B)
	t_1 = x / math.tan(B)
	tmp = 0
	if F <= -100000000.0:
		tmp = t_0 - t_1
	elif F <= 14600000000.0:
		tmp = ((F / math.sin(B)) * math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - t_1
	else:
		tmp = (1.0 / math.sin(B)) + (x * (t_0 * math.cos(B)))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(-1.0 / sin(B))
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -100000000.0)
		tmp = Float64(t_0 - t_1);
	elseif (F <= 14600000000.0)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5)) - t_1);
	else
		tmp = Float64(Float64(1.0 / sin(B)) + Float64(x * Float64(t_0 * cos(B))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = -1.0 / sin(B);
	t_1 = x / tan(B);
	tmp = 0.0;
	if (F <= -100000000.0)
		tmp = t_0 - t_1;
	elseif (F <= 14600000000.0)
		tmp = ((F / sin(B)) * ((((F * F) + 2.0) + (x * 2.0)) ^ -0.5)) - t_1;
	else
		tmp = (1.0 / sin(B)) + (x * (t_0 * cos(B)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -100000000.0], N[(t$95$0 - t$95$1), $MachinePrecision], If[LessEqual[F, 14600000000.0], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[(x * N[(t$95$0 * N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1e8

   1. Initial program 68.3%

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

   2. Taylor expanded in F around -inf 99.7%

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

      1. +-commutative 99.7%

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

      2. div-inv 99.9%

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

      3. unsub-neg 99.9%

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

   4. Applied egg-rr 99.9%

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

   if -1e8 < F < 1.46e10

   1. Initial program 99.4%

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

      1. div-inv 99.6%

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

      2. expm1-log1p-u 74.5%

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

      3. expm1-udef 55.3%

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

   3. Applied egg-rr 55.3%

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

      1. expm1-def 74.5%

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

      2. expm1-log1p 99.6%

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

   5. Simplified 99.6%

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

   if 1.46e10 < F

   1. Initial program 51.6%

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

   2. Taylor expanded in F around inf 99.8%

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

      1. tan-quot 99.7%

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

      2. associate-/r/ 99.8%

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

   4. Applied egg-rr 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -100000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 14600000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} + x \cdot \left(\frac{-1}{\sin B} \cdot \cos B\right)\\ \end{array} \]

Alternative 2: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.42:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1 + \frac{1}{{F}^{2}}}{\sin B}\\ \mathbf{elif}\;F \leq 1.45:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{2}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} + x \cdot \left(\frac{-1}{\sin B} \cdot \cos B\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.42)
   (+ (* x (/ -1.0 (tan B))) (/ (+ -1.0 (/ 1.0 (pow F 2.0))) (sin B)))
   (if (<= F 1.45)
     (- (/ F (* (sin B) (sqrt 2.0))) (/ x (tan B)))
     (+ (/ 1.0 (sin B)) (* x (* (/ -1.0 (sin B)) (cos B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.42) {
		tmp = (x * (-1.0 / tan(B))) + ((-1.0 + (1.0 / pow(F, 2.0))) / sin(B));
	} else if (F <= 1.45) {
		tmp = (F / (sin(B) * sqrt(2.0))) - (x / tan(B));
	} else {
		tmp = (1.0 / sin(B)) + (x * ((-1.0 / sin(B)) * cos(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 <= (-1.42d0)) then
        tmp = (x * ((-1.0d0) / tan(b))) + (((-1.0d0) + (1.0d0 / (f ** 2.0d0))) / sin(b))
    else if (f <= 1.45d0) then
        tmp = (f / (sin(b) * sqrt(2.0d0))) - (x / tan(b))
    else
        tmp = (1.0d0 / sin(b)) + (x * (((-1.0d0) / sin(b)) * cos(b)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.42) {
		tmp = (x * (-1.0 / Math.tan(B))) + ((-1.0 + (1.0 / Math.pow(F, 2.0))) / Math.sin(B));
	} else if (F <= 1.45) {
		tmp = (F / (Math.sin(B) * Math.sqrt(2.0))) - (x / Math.tan(B));
	} else {
		tmp = (1.0 / Math.sin(B)) + (x * ((-1.0 / Math.sin(B)) * Math.cos(B)));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.42:
		tmp = (x * (-1.0 / math.tan(B))) + ((-1.0 + (1.0 / math.pow(F, 2.0))) / math.sin(B))
	elif F <= 1.45:
		tmp = (F / (math.sin(B) * math.sqrt(2.0))) - (x / math.tan(B))
	else:
		tmp = (1.0 / math.sin(B)) + (x * ((-1.0 / math.sin(B)) * math.cos(B)))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.42)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(Float64(-1.0 + Float64(1.0 / (F ^ 2.0))) / sin(B)));
	elseif (F <= 1.45)
		tmp = Float64(Float64(F / Float64(sin(B) * sqrt(2.0))) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) + Float64(x * Float64(Float64(-1.0 / sin(B)) * cos(B))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.42)
		tmp = (x * (-1.0 / tan(B))) + ((-1.0 + (1.0 / (F ^ 2.0))) / sin(B));
	elseif (F <= 1.45)
		tmp = (F / (sin(B) * sqrt(2.0))) - (x / tan(B));
	else
		tmp = (1.0 / sin(B)) + (x * ((-1.0 / sin(B)) * cos(B)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.42], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 + N[(1.0 / N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.45], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.4199999999999999

   1. Initial program 69.0%

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

      1. associate-*l/ 78.3%

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

      2. +-commutative 78.3%

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

      3. *-commutative 78.3%

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

      4. fma-udef 78.3%

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

      5. fma-def 78.3%

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

      6. metadata-eval 78.3%

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

      7. metadata-eval 78.3%

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

      8. fma-def 78.3%

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

      9. fma-udef 78.3%

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

      10. *-commutative 78.3%

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

      11. fma-def 78.3%

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

      12. fma-def 78.3%

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

   3. Applied egg-rr 78.3%

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

   4. Taylor expanded in x around 0 78.3%

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

      1. +-commutative 78.3%

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

      2. unpow2 78.3%

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

      3. fma-udef 78.3%

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

   6. Simplified 78.3%

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

   7. Taylor expanded in F around -inf 99.7%

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

   if -1.4199999999999999 < F < 1.44999999999999996

   1. Initial program 99.4%

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

   2. Taylor expanded in F around 0 98.7%

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

      1. +-commutative 98.7%

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

      2. div-inv 98.9%

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

      3. unsub-neg 98.9%

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

      4. *-commutative 98.9%

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

      5. sqrt-div 98.8%

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

      6. metadata-eval 98.8%

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

      7. frac-times 98.9%

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

      8. *-un-lft-identity 98.9%

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

      9. +-commutative 98.9%

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

      10. fma-def 98.9%

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

   4. Applied egg-rr 98.9%

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

   5. Taylor expanded in x around 0 98.9%

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

   if 1.44999999999999996 < F

   1. Initial program 52.4%

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

   2. Taylor expanded in F around inf 98.8%

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

      1. tan-quot 98.7%

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

      2. associate-/r/ 98.8%

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

   4. Applied egg-rr 98.8%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.42:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1 + \frac{1}{{F}^{2}}}{\sin B}\\ \mathbf{elif}\;F \leq 1.45:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{2}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} + x \cdot \left(\frac{-1}{\sin B} \cdot \cos B\right)\\ \end{array} \]

Alternative 3: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{\sin B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.42:\\ \;\;\;\;t_0 - t_1\\ \mathbf{elif}\;F \leq 1.5:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{2}} - t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} + x \cdot \left(t_0 \cdot \cos B\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ -1.0 (sin B))) (t_1 (/ x (tan B))))
   (if (<= F -1.42)
     (- t_0 t_1)
     (if (<= F 1.5)
       (- (/ F (* (sin B) (sqrt 2.0))) t_1)
       (+ (/ 1.0 (sin B)) (* x (* t_0 (cos B))))))))

C

double code(double F, double B, double x) {
	double t_0 = -1.0 / sin(B);
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -1.42) {
		tmp = t_0 - t_1;
	} else if (F <= 1.5) {
		tmp = (F / (sin(B) * sqrt(2.0))) - t_1;
	} else {
		tmp = (1.0 / sin(B)) + (x * (t_0 * cos(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 = (-1.0d0) / sin(b)
    t_1 = x / tan(b)
    if (f <= (-1.42d0)) then
        tmp = t_0 - t_1
    else if (f <= 1.5d0) then
        tmp = (f / (sin(b) * sqrt(2.0d0))) - t_1
    else
        tmp = (1.0d0 / sin(b)) + (x * (t_0 * cos(b)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = -1.0 / Math.sin(B);
	double t_1 = x / Math.tan(B);
	double tmp;
	if (F <= -1.42) {
		tmp = t_0 - t_1;
	} else if (F <= 1.5) {
		tmp = (F / (Math.sin(B) * Math.sqrt(2.0))) - t_1;
	} else {
		tmp = (1.0 / Math.sin(B)) + (x * (t_0 * Math.cos(B)));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -1.0 / math.sin(B)
	t_1 = x / math.tan(B)
	tmp = 0
	if F <= -1.42:
		tmp = t_0 - t_1
	elif F <= 1.5:
		tmp = (F / (math.sin(B) * math.sqrt(2.0))) - t_1
	else:
		tmp = (1.0 / math.sin(B)) + (x * (t_0 * math.cos(B)))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(-1.0 / sin(B))
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.42)
		tmp = Float64(t_0 - t_1);
	elseif (F <= 1.5)
		tmp = Float64(Float64(F / Float64(sin(B) * sqrt(2.0))) - t_1);
	else
		tmp = Float64(Float64(1.0 / sin(B)) + Float64(x * Float64(t_0 * cos(B))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = -1.0 / sin(B);
	t_1 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.42)
		tmp = t_0 - t_1;
	elseif (F <= 1.5)
		tmp = (F / (sin(B) * sqrt(2.0))) - t_1;
	else
		tmp = (1.0 / sin(B)) + (x * (t_0 * cos(B)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.42], N[(t$95$0 - t$95$1), $MachinePrecision], If[LessEqual[F, 1.5], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[(x * N[(t$95$0 * N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.4199999999999999

   1. Initial program 69.0%

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

   2. Taylor expanded in F around -inf 98.9%

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

      1. +-commutative 98.9%

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

      2. div-inv 99.1%

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

      3. unsub-neg 99.1%

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

   4. Applied egg-rr 99.1%

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

   if -1.4199999999999999 < F < 1.5

   1. Initial program 99.4%

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

   2. Taylor expanded in F around 0 98.7%

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

      1. +-commutative 98.7%

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

      2. div-inv 98.9%

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

      3. unsub-neg 98.9%

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

      4. *-commutative 98.9%

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

      5. sqrt-div 98.8%

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

      6. metadata-eval 98.8%

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

      7. frac-times 98.9%

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

      8. *-un-lft-identity 98.9%

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

      9. +-commutative 98.9%

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

      10. fma-def 98.9%

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

   4. Applied egg-rr 98.9%

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

   5. Taylor expanded in x around 0 98.9%

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

   if 1.5 < F

   1. Initial program 52.4%

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

   2. Taylor expanded in F around inf 98.8%

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

      1. tan-quot 98.7%

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

      2. associate-/r/ 98.8%

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

   4. Applied egg-rr 98.8%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.42:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.5:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{2}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} + x \cdot \left(\frac{-1}{\sin B} \cdot \cos B\right)\\ \end{array} \]

Alternative 4: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.42:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 1.5:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{2}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} + x \cdot \frac{-1}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.42)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.5)
       (- (/ F (* (sin B) (sqrt 2.0))) t_0)
       (+ (/ 1.0 (sin B)) (* x (/ -1.0 (tan B))))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.42) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.5) {
		tmp = (F / (sin(B) * sqrt(2.0))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) + (x * (-1.0 / tan(B)));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.42d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.5d0) then
        tmp = (f / (sin(b) * sqrt(2.0d0))) - t_0
    else
        tmp = (1.0d0 / sin(b)) + (x * ((-1.0d0) / tan(b)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.42) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.5) {
		tmp = (F / (Math.sin(B) * Math.sqrt(2.0))) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) + (x * (-1.0 / Math.tan(B)));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.42:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.5:
		tmp = (F / (math.sin(B) * math.sqrt(2.0))) - t_0
	else:
		tmp = (1.0 / math.sin(B)) + (x * (-1.0 / math.tan(B)))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.42)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.5)
		tmp = Float64(Float64(F / Float64(sin(B) * sqrt(2.0))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) + Float64(x * Float64(-1.0 / tan(B))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.42)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.5)
		tmp = (F / (sin(B) * sqrt(2.0))) - t_0;
	else
		tmp = (1.0 / sin(B)) + (x * (-1.0 / tan(B)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.42], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.5], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.4199999999999999

   1. Initial program 69.0%

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

   2. Taylor expanded in F around -inf 98.9%

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

      1. +-commutative 98.9%

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

      2. div-inv 99.1%

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

      3. unsub-neg 99.1%

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

   4. Applied egg-rr 99.1%

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

   if -1.4199999999999999 < F < 1.5

   1. Initial program 99.4%

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

   2. Taylor expanded in F around 0 98.7%

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

      1. +-commutative 98.7%

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

      2. div-inv 98.9%

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

      3. unsub-neg 98.9%

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

      4. *-commutative 98.9%

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

      5. sqrt-div 98.8%

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

      6. metadata-eval 98.8%

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

      7. frac-times 98.9%

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

      8. *-un-lft-identity 98.9%

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

      9. +-commutative 98.9%

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

      10. fma-def 98.9%

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

   4. Applied egg-rr 98.9%

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

   5. Taylor expanded in x around 0 98.9%

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

   if 1.5 < F

   1. Initial program 52.4%

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

   2. Taylor expanded in F around inf 98.8%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.42:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.5:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{2}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} + x \cdot \frac{-1}{\tan B}\\ \end{array} \]

Alternative 5: 92.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1 \cdot 10^{-21}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 4.7 \cdot 10^{-106}:\\ \;\;\;\;\frac{F}{B \cdot \sqrt{2 + x \cdot 2}} - t_0\\ \mathbf{elif}\;F \leq 8000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} + x \cdot \frac{-1}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1e-21)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 4.7e-106)
       (- (/ F (* B (sqrt (+ 2.0 (* x 2.0))))) t_0)
       (if (<= F 8000000.0)
         (- (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5)) (/ x B))
         (+ (/ 1.0 (sin B)) (* x (/ -1.0 (tan B)))))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1e-21) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 4.7e-106) {
		tmp = (F / (B * sqrt((2.0 + (x * 2.0))))) - t_0;
	} else if (F <= 8000000.0) {
		tmp = ((F / sin(B)) * pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - (x / B);
	} else {
		tmp = (1.0 / sin(B)) + (x * (-1.0 / tan(B)));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1d-21)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 4.7d-106) then
        tmp = (f / (b * sqrt((2.0d0 + (x * 2.0d0))))) - t_0
    else if (f <= 8000000.0d0) then
        tmp = ((f / sin(b)) * ((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0))) - (x / b)
    else
        tmp = (1.0d0 / sin(b)) + (x * ((-1.0d0) / tan(b)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1e-21) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 4.7e-106) {
		tmp = (F / (B * Math.sqrt((2.0 + (x * 2.0))))) - t_0;
	} else if (F <= 8000000.0) {
		tmp = ((F / Math.sin(B)) * Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - (x / B);
	} else {
		tmp = (1.0 / Math.sin(B)) + (x * (-1.0 / Math.tan(B)));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1e-21:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 4.7e-106:
		tmp = (F / (B * math.sqrt((2.0 + (x * 2.0))))) - t_0
	elif F <= 8000000.0:
		tmp = ((F / math.sin(B)) * math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - (x / B)
	else:
		tmp = (1.0 / math.sin(B)) + (x * (-1.0 / math.tan(B)))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1e-21)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 4.7e-106)
		tmp = Float64(Float64(F / Float64(B * sqrt(Float64(2.0 + Float64(x * 2.0))))) - t_0);
	elseif (F <= 8000000.0)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) + Float64(x * Float64(-1.0 / tan(B))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1e-21)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 4.7e-106)
		tmp = (F / (B * sqrt((2.0 + (x * 2.0))))) - t_0;
	elseif (F <= 8000000.0)
		tmp = ((F / sin(B)) * ((((F * F) + 2.0) + (x * 2.0)) ^ -0.5)) - (x / B);
	else
		tmp = (1.0 / sin(B)) + (x * (-1.0 / tan(B)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1e-21], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 4.7e-106], N[(N[(F / N[(B * N[Sqrt[N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 8000000.0], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -9.99999999999999908e-22

   1. Initial program 70.0%

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

   2. Taylor expanded in F around -inf 97.0%

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

      1. +-commutative 97.0%

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

      2. div-inv 97.1%

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

      3. unsub-neg 97.1%

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

   4. Applied egg-rr 97.1%

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

   if -9.99999999999999908e-22 < F < 4.6999999999999999e-106

   1. Initial program 99.5%

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

   2. Taylor expanded in F around 0 99.5%

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

      1. +-commutative 99.5%

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

      2. div-inv 99.7%

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

      3. unsub-neg 99.7%

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

      4. *-commutative 99.7%

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

      5. sqrt-div 99.6%

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

      6. metadata-eval 99.6%

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

      7. frac-times 99.7%

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

      8. *-un-lft-identity 99.7%

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

      9. +-commutative 99.7%

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

      10. fma-def 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in B around 0 85.4%

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

   if 4.6999999999999999e-106 < F < 8e6

   1. Initial program 99.2%

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

   2. Taylor expanded in B around 0 91.3%

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

   if 8e6 < F

   1. Initial program 51.6%

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

   2. Taylor expanded in F around inf 99.8%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{-21}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 4.7 \cdot 10^{-106}:\\ \;\;\;\;\frac{F}{B \cdot \sqrt{2 + x \cdot 2}} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 8000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} + x \cdot \frac{-1}{\tan B}\\ \end{array} \]

Alternative 6: 92.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ t_1 := {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{if}\;F \leq -128000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 3.65 \cdot 10^{-109}:\\ \;\;\;\;t_0 + t_1 \cdot \frac{F}{B}\\ \mathbf{elif}\;F \leq 45000:\\ \;\;\;\;\frac{F}{\sin B} \cdot t_1 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} + t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B))))
        (t_1 (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5)))
   (if (<= F -128000.0)
     (- (/ -1.0 (sin B)) (/ x (tan B)))
     (if (<= F 3.65e-109)
       (+ t_0 (* t_1 (/ F B)))
       (if (<= F 45000.0)
         (- (* (/ F (sin B)) t_1) (/ x B))
         (+ (/ 1.0 (sin B)) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double t_1 = pow((((F * F) + 2.0) + (x * 2.0)), -0.5);
	double tmp;
	if (F <= -128000.0) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 3.65e-109) {
		tmp = t_0 + (t_1 * (F / B));
	} else if (F <= 45000.0) {
		tmp = ((F / sin(B)) * t_1) - (x / B);
	} else {
		tmp = (1.0 / sin(B)) + t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x * ((-1.0d0) / tan(b))
    t_1 = (((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0)
    if (f <= (-128000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 3.65d-109) then
        tmp = t_0 + (t_1 * (f / b))
    else if (f <= 45000.0d0) then
        tmp = ((f / sin(b)) * t_1) - (x / b)
    else
        tmp = (1.0d0 / sin(b)) + t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / Math.tan(B));
	double t_1 = Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5);
	double tmp;
	if (F <= -128000.0) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 3.65e-109) {
		tmp = t_0 + (t_1 * (F / B));
	} else if (F <= 45000.0) {
		tmp = ((F / Math.sin(B)) * t_1) - (x / B);
	} else {
		tmp = (1.0 / Math.sin(B)) + t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * (-1.0 / math.tan(B))
	t_1 = math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)
	tmp = 0
	if F <= -128000.0:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 3.65e-109:
		tmp = t_0 + (t_1 * (F / B))
	elif F <= 45000.0:
		tmp = ((F / math.sin(B)) * t_1) - (x / B)
	else:
		tmp = (1.0 / math.sin(B)) + t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	t_1 = Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5
	tmp = 0.0
	if (F <= -128000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 3.65e-109)
		tmp = Float64(t_0 + Float64(t_1 * Float64(F / B)));
	elseif (F <= 45000.0)
		tmp = Float64(Float64(Float64(F / sin(B)) * t_1) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x * (-1.0 / tan(B));
	t_1 = (((F * F) + 2.0) + (x * 2.0)) ^ -0.5;
	tmp = 0.0;
	if (F <= -128000.0)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 3.65e-109)
		tmp = t_0 + (t_1 * (F / B));
	elseif (F <= 45000.0)
		tmp = ((F / sin(B)) * t_1) - (x / B);
	else
		tmp = (1.0 / sin(B)) + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]}, If[LessEqual[F, -128000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.65e-109], N[(t$95$0 + N[(t$95$1 * N[(F / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 45000.0], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -128000

   1. Initial program 68.3%

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

   2. Taylor expanded in F around -inf 99.7%

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

      1. +-commutative 99.7%

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

      2. div-inv 99.9%

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

      3. unsub-neg 99.9%

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

   4. Applied egg-rr 99.9%

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

   if -128000 < F < 3.6500000000000002e-109

   1. Initial program 99.4%

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

   2. Taylor expanded in B around 0 83.9%

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

   if 3.6500000000000002e-109 < F < 45000

   1. Initial program 99.2%

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

   2. Taylor expanded in B around 0 91.3%

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

   if 45000 < F

   1. Initial program 51.6%

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

   2. Taylor expanded in F around inf 99.8%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -128000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 3.65 \cdot 10^{-109}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B}\\ \mathbf{elif}\;F \leq 45000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} + x \cdot \frac{-1}{\tan B}\\ \end{array} \]

Alternative 7: 89.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F}{\frac{\sin B}{\sqrt{0.5}}} - \frac{x}{B}\\ \mathbf{if}\;F \leq -0.45:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -5 \cdot 10^{-77}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 8 \cdot 10^{-187}:\\ \;\;\;\;\frac{\cos B \cdot \left(-x\right)}{\sin B}\\ \mathbf{elif}\;F \leq 3.8 \cdot 10^{-16}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} + x \cdot \frac{-1}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ F (/ (sin B) (sqrt 0.5))) (/ x B))))
   (if (<= F -0.45)
     (- (/ -1.0 (sin B)) (/ x (tan B)))
     (if (<= F -5e-77)
       t_0
       (if (<= F 8e-187)
         (/ (* (cos B) (- x)) (sin B))
         (if (<= F 3.8e-16)
           t_0
           (+ (/ 1.0 (sin B)) (* x (/ -1.0 (tan B))))))))))

C

double code(double F, double B, double x) {
	double t_0 = (F / (sin(B) / sqrt(0.5))) - (x / B);
	double tmp;
	if (F <= -0.45) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= -5e-77) {
		tmp = t_0;
	} else if (F <= 8e-187) {
		tmp = (cos(B) * -x) / sin(B);
	} else if (F <= 3.8e-16) {
		tmp = t_0;
	} else {
		tmp = (1.0 / sin(B)) + (x * (-1.0 / tan(B)));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (f / (sin(b) / sqrt(0.5d0))) - (x / b)
    if (f <= (-0.45d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= (-5d-77)) then
        tmp = t_0
    else if (f <= 8d-187) then
        tmp = (cos(b) * -x) / sin(b)
    else if (f <= 3.8d-16) then
        tmp = t_0
    else
        tmp = (1.0d0 / sin(b)) + (x * ((-1.0d0) / tan(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.sqrt(0.5))) - (x / B);
	double tmp;
	if (F <= -0.45) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= -5e-77) {
		tmp = t_0;
	} else if (F <= 8e-187) {
		tmp = (Math.cos(B) * -x) / Math.sin(B);
	} else if (F <= 3.8e-16) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) + (x * (-1.0 / Math.tan(B)));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (F / (math.sin(B) / math.sqrt(0.5))) - (x / B)
	tmp = 0
	if F <= -0.45:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= -5e-77:
		tmp = t_0
	elif F <= 8e-187:
		tmp = (math.cos(B) * -x) / math.sin(B)
	elif F <= 3.8e-16:
		tmp = t_0
	else:
		tmp = (1.0 / math.sin(B)) + (x * (-1.0 / math.tan(B)))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(F / Float64(sin(B) / sqrt(0.5))) - Float64(x / B))
	tmp = 0.0
	if (F <= -0.45)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= -5e-77)
		tmp = t_0;
	elseif (F <= 8e-187)
		tmp = Float64(Float64(cos(B) * Float64(-x)) / sin(B));
	elseif (F <= 3.8e-16)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / sin(B)) + Float64(x * Float64(-1.0 / tan(B))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (F / (sin(B) / sqrt(0.5))) - (x / B);
	tmp = 0.0;
	if (F <= -0.45)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= -5e-77)
		tmp = t_0;
	elseif (F <= 8e-187)
		tmp = (cos(B) * -x) / sin(B);
	elseif (F <= 3.8e-16)
		tmp = t_0;
	else
		tmp = (1.0 / sin(B)) + (x * (-1.0 / tan(B)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(F / N[(N[Sin[B], $MachinePrecision] / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.45], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -5e-77], t$95$0, If[LessEqual[F, 8e-187], N[(N[(N[Cos[B], $MachinePrecision] * (-x)), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.8e-16], t$95$0, N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.450000000000000011

   1. Initial program 69.0%

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

   2. Taylor expanded in F around -inf 98.9%

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

      1. +-commutative 98.9%

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

      2. div-inv 99.1%

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

      3. unsub-neg 99.1%

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

   4. Applied egg-rr 99.1%

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

   if -0.450000000000000011 < F < -4.99999999999999963e-77 or 8.0000000000000001e-187 < F < 3.80000000000000012e-16

   1. Initial program 99.4%

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

   2. Taylor expanded in F around 0 98.1%

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

   3. Taylor expanded in x around 0 98.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot \sqrt{0.5}}{\sin B}} \]
   4. Step-by-step derivation

      1. associate-/l* 98.1%

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

   5. Simplified 98.1%

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

   6. Taylor expanded in B around 0 76.2%

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

   if -4.99999999999999963e-77 < F < 8.0000000000000001e-187

   1. Initial program 99.4%

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

   2. Taylor expanded in F around -inf 39.2%

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

   3. Taylor expanded in x around inf 83.0%

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

      1. associate-*r/ 83.0%

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

      2. *-commutative 83.0%

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

      3. neg-mul-1 83.0%

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

      4. distribute-lft-neg-in 83.0%

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

   5. Simplified 83.0%

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

   if 3.80000000000000012e-16 < F

   1. Initial program 54.6%

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

   2. Taylor expanded in F around inf 97.5%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.45:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -5 \cdot 10^{-77}:\\ \;\;\;\;\frac{F}{\frac{\sin B}{\sqrt{0.5}}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 8 \cdot 10^{-187}:\\ \;\;\;\;\frac{\cos B \cdot \left(-x\right)}{\sin B}\\ \mathbf{elif}\;F \leq 3.8 \cdot 10^{-16}:\\ \;\;\;\;\frac{F}{\frac{\sin B}{\sqrt{0.5}}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} + x \cdot \frac{-1}{\tan B}\\ \end{array} \]

Alternative 8: 91.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1 \cdot 10^{-21}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 2.05 \cdot 10^{-29}:\\ \;\;\;\;\frac{F}{B \cdot \sqrt{2 + x \cdot 2}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} + x \cdot \frac{-1}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1e-21)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 2.05e-29)
       (- (/ F (* B (sqrt (+ 2.0 (* x 2.0))))) t_0)
       (+ (/ 1.0 (sin B)) (* x (/ -1.0 (tan B))))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1e-21) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 2.05e-29) {
		tmp = (F / (B * sqrt((2.0 + (x * 2.0))))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) + (x * (-1.0 / tan(B)));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1d-21)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 2.05d-29) then
        tmp = (f / (b * sqrt((2.0d0 + (x * 2.0d0))))) - t_0
    else
        tmp = (1.0d0 / sin(b)) + (x * ((-1.0d0) / tan(b)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1e-21) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 2.05e-29) {
		tmp = (F / (B * Math.sqrt((2.0 + (x * 2.0))))) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) + (x * (-1.0 / Math.tan(B)));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1e-21:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 2.05e-29:
		tmp = (F / (B * math.sqrt((2.0 + (x * 2.0))))) - t_0
	else:
		tmp = (1.0 / math.sin(B)) + (x * (-1.0 / math.tan(B)))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1e-21)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 2.05e-29)
		tmp = Float64(Float64(F / Float64(B * sqrt(Float64(2.0 + Float64(x * 2.0))))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) + Float64(x * Float64(-1.0 / tan(B))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1e-21)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 2.05e-29)
		tmp = (F / (B * sqrt((2.0 + (x * 2.0))))) - t_0;
	else
		tmp = (1.0 / sin(B)) + (x * (-1.0 / tan(B)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1e-21], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 2.05e-29], N[(N[(F / N[(B * N[Sqrt[N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -9.99999999999999908e-22

   1. Initial program 70.0%

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

   2. Taylor expanded in F around -inf 97.0%

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

      1. +-commutative 97.0%

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

      2. div-inv 97.1%

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

      3. unsub-neg 97.1%

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

   4. Applied egg-rr 97.1%

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

   if -9.99999999999999908e-22 < F < 2.0499999999999999e-29

   1. Initial program 99.4%

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

   2. Taylor expanded in F around 0 99.4%

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

      1. +-commutative 99.4%

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

      2. div-inv 99.7%

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

      3. unsub-neg 99.7%

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

      4. *-commutative 99.7%

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

      5. sqrt-div 99.6%

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

      6. metadata-eval 99.6%

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

      7. frac-times 99.7%

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

      8. *-un-lft-identity 99.7%

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

      9. +-commutative 99.7%

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

      10. fma-def 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in B around 0 83.6%

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

   if 2.0499999999999999e-29 < F

   1. Initial program 55.3%

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

   2. Taylor expanded in F around inf 96.1%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{-21}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.05 \cdot 10^{-29}:\\ \;\;\;\;\frac{F}{B \cdot \sqrt{2 + x \cdot 2}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} + x \cdot \frac{-1}{\tan B}\\ \end{array} \]

Alternative 9: 71.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3.9 \cdot 10^{+77}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -2.25 \cdot 10^{+27}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.7 \cdot 10^{-10}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -7.2 \cdot 10^{-76}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{0.5}\\ \mathbf{elif}\;F \leq 4.1 \cdot 10^{+18}:\\ \;\;\;\;\cos B \cdot \frac{x}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ -1.0 B) (/ x (tan B)))))
   (if (<= F -3.9e+77)
     t_0
     (if (<= F -2.25e+27)
       (- (/ -1.0 (sin B)) (/ x B))
       (if (<= F -1.7e-10)
         t_0
         (if (<= F -7.2e-76)
           (* (/ F (sin B)) (sqrt 0.5))
           (if (<= F 4.1e+18)
             (* (cos B) (/ x (- (sin B))))
             (- (/ 1.0 (sin B)) (/ x B)))))))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 / B) - (x / tan(B));
	double tmp;
	if (F <= -3.9e+77) {
		tmp = t_0;
	} else if (F <= -2.25e+27) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -1.7e-10) {
		tmp = t_0;
	} else if (F <= -7.2e-76) {
		tmp = (F / sin(B)) * sqrt(0.5);
	} else if (F <= 4.1e+18) {
		tmp = cos(B) * (x / -sin(B));
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-1.0d0) / b) - (x / tan(b))
    if (f <= (-3.9d+77)) then
        tmp = t_0
    else if (f <= (-2.25d+27)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-1.7d-10)) then
        tmp = t_0
    else if (f <= (-7.2d-76)) then
        tmp = (f / sin(b)) * sqrt(0.5d0)
    else if (f <= 4.1d+18) then
        tmp = cos(b) * (x / -sin(b))
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (-1.0 / B) - (x / Math.tan(B));
	double tmp;
	if (F <= -3.9e+77) {
		tmp = t_0;
	} else if (F <= -2.25e+27) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -1.7e-10) {
		tmp = t_0;
	} else if (F <= -7.2e-76) {
		tmp = (F / Math.sin(B)) * Math.sqrt(0.5);
	} else if (F <= 4.1e+18) {
		tmp = Math.cos(B) * (x / -Math.sin(B));
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (-1.0 / B) - (x / math.tan(B))
	tmp = 0
	if F <= -3.9e+77:
		tmp = t_0
	elif F <= -2.25e+27:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -1.7e-10:
		tmp = t_0
	elif F <= -7.2e-76:
		tmp = (F / math.sin(B)) * math.sqrt(0.5)
	elif F <= 4.1e+18:
		tmp = math.cos(B) * (x / -math.sin(B))
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 / B) - Float64(x / tan(B)))
	tmp = 0.0
	if (F <= -3.9e+77)
		tmp = t_0;
	elseif (F <= -2.25e+27)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -1.7e-10)
		tmp = t_0;
	elseif (F <= -7.2e-76)
		tmp = Float64(Float64(F / sin(B)) * sqrt(0.5));
	elseif (F <= 4.1e+18)
		tmp = Float64(cos(B) * Float64(x / Float64(-sin(B))));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (-1.0 / B) - (x / tan(B));
	tmp = 0.0;
	if (F <= -3.9e+77)
		tmp = t_0;
	elseif (F <= -2.25e+27)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -1.7e-10)
		tmp = t_0;
	elseif (F <= -7.2e-76)
		tmp = (F / sin(B)) * sqrt(0.5);
	elseif (F <= 4.1e+18)
		tmp = cos(B) * (x / -sin(B));
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -3.9e+77], t$95$0, If[LessEqual[F, -2.25e+27], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.7e-10], t$95$0, If[LessEqual[F, -7.2e-76], N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.1e+18], N[(N[Cos[B], $MachinePrecision] * N[(x / (-N[Sin[B], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -3.8999999999999998e77 or -2.25e27 < F < -1.70000000000000007e-10

   1. Initial program 66.3%

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

   2. Taylor expanded in F around -inf 97.8%

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

      1. +-commutative 97.8%

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

      2. div-inv 97.9%

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

      3. unsub-neg 97.9%

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

   4. Applied egg-rr 97.9%

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

   5. Taylor expanded in B around 0 79.2%

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

   if -3.8999999999999998e77 < F < -2.25e27

   1. Initial program 99.7%

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

   2. Taylor expanded in F around -inf 99.8%

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

   3. Taylor expanded in B around 0 99.8%

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

   if -1.70000000000000007e-10 < F < -7.2000000000000001e-76

   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 99.5%

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

   3. Taylor expanded in x around 0 99.5%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot \sqrt{0.5}}{\sin B}} \]
   4. Step-by-step derivation

      1. associate-/l* 99.1%

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

   5. Simplified 99.1%

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

   6. Taylor expanded in x around 0 64.5%

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

      1. associate-*l/ 64.5%

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

      2. *-commutative 64.5%

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

   8. Simplified 64.5%

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

   if -7.2000000000000001e-76 < F < 4.1e18

   1. Initial program 98.4%

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

   2. Taylor expanded in F around -inf 39.9%

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

   3. Taylor expanded in x around inf 74.6%

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

      1. mul-1-neg 74.6%

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

      2. *-commutative 74.6%

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

      3. associate-*r/ 74.6%

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

      4. distribute-rgt-neg-in 74.6%

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

      5. *-lft-identity 74.6%

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

      6. metadata-eval 74.6%

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

      7. times-frac 74.6%

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

      8. neg-mul-1 74.6%

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

      9. neg-mul-1 74.6%

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

      10. distribute-frac-neg 74.6%

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

      11. remove-double-neg 74.6%

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

   5. Simplified 74.6%

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

   if 4.1e18 < F

   1. Initial program 51.7%

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

   2. Taylor expanded in F around inf 99.8%

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

   3. Taylor expanded in B around 0 81.5%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.9 \cdot 10^{+77}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -2.25 \cdot 10^{+27}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.7 \cdot 10^{-10}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -7.2 \cdot 10^{-76}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{0.5}\\ \mathbf{elif}\;F \leq 4.1 \cdot 10^{+18}:\\ \;\;\;\;\cos B \cdot \frac{x}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 10: 71.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -6.6 \cdot 10^{+77}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -5.8 \cdot 10^{+28}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -8.8 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -7.5 \cdot 10^{-77}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{0.5}\\ \mathbf{elif}\;F \leq 4.1 \cdot 10^{+18}:\\ \;\;\;\;\frac{\cos B \cdot \left(-x\right)}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ -1.0 B) (/ x (tan B)))))
   (if (<= F -6.6e+77)
     t_0
     (if (<= F -5.8e+28)
       (- (/ -1.0 (sin B)) (/ x B))
       (if (<= F -8.8e-7)
         t_0
         (if (<= F -7.5e-77)
           (* (/ F (sin B)) (sqrt 0.5))
           (if (<= F 4.1e+18)
             (/ (* (cos B) (- x)) (sin B))
             (- (/ 1.0 (sin B)) (/ x B)))))))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 / B) - (x / tan(B));
	double tmp;
	if (F <= -6.6e+77) {
		tmp = t_0;
	} else if (F <= -5.8e+28) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -8.8e-7) {
		tmp = t_0;
	} else if (F <= -7.5e-77) {
		tmp = (F / sin(B)) * sqrt(0.5);
	} else if (F <= 4.1e+18) {
		tmp = (cos(B) * -x) / sin(B);
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-1.0d0) / b) - (x / tan(b))
    if (f <= (-6.6d+77)) then
        tmp = t_0
    else if (f <= (-5.8d+28)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-8.8d-7)) then
        tmp = t_0
    else if (f <= (-7.5d-77)) then
        tmp = (f / sin(b)) * sqrt(0.5d0)
    else if (f <= 4.1d+18) then
        tmp = (cos(b) * -x) / sin(b)
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (-1.0 / B) - (x / Math.tan(B));
	double tmp;
	if (F <= -6.6e+77) {
		tmp = t_0;
	} else if (F <= -5.8e+28) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -8.8e-7) {
		tmp = t_0;
	} else if (F <= -7.5e-77) {
		tmp = (F / Math.sin(B)) * Math.sqrt(0.5);
	} else if (F <= 4.1e+18) {
		tmp = (Math.cos(B) * -x) / Math.sin(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (-1.0 / B) - (x / math.tan(B))
	tmp = 0
	if F <= -6.6e+77:
		tmp = t_0
	elif F <= -5.8e+28:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -8.8e-7:
		tmp = t_0
	elif F <= -7.5e-77:
		tmp = (F / math.sin(B)) * math.sqrt(0.5)
	elif F <= 4.1e+18:
		tmp = (math.cos(B) * -x) / math.sin(B)
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 / B) - Float64(x / tan(B)))
	tmp = 0.0
	if (F <= -6.6e+77)
		tmp = t_0;
	elseif (F <= -5.8e+28)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -8.8e-7)
		tmp = t_0;
	elseif (F <= -7.5e-77)
		tmp = Float64(Float64(F / sin(B)) * sqrt(0.5));
	elseif (F <= 4.1e+18)
		tmp = Float64(Float64(cos(B) * Float64(-x)) / sin(B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (-1.0 / B) - (x / tan(B));
	tmp = 0.0;
	if (F <= -6.6e+77)
		tmp = t_0;
	elseif (F <= -5.8e+28)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -8.8e-7)
		tmp = t_0;
	elseif (F <= -7.5e-77)
		tmp = (F / sin(B)) * sqrt(0.5);
	elseif (F <= 4.1e+18)
		tmp = (cos(B) * -x) / sin(B);
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -6.6e+77], t$95$0, If[LessEqual[F, -5.8e+28], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -8.8e-7], t$95$0, If[LessEqual[F, -7.5e-77], N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.1e+18], N[(N[(N[Cos[B], $MachinePrecision] * (-x)), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -6.5999999999999996e77 or -5.8000000000000002e28 < F < -8.8000000000000004e-7

   1. Initial program 66.3%

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

   2. Taylor expanded in F around -inf 97.8%

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

      1. +-commutative 97.8%

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

      2. div-inv 97.9%

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

      3. unsub-neg 97.9%

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

   4. Applied egg-rr 97.9%

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

   5. Taylor expanded in B around 0 79.2%

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

   if -6.5999999999999996e77 < F < -5.8000000000000002e28

   1. Initial program 99.7%

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

   2. Taylor expanded in F around -inf 99.8%

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

   3. Taylor expanded in B around 0 99.8%

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

   if -8.8000000000000004e-7 < F < -7.5000000000000006e-77

   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 99.5%

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

   3. Taylor expanded in x around 0 99.5%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot \sqrt{0.5}}{\sin B}} \]
   4. Step-by-step derivation

      1. associate-/l* 99.1%

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

   5. Simplified 99.1%

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

   6. Taylor expanded in x around 0 64.5%

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

      1. associate-*l/ 64.5%

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

      2. *-commutative 64.5%

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

   8. Simplified 64.5%

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

   if -7.5000000000000006e-77 < F < 4.1e18

   1. Initial program 98.4%

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

   2. Taylor expanded in F around -inf 39.9%

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

   3. Taylor expanded in x around inf 74.6%

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

      1. associate-*r/ 74.6%

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

      2. *-commutative 74.6%

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

      3. neg-mul-1 74.6%

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

      4. distribute-lft-neg-in 74.6%

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

   5. Simplified 74.6%

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

   if 4.1e18 < F

   1. Initial program 51.7%

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

   2. Taylor expanded in F around inf 99.8%

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

   3. Taylor expanded in B around 0 81.5%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6.6 \cdot 10^{+77}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -5.8 \cdot 10^{+28}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -8.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -7.5 \cdot 10^{-77}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{0.5}\\ \mathbf{elif}\;F \leq 4.1 \cdot 10^{+18}:\\ \;\;\;\;\frac{\cos B \cdot \left(-x\right)}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 11: 85.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -9 \cdot 10^{-7}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.02 \cdot 10^{-75}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{0.5}\\ \mathbf{elif}\;F \leq 3.7 \cdot 10^{-31}:\\ \;\;\;\;\frac{\cos B \cdot \left(-x\right)}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} + x \cdot \frac{-1}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -9e-7)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F -1.02e-75)
     (* (/ F (sin B)) (sqrt 0.5))
     (if (<= F 3.7e-31)
       (/ (* (cos B) (- x)) (sin B))
       (+ (/ 1.0 (sin B)) (* x (/ -1.0 (tan B))))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -9e-7) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= -1.02e-75) {
		tmp = (F / sin(B)) * sqrt(0.5);
	} else if (F <= 3.7e-31) {
		tmp = (cos(B) * -x) / sin(B);
	} else {
		tmp = (1.0 / sin(B)) + (x * (-1.0 / tan(B)));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-9d-7)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= (-1.02d-75)) then
        tmp = (f / sin(b)) * sqrt(0.5d0)
    else if (f <= 3.7d-31) then
        tmp = (cos(b) * -x) / sin(b)
    else
        tmp = (1.0d0 / sin(b)) + (x * ((-1.0d0) / tan(b)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -9e-7) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= -1.02e-75) {
		tmp = (F / Math.sin(B)) * Math.sqrt(0.5);
	} else if (F <= 3.7e-31) {
		tmp = (Math.cos(B) * -x) / Math.sin(B);
	} else {
		tmp = (1.0 / Math.sin(B)) + (x * (-1.0 / Math.tan(B)));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -9e-7:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= -1.02e-75:
		tmp = (F / math.sin(B)) * math.sqrt(0.5)
	elif F <= 3.7e-31:
		tmp = (math.cos(B) * -x) / math.sin(B)
	else:
		tmp = (1.0 / math.sin(B)) + (x * (-1.0 / math.tan(B)))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -9e-7)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= -1.02e-75)
		tmp = Float64(Float64(F / sin(B)) * sqrt(0.5));
	elseif (F <= 3.7e-31)
		tmp = Float64(Float64(cos(B) * Float64(-x)) / sin(B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) + Float64(x * Float64(-1.0 / tan(B))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -9e-7)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= -1.02e-75)
		tmp = (F / sin(B)) * sqrt(0.5);
	elseif (F <= 3.7e-31)
		tmp = (cos(B) * -x) / sin(B);
	else
		tmp = (1.0 / sin(B)) + (x * (-1.0 / tan(B)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -9e-7], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.02e-75], N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.7e-31], N[(N[(N[Cos[B], $MachinePrecision] * (-x)), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -8.99999999999999959e-7

   1. Initial program 69.7%

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

   2. Taylor expanded in F around -inf 98.0%

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

      1. +-commutative 98.0%

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

      2. div-inv 98.1%

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

      3. unsub-neg 98.1%

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

   4. Applied egg-rr 98.1%

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

   if -8.99999999999999959e-7 < F < -1.01999999999999997e-75

   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 99.5%

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

   3. Taylor expanded in x around 0 99.5%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot \sqrt{0.5}}{\sin B}} \]
   4. Step-by-step derivation

      1. associate-/l* 99.1%

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

   5. Simplified 99.1%

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

   6. Taylor expanded in x around 0 64.5%

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

      1. associate-*l/ 64.5%

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

      2. *-commutative 64.5%

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

   8. Simplified 64.5%

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

   if -1.01999999999999997e-75 < F < 3.6999999999999998e-31

   1. Initial program 99.4%

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

   2. Taylor expanded in F around -inf 38.5%

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

   3. Taylor expanded in x around inf 75.9%

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

      1. associate-*r/ 75.9%

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

      2. *-commutative 75.9%

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

      3. neg-mul-1 75.9%

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

      4. distribute-lft-neg-in 75.9%

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

   5. Simplified 75.9%

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

   if 3.6999999999999998e-31 < F

   1. Initial program 55.3%

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

   2. Taylor expanded in F around inf 96.1%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -9 \cdot 10^{-7}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.02 \cdot 10^{-75}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{0.5}\\ \mathbf{elif}\;F \leq 3.7 \cdot 10^{-31}:\\ \;\;\;\;\frac{\cos B \cdot \left(-x\right)}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} + x \cdot \frac{-1}{\tan B}\\ \end{array} \]

Alternative 12: 78.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.7 \cdot 10^{-8}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.2 \cdot 10^{-76}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{0.5}\\ \mathbf{elif}\;F \leq 4.1 \cdot 10^{+18}:\\ \;\;\;\;\frac{\cos B \cdot \left(-x\right)}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.7e-8)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F -1.2e-76)
     (* (/ F (sin B)) (sqrt 0.5))
     (if (<= F 4.1e+18)
       (/ (* (cos B) (- x)) (sin B))
       (- (/ 1.0 (sin B)) (/ x B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.7e-8) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= -1.2e-76) {
		tmp = (F / sin(B)) * sqrt(0.5);
	} else if (F <= 4.1e+18) {
		tmp = (cos(B) * -x) / sin(B);
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-3.7d-8)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= (-1.2d-76)) then
        tmp = (f / sin(b)) * sqrt(0.5d0)
    else if (f <= 4.1d+18) then
        tmp = (cos(b) * -x) / sin(b)
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.7e-8) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= -1.2e-76) {
		tmp = (F / Math.sin(B)) * Math.sqrt(0.5);
	} else if (F <= 4.1e+18) {
		tmp = (Math.cos(B) * -x) / Math.sin(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.7e-8:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= -1.2e-76:
		tmp = (F / math.sin(B)) * math.sqrt(0.5)
	elif F <= 4.1e+18:
		tmp = (math.cos(B) * -x) / math.sin(B)
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.7e-8)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= -1.2e-76)
		tmp = Float64(Float64(F / sin(B)) * sqrt(0.5));
	elseif (F <= 4.1e+18)
		tmp = Float64(Float64(cos(B) * Float64(-x)) / sin(B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -3.7e-8)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= -1.2e-76)
		tmp = (F / sin(B)) * sqrt(0.5);
	elseif (F <= 4.1e+18)
		tmp = (cos(B) * -x) / sin(B);
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.7e-8], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.2e-76], N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.1e+18], N[(N[(N[Cos[B], $MachinePrecision] * (-x)), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.7e-8

   1. Initial program 69.7%

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

   2. Taylor expanded in F around -inf 98.0%

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

      1. +-commutative 98.0%

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

      2. div-inv 98.1%

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

      3. unsub-neg 98.1%

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

   4. Applied egg-rr 98.1%

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

   if -3.7e-8 < F < -1.20000000000000007e-76

   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 99.5%

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

   3. Taylor expanded in x around 0 99.5%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot \sqrt{0.5}}{\sin B}} \]
   4. Step-by-step derivation

      1. associate-/l* 99.1%

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

   5. Simplified 99.1%

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

   6. Taylor expanded in x around 0 64.5%

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

      1. associate-*l/ 64.5%

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

      2. *-commutative 64.5%

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

   8. Simplified 64.5%

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

   if -1.20000000000000007e-76 < F < 4.1e18

   1. Initial program 98.4%

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

   2. Taylor expanded in F around -inf 39.9%

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

   3. Taylor expanded in x around inf 74.6%

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

      1. associate-*r/ 74.6%

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

      2. *-commutative 74.6%

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

      3. neg-mul-1 74.6%

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

      4. distribute-lft-neg-in 74.6%

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

   5. Simplified 74.6%

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

   if 4.1e18 < F

   1. Initial program 51.7%

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

   2. Taylor expanded in F around inf 99.8%

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

   3. Taylor expanded in B around 0 81.5%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.7 \cdot 10^{-8}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.2 \cdot 10^{-76}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{0.5}\\ \mathbf{elif}\;F \leq 4.1 \cdot 10^{+18}:\\ \;\;\;\;\frac{\cos B \cdot \left(-x\right)}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 13: 58.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{\sin B}\\ t_1 := \frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;x \leq -2.7 \cdot 10^{-84}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-301}:\\ \;\;\;\;t_0 - \frac{x}{B}\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-251}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{0.5}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-48}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ -1.0 (sin B))) (t_1 (- (/ 1.0 B) (* x (/ 1.0 (tan B))))))
   (if (<= x -2.7e-84)
     t_1
     (if (<= x -2.2e-301)
       (- t_0 (/ x B))
       (if (<= x 6e-251)
         (* (/ F (sin B)) (sqrt 0.5))
         (if (<= x 6.5e-48) t_0 t_1))))))

C

double code(double F, double B, double x) {
	double t_0 = -1.0 / sin(B);
	double t_1 = (1.0 / B) - (x * (1.0 / tan(B)));
	double tmp;
	if (x <= -2.7e-84) {
		tmp = t_1;
	} else if (x <= -2.2e-301) {
		tmp = t_0 - (x / B);
	} else if (x <= 6e-251) {
		tmp = (F / sin(B)) * sqrt(0.5);
	} else if (x <= 6.5e-48) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (-1.0d0) / sin(b)
    t_1 = (1.0d0 / b) - (x * (1.0d0 / tan(b)))
    if (x <= (-2.7d-84)) then
        tmp = t_1
    else if (x <= (-2.2d-301)) then
        tmp = t_0 - (x / b)
    else if (x <= 6d-251) then
        tmp = (f / sin(b)) * sqrt(0.5d0)
    else if (x <= 6.5d-48) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = -1.0 / Math.sin(B);
	double t_1 = (1.0 / B) - (x * (1.0 / Math.tan(B)));
	double tmp;
	if (x <= -2.7e-84) {
		tmp = t_1;
	} else if (x <= -2.2e-301) {
		tmp = t_0 - (x / B);
	} else if (x <= 6e-251) {
		tmp = (F / Math.sin(B)) * Math.sqrt(0.5);
	} else if (x <= 6.5e-48) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -1.0 / math.sin(B)
	t_1 = (1.0 / B) - (x * (1.0 / math.tan(B)))
	tmp = 0
	if x <= -2.7e-84:
		tmp = t_1
	elif x <= -2.2e-301:
		tmp = t_0 - (x / B)
	elif x <= 6e-251:
		tmp = (F / math.sin(B)) * math.sqrt(0.5)
	elif x <= 6.5e-48:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(-1.0 / sin(B))
	t_1 = Float64(Float64(1.0 / B) - Float64(x * Float64(1.0 / tan(B))))
	tmp = 0.0
	if (x <= -2.7e-84)
		tmp = t_1;
	elseif (x <= -2.2e-301)
		tmp = Float64(t_0 - Float64(x / B));
	elseif (x <= 6e-251)
		tmp = Float64(Float64(F / sin(B)) * sqrt(0.5));
	elseif (x <= 6.5e-48)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = -1.0 / sin(B);
	t_1 = (1.0 / B) - (x * (1.0 / tan(B)));
	tmp = 0.0;
	if (x <= -2.7e-84)
		tmp = t_1;
	elseif (x <= -2.2e-301)
		tmp = t_0 - (x / B);
	elseif (x <= 6e-251)
		tmp = (F / sin(B)) * sqrt(0.5);
	elseif (x <= 6.5e-48)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.7e-84], t$95$1, If[LessEqual[x, -2.2e-301], N[(t$95$0 - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6e-251], N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.5e-48], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.6999999999999999e-84 or 6.5e-48 < x

   1. Initial program 85.7%

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

   2. Taylor expanded in F around inf 85.8%

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

   3. Taylor expanded in B around 0 87.8%

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

   if -2.6999999999999999e-84 < x < -2.2e-301

   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 F around -inf 41.6%

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

   3. Taylor expanded in B around 0 41.6%

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

   if -2.2e-301 < x < 5.9999999999999997e-251

   1. Initial program 77.3%

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

   2. Taylor expanded in F around 0 63.6%

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

   3. Taylor expanded in x around 0 63.8%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot \sqrt{0.5}}{\sin B}} \]
   4. Step-by-step derivation

      1. associate-/l* 63.9%

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

   5. Simplified 63.9%

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

   6. Taylor expanded in x around 0 55.7%

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

      1. associate-*l/ 55.5%

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

      2. *-commutative 55.5%

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

   8. Simplified 55.5%

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

   if 5.9999999999999997e-251 < x < 6.5e-48

   1. Initial program 66.2%

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

   2. Taylor expanded in F around -inf 38.6%

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

      1. +-commutative 38.6%

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

      2. div-inv 38.6%

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

      3. unsub-neg 38.6%

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

   4. Applied egg-rr 38.6%

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

   5. Taylor expanded in x around 0 38.6%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-84}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-301}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-251}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{0.5}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-48}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \end{array} \]

Alternative 14: 58.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{\sin B}\\ t_1 := \frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;x \leq -6.8 \cdot 10^{-83}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-303}:\\ \;\;\;\;t_0 - \frac{x}{B}\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{-254}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-43}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ -1.0 (sin B))) (t_1 (- (/ 1.0 B) (* x (/ 1.0 (tan B))))))
   (if (<= x -6.8e-83)
     t_1
     (if (<= x -1.7e-303)
       (- t_0 (/ x B))
       (if (<= x 7.6e-254)
         (/ (* F (sqrt 0.5)) (sin B))
         (if (<= x 6.8e-43) t_0 t_1))))))

C

double code(double F, double B, double x) {
	double t_0 = -1.0 / sin(B);
	double t_1 = (1.0 / B) - (x * (1.0 / tan(B)));
	double tmp;
	if (x <= -6.8e-83) {
		tmp = t_1;
	} else if (x <= -1.7e-303) {
		tmp = t_0 - (x / B);
	} else if (x <= 7.6e-254) {
		tmp = (F * sqrt(0.5)) / sin(B);
	} else if (x <= 6.8e-43) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (-1.0d0) / sin(b)
    t_1 = (1.0d0 / b) - (x * (1.0d0 / tan(b)))
    if (x <= (-6.8d-83)) then
        tmp = t_1
    else if (x <= (-1.7d-303)) then
        tmp = t_0 - (x / b)
    else if (x <= 7.6d-254) then
        tmp = (f * sqrt(0.5d0)) / sin(b)
    else if (x <= 6.8d-43) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = -1.0 / Math.sin(B);
	double t_1 = (1.0 / B) - (x * (1.0 / Math.tan(B)));
	double tmp;
	if (x <= -6.8e-83) {
		tmp = t_1;
	} else if (x <= -1.7e-303) {
		tmp = t_0 - (x / B);
	} else if (x <= 7.6e-254) {
		tmp = (F * Math.sqrt(0.5)) / Math.sin(B);
	} else if (x <= 6.8e-43) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -1.0 / math.sin(B)
	t_1 = (1.0 / B) - (x * (1.0 / math.tan(B)))
	tmp = 0
	if x <= -6.8e-83:
		tmp = t_1
	elif x <= -1.7e-303:
		tmp = t_0 - (x / B)
	elif x <= 7.6e-254:
		tmp = (F * math.sqrt(0.5)) / math.sin(B)
	elif x <= 6.8e-43:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(-1.0 / sin(B))
	t_1 = Float64(Float64(1.0 / B) - Float64(x * Float64(1.0 / tan(B))))
	tmp = 0.0
	if (x <= -6.8e-83)
		tmp = t_1;
	elseif (x <= -1.7e-303)
		tmp = Float64(t_0 - Float64(x / B));
	elseif (x <= 7.6e-254)
		tmp = Float64(Float64(F * sqrt(0.5)) / sin(B));
	elseif (x <= 6.8e-43)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = -1.0 / sin(B);
	t_1 = (1.0 / B) - (x * (1.0 / tan(B)));
	tmp = 0.0;
	if (x <= -6.8e-83)
		tmp = t_1;
	elseif (x <= -1.7e-303)
		tmp = t_0 - (x / B);
	elseif (x <= 7.6e-254)
		tmp = (F * sqrt(0.5)) / sin(B);
	elseif (x <= 6.8e-43)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.8e-83], t$95$1, If[LessEqual[x, -1.7e-303], N[(t$95$0 - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.6e-254], N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.8e-43], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -6.7999999999999995e-83 or 6.8000000000000001e-43 < x

   1. Initial program 85.7%

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

   2. Taylor expanded in F around inf 85.8%

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

   3. Taylor expanded in B around 0 87.8%

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

   if -6.7999999999999995e-83 < x < -1.7e-303

   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 F around -inf 41.6%

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

   3. Taylor expanded in B around 0 41.6%

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

   if -1.7e-303 < x < 7.6000000000000002e-254

   1. Initial program 77.3%

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

   2. Taylor expanded in F around 0 63.6%

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

   3. Taylor expanded in x around 0 63.8%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot \sqrt{0.5}}{\sin B}} \]
   4. Step-by-step derivation

      1. associate-/l* 63.9%

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

   5. Simplified 63.9%

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

   6. Taylor expanded in x around 0 55.7%

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

   if 7.6000000000000002e-254 < x < 6.8000000000000001e-43

   1. Initial program 66.2%

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

   2. Taylor expanded in F around -inf 38.6%

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

      1. +-commutative 38.6%

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

      2. div-inv 38.6%

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

      3. unsub-neg 38.6%

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

   4. Applied egg-rr 38.6%

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

   5. Taylor expanded in x around 0 38.6%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{-83}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-303}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{-254}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-43}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \end{array} \]

Alternative 15: 54.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;x \leq -9.2 \cdot 10^{-52}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-75}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{-6}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+50}:\\ \;\;\;\;\frac{1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ -1.0 B) (/ x (tan B)))))
   (if (<= x -9.2e-52)
     t_0
     (if (<= x 4e-75)
       (/ -1.0 (sin B))
       (if (<= x 2.05e-6)
         (/ (- (* F (sqrt 0.5)) x) B)
         (if (<= x 2.9e+50) (/ (- 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 (x <= -9.2e-52) {
		tmp = t_0;
	} else if (x <= 4e-75) {
		tmp = -1.0 / sin(B);
	} else if (x <= 2.05e-6) {
		tmp = ((F * sqrt(0.5)) - x) / B;
	} else if (x <= 2.9e+50) {
		tmp = (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 (x <= (-9.2d-52)) then
        tmp = t_0
    else if (x <= 4d-75) then
        tmp = (-1.0d0) / sin(b)
    else if (x <= 2.05d-6) then
        tmp = ((f * sqrt(0.5d0)) - x) / b
    else if (x <= 2.9d+50) then
        tmp = (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 (x <= -9.2e-52) {
		tmp = t_0;
	} else if (x <= 4e-75) {
		tmp = -1.0 / Math.sin(B);
	} else if (x <= 2.05e-6) {
		tmp = ((F * Math.sqrt(0.5)) - x) / B;
	} else if (x <= 2.9e+50) {
		tmp = (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 x <= -9.2e-52:
		tmp = t_0
	elif x <= 4e-75:
		tmp = -1.0 / math.sin(B)
	elif x <= 2.05e-6:
		tmp = ((F * math.sqrt(0.5)) - x) / B
	elif x <= 2.9e+50:
		tmp = (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 (x <= -9.2e-52)
		tmp = t_0;
	elseif (x <= 4e-75)
		tmp = Float64(-1.0 / sin(B));
	elseif (x <= 2.05e-6)
		tmp = Float64(Float64(Float64(F * sqrt(0.5)) - x) / B);
	elseif (x <= 2.9e+50)
		tmp = 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 (x <= -9.2e-52)
		tmp = t_0;
	elseif (x <= 4e-75)
		tmp = -1.0 / sin(B);
	elseif (x <= 2.05e-6)
		tmp = ((F * sqrt(0.5)) - x) / B;
	elseif (x <= 2.9e+50)
		tmp = (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[x, -9.2e-52], t$95$0, If[LessEqual[x, 4e-75], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.05e-6], N[(N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[x, 2.9e+50], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -9.19999999999999977e-52 or 2.9e50 < x

   1. Initial program 88.3%

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

   2. Taylor expanded in F around -inf 90.5%

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

      1. +-commutative 90.5%

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

      2. div-inv 90.7%

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

      3. unsub-neg 90.7%

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

   4. Applied egg-rr 90.7%

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

   5. Taylor expanded in B around 0 93.2%

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

   if -9.19999999999999977e-52 < x < 3.9999999999999998e-75

   1. Initial program 68.0%

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

   2. Taylor expanded in F around -inf 36.5%

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

      1. +-commutative 36.5%

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

      2. div-inv 36.5%

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

      3. unsub-neg 36.5%

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

   4. Applied egg-rr 36.5%

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

   5. Taylor expanded in x around 0 36.5%

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

   if 3.9999999999999998e-75 < x < 2.0499999999999999e-6

   1. Initial program 88.4%

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

   2. Taylor expanded in F around 0 68.2%

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

   3. Taylor expanded in x around 0 68.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot \sqrt{0.5}}{\sin B}} \]
   4. Step-by-step derivation

      1. associate-/l* 68.2%

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

   5. Simplified 68.2%

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

   6. Taylor expanded in B around 0 47.1%

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

   if 2.0499999999999999e-6 < x < 2.9e50

   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 F around inf 83.0%

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

   3. Taylor expanded in B around 0 76.1%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{-52}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-75}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{-6}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+50}:\\ \;\;\;\;\frac{1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 16: 62.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -8 \cdot 10^{+76}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -3.55 \cdot 10^{+28}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4.1 \cdot 10^{+18}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ -1.0 B) (/ x (tan B)))))
   (if (<= F -8e+76)
     t_0
     (if (<= F -3.55e+28)
       (- (/ -1.0 (sin B)) (/ x B))
       (if (<= F 4.1e+18) t_0 (- (/ 1.0 (sin B)) (/ x B)))))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 / B) - (x / tan(B));
	double tmp;
	if (F <= -8e+76) {
		tmp = t_0;
	} else if (F <= -3.55e+28) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 4.1e+18) {
		tmp = t_0;
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-1.0d0) / b) - (x / tan(b))
    if (f <= (-8d+76)) then
        tmp = t_0
    else if (f <= (-3.55d+28)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 4.1d+18) then
        tmp = t_0
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (-1.0 / B) - (x / Math.tan(B));
	double tmp;
	if (F <= -8e+76) {
		tmp = t_0;
	} else if (F <= -3.55e+28) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 4.1e+18) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (-1.0 / B) - (x / math.tan(B))
	tmp = 0
	if F <= -8e+76:
		tmp = t_0
	elif F <= -3.55e+28:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 4.1e+18:
		tmp = t_0
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 / B) - Float64(x / tan(B)))
	tmp = 0.0
	if (F <= -8e+76)
		tmp = t_0;
	elseif (F <= -3.55e+28)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 4.1e+18)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (-1.0 / B) - (x / tan(B));
	tmp = 0.0;
	if (F <= -8e+76)
		tmp = t_0;
	elseif (F <= -3.55e+28)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 4.1e+18)
		tmp = t_0;
	else
		tmp = (1.0 / sin(B)) - (x / B);
	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, -8e+76], t$95$0, If[LessEqual[F, -3.55e+28], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.1e+18], t$95$0, N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -8.0000000000000004e76 or -3.55e28 < F < 4.1e18

   1. Initial program 85.1%

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

   2. Taylor expanded in F around -inf 63.2%

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

      1. +-commutative 63.2%

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

      2. div-inv 63.3%

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

      3. unsub-neg 63.3%

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

   4. Applied egg-rr 63.3%

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

   5. Taylor expanded in B around 0 59.6%

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

   if -8.0000000000000004e76 < F < -3.55e28

   1. Initial program 99.7%

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

   2. Taylor expanded in F around -inf 99.8%

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

   3. Taylor expanded in B around 0 99.8%

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

   if 4.1e18 < F

   1. Initial program 51.7%

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

   2. Taylor expanded in F around inf 99.8%

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

   3. Taylor expanded in B around 0 81.5%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -8 \cdot 10^{+76}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -3.55 \cdot 10^{+28}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4.1 \cdot 10^{+18}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 17: 57.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-81} \lor \neg \left(x \leq 1.7 \cdot 10^{-45}\right):\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (or (<= x -1.35e-81) (not (<= x 1.7e-45)))
   (- (/ 1.0 B) (* x (/ 1.0 (tan B))))
   (/ -1.0 (sin B))))

C

double code(double F, double B, double x) {
	double tmp;
	if ((x <= -1.35e-81) || !(x <= 1.7e-45)) {
		tmp = (1.0 / B) - (x * (1.0 / tan(B)));
	} else {
		tmp = -1.0 / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.35d-81)) .or. (.not. (x <= 1.7d-45))) then
        tmp = (1.0d0 / b) - (x * (1.0d0 / tan(b)))
    else
        tmp = (-1.0d0) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if ((x <= -1.35e-81) || !(x <= 1.7e-45)) {
		tmp = (1.0 / B) - (x * (1.0 / Math.tan(B)));
	} else {
		tmp = -1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if (x <= -1.35e-81) or not (x <= 1.7e-45):
		tmp = (1.0 / B) - (x * (1.0 / math.tan(B)))
	else:
		tmp = -1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if ((x <= -1.35e-81) || !(x <= 1.7e-45))
		tmp = Float64(Float64(1.0 / B) - Float64(x * Float64(1.0 / tan(B))));
	else
		tmp = Float64(-1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if ((x <= -1.35e-81) || ~((x <= 1.7e-45)))
		tmp = (1.0 / B) - (x * (1.0 / tan(B)));
	else
		tmp = -1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[Or[LessEqual[x, -1.35e-81], N[Not[LessEqual[x, 1.7e-45]], $MachinePrecision]], N[(N[(1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.34999999999999995e-81 or 1.70000000000000002e-45 < x

   1. Initial program 85.7%

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

   2. Taylor expanded in F around inf 85.8%

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

   3. Taylor expanded in B around 0 87.8%

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

   if -1.34999999999999995e-81 < x < 1.70000000000000002e-45

   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 F around -inf 36.4%

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

      1. +-commutative 36.4%

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

      2. div-inv 36.4%

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

      3. unsub-neg 36.4%

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

   4. Applied egg-rr 36.4%

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

   5. Taylor expanded in x around 0 36.4%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-81} \lor \neg \left(x \leq 1.7 \cdot 10^{-45}\right):\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \end{array} \]

Alternative 18: 57.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-52} \lor \neg \left(x \leq 2.7 \cdot 10^{-89}\right):\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (or (<= x -8e-52) (not (<= x 2.7e-89)))
   (- (/ -1.0 B) (/ x (tan B)))
   (/ -1.0 (sin B))))

C

double code(double F, double B, double x) {
	double tmp;
	if ((x <= -8e-52) || !(x <= 2.7e-89)) {
		tmp = (-1.0 / B) - (x / tan(B));
	} else {
		tmp = -1.0 / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-8d-52)) .or. (.not. (x <= 2.7d-89))) then
        tmp = ((-1.0d0) / b) - (x / tan(b))
    else
        tmp = (-1.0d0) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if ((x <= -8e-52) || !(x <= 2.7e-89)) {
		tmp = (-1.0 / B) - (x / Math.tan(B));
	} else {
		tmp = -1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if (x <= -8e-52) or not (x <= 2.7e-89):
		tmp = (-1.0 / B) - (x / math.tan(B))
	else:
		tmp = -1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if ((x <= -8e-52) || !(x <= 2.7e-89))
		tmp = Float64(Float64(-1.0 / B) - Float64(x / tan(B)));
	else
		tmp = Float64(-1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if ((x <= -8e-52) || ~((x <= 2.7e-89)))
		tmp = (-1.0 / B) - (x / tan(B));
	else
		tmp = -1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[Or[LessEqual[x, -8e-52], N[Not[LessEqual[x, 2.7e-89]], $MachinePrecision]], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.0000000000000001e-52 or 2.69999999999999988e-89 < x

   1. Initial program 86.3%

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

   2. Taylor expanded in F around -inf 79.1%

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

      1. +-commutative 79.1%

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

      2. div-inv 79.3%

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

      3. unsub-neg 79.3%

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

   4. Applied egg-rr 79.3%

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

   5. Taylor expanded in B around 0 82.3%

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

   if -8.0000000000000001e-52 < x < 2.69999999999999988e-89

   1. Initial program 68.3%

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

   2. Taylor expanded in F around -inf 36.2%

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

      1. +-commutative 36.2%

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

      2. div-inv 36.2%

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

      3. unsub-neg 36.2%

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

   4. Applied egg-rr 36.2%

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

   5. Taylor expanded in x around 0 36.2%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-52} \lor \neg \left(x \leq 2.7 \cdot 10^{-89}\right):\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \end{array} \]

Alternative 19: 43.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.00185:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 4.1 \cdot 10^{+18}:\\ \;\;\;\;x \cdot \left(\frac{-1}{B} + B \cdot 0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.00185)
   (/ -1.0 (sin B))
   (if (<= F 4.1e+18)
     (* x (+ (/ -1.0 B) (* B 0.3333333333333333)))
     (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.00185) {
		tmp = -1.0 / sin(B);
	} else if (F <= 4.1e+18) {
		tmp = x * ((-1.0 / B) + (B * 0.3333333333333333));
	} 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 <= (-0.00185d0)) then
        tmp = (-1.0d0) / sin(b)
    else if (f <= 4.1d+18) then
        tmp = x * (((-1.0d0) / b) + (b * 0.3333333333333333d0))
    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 <= -0.00185) {
		tmp = -1.0 / Math.sin(B);
	} else if (F <= 4.1e+18) {
		tmp = x * ((-1.0 / B) + (B * 0.3333333333333333));
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -0.00185:
		tmp = -1.0 / math.sin(B)
	elif F <= 4.1e+18:
		tmp = x * ((-1.0 / B) + (B * 0.3333333333333333))
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.00185)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 4.1e+18)
		tmp = Float64(x * Float64(Float64(-1.0 / B) + Float64(B * 0.3333333333333333)));
	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 <= -0.00185)
		tmp = -1.0 / sin(B);
	elseif (F <= 4.1e+18)
		tmp = x * ((-1.0 / B) + (B * 0.3333333333333333));
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.00185], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.1e+18], N[(x * N[(N[(-1.0 / B), $MachinePrecision] + N[(B * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.0018500000000000001

   1. Initial program 69.3%

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

   2. Taylor expanded in F around -inf 98.0%

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

      1. +-commutative 98.0%

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

      2. div-inv 98.1%

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

      3. unsub-neg 98.1%

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

   4. Applied egg-rr 98.1%

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

   5. Taylor expanded in x around 0 54.3%

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

   if -0.0018500000000000001 < F < 4.1e18

   1. Initial program 98.6%

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

   2. Taylor expanded in F around -inf 38.9%

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

   3. Taylor expanded in B around 0 18.6%

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

   4. Taylor expanded in x around inf 32.2%

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

      1. associate-*r* 32.2%

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

      2. neg-mul-1 32.2%

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

      3. +-commutative 32.2%

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

      4. *-commutative 32.2%

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

      5. distribute-rgt-in 32.2%

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

      6. distribute-rgt-neg-in 32.2%

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

      7. distribute-lft-neg-in 32.2%

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

      8. distribute-rgt-neg-in 32.2%

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

      9. distribute-lft-neg-in 32.2%

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

      10. distribute-rgt-out 32.2%

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

      11. distribute-neg-frac 32.2%

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

      12. metadata-eval 32.2%

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

      13. distribute-rgt-neg-in 32.2%

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

      14. metadata-eval 32.2%

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

   6. Simplified 32.2%

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

   if 4.1e18 < F

   1. Initial program 51.7%

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

   2. Taylor expanded in F around inf 99.8%

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

   3. Taylor expanded in B around 0 58.5%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.00185:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 4.1 \cdot 10^{+18}:\\ \;\;\;\;x \cdot \left(\frac{-1}{B} + B \cdot 0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 20: 43.1% accurate, 21.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.1 \cdot 10^{-24}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333 - 0.16666666666666666\right) + \frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 4.1 \cdot 10^{+18}:\\ \;\;\;\;x \cdot \left(\frac{-1}{B} + B \cdot 0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.1e-24)
   (+ (* B (- (* x 0.3333333333333333) 0.16666666666666666)) (/ (- -1.0 x) B))
   (if (<= F 4.1e+18)
     (* x (+ (/ -1.0 B) (* B 0.3333333333333333)))
     (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.1e-24) {
		tmp = (B * ((x * 0.3333333333333333) - 0.16666666666666666)) + ((-1.0 - x) / B);
	} else if (F <= 4.1e+18) {
		tmp = x * ((-1.0 / B) + (B * 0.3333333333333333));
	} 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 <= (-1.1d-24)) then
        tmp = (b * ((x * 0.3333333333333333d0) - 0.16666666666666666d0)) + (((-1.0d0) - x) / b)
    else if (f <= 4.1d+18) then
        tmp = x * (((-1.0d0) / b) + (b * 0.3333333333333333d0))
    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 <= -1.1e-24) {
		tmp = (B * ((x * 0.3333333333333333) - 0.16666666666666666)) + ((-1.0 - x) / B);
	} else if (F <= 4.1e+18) {
		tmp = x * ((-1.0 / B) + (B * 0.3333333333333333));
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.1e-24:
		tmp = (B * ((x * 0.3333333333333333) - 0.16666666666666666)) + ((-1.0 - x) / B)
	elif F <= 4.1e+18:
		tmp = x * ((-1.0 / B) + (B * 0.3333333333333333))
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.1e-24)
		tmp = Float64(Float64(B * Float64(Float64(x * 0.3333333333333333) - 0.16666666666666666)) + Float64(Float64(-1.0 - x) / B));
	elseif (F <= 4.1e+18)
		tmp = Float64(x * Float64(Float64(-1.0 / B) + Float64(B * 0.3333333333333333)));
	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 <= -1.1e-24)
		tmp = (B * ((x * 0.3333333333333333) - 0.16666666666666666)) + ((-1.0 - x) / B);
	elseif (F <= 4.1e+18)
		tmp = x * ((-1.0 / B) + (B * 0.3333333333333333));
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.1e-24], N[(N[(B * N[(N[(x * 0.3333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.1e+18], N[(x * N[(N[(-1.0 / B), $MachinePrecision] + N[(B * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.10000000000000001e-24

   1. Initial program 70.3%

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

   2. Taylor expanded in F around -inf 96.0%

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

   3. Taylor expanded in B around 0 50.7%

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

   if -1.10000000000000001e-24 < F < 4.1e18

   1. Initial program 98.5%

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

   2. Taylor expanded in F around -inf 38.9%

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

   3. Taylor expanded in B around 0 18.1%

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

   4. Taylor expanded in x around inf 32.1%

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

      1. associate-*r* 32.1%

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

      2. neg-mul-1 32.1%

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

      3. +-commutative 32.1%

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

      4. *-commutative 32.1%

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

      5. distribute-rgt-in 32.1%

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

      6. distribute-rgt-neg-in 32.1%

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

      7. distribute-lft-neg-in 32.1%

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

      8. distribute-rgt-neg-in 32.1%

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

      9. distribute-lft-neg-in 32.1%

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

      10. distribute-rgt-out 32.1%

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

      11. distribute-neg-frac 32.1%

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

      12. metadata-eval 32.1%

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

      13. distribute-rgt-neg-in 32.1%

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

      14. metadata-eval 32.1%

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

   6. Simplified 32.1%

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

   if 4.1e18 < F

   1. Initial program 51.7%

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

   2. Taylor expanded in F around inf 99.8%

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

   3. Taylor expanded in B around 0 58.5%

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

4. Final simplification 44.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.1 \cdot 10^{-24}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333 - 0.16666666666666666\right) + \frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 4.1 \cdot 10^{+18}:\\ \;\;\;\;x \cdot \left(\frac{-1}{B} + B \cdot 0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 21: 43.0% accurate, 24.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -7.2 \cdot 10^{-45}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 4.1 \cdot 10^{+18}:\\ \;\;\;\;x \cdot \left(\frac{-1}{B} + B \cdot 0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -7.2e-45)
   (/ (- -1.0 x) B)
   (if (<= F 4.1e+18)
     (* x (+ (/ -1.0 B) (* B 0.3333333333333333)))
     (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -7.2e-45) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 4.1e+18) {
		tmp = x * ((-1.0 / B) + (B * 0.3333333333333333));
	} 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 <= (-7.2d-45)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 4.1d+18) then
        tmp = x * (((-1.0d0) / b) + (b * 0.3333333333333333d0))
    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 <= -7.2e-45) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 4.1e+18) {
		tmp = x * ((-1.0 / B) + (B * 0.3333333333333333));
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -7.2e-45:
		tmp = (-1.0 - x) / B
	elif F <= 4.1e+18:
		tmp = x * ((-1.0 / B) + (B * 0.3333333333333333))
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -7.2e-45)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 4.1e+18)
		tmp = Float64(x * Float64(Float64(-1.0 / B) + Float64(B * 0.3333333333333333)));
	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 <= -7.2e-45)
		tmp = (-1.0 - x) / B;
	elseif (F <= 4.1e+18)
		tmp = x * ((-1.0 / B) + (B * 0.3333333333333333));
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -7.2e-45], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 4.1e+18], N[(x * N[(N[(-1.0 / B), $MachinePrecision] + N[(B * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -7.20000000000000001e-45

   1. Initial program 71.3%

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

   2. Taylor expanded in F around -inf 94.6%

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

   3. Taylor expanded in B around 0 48.9%

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

      1. associate-*r/ 48.9%

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

      2. mul-1-neg 48.9%

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

   5. Simplified 48.9%

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

   if -7.20000000000000001e-45 < F < 4.1e18

   1. Initial program 98.5%

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

   2. Taylor expanded in F around -inf 38.5%

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

   3. Taylor expanded in B around 0 18.5%

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

   4. Taylor expanded in x around inf 33.0%

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

      1. associate-*r* 33.0%

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

      2. neg-mul-1 33.0%

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

      3. +-commutative 33.0%

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

      4. *-commutative 33.0%

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

      5. distribute-rgt-in 33.0%

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

      6. distribute-rgt-neg-in 33.0%

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

      7. distribute-lft-neg-in 33.0%

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

      8. distribute-rgt-neg-in 33.0%

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

      9. distribute-lft-neg-in 33.0%

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

      10. distribute-rgt-out 33.0%

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

      11. distribute-neg-frac 33.0%

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

      12. metadata-eval 33.0%

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

      13. distribute-rgt-neg-in 33.0%

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

      14. metadata-eval 33.0%

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

   6. Simplified 33.0%

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

   if 4.1e18 < F

   1. Initial program 51.7%

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

   2. Taylor expanded in F around inf 99.8%

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

   3. Taylor expanded in B around 0 58.5%

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

4. Final simplification 44.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7.2 \cdot 10^{-45}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 4.1 \cdot 10^{+18}:\\ \;\;\;\;x \cdot \left(\frac{-1}{B} + B \cdot 0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 22: 37.0% accurate, 35.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.0054:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq 3.6 \cdot 10^{-35}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.0054)
   (/ -1.0 B)
   (if (<= F 3.6e-35) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.0054) {
		tmp = -1.0 / B;
	} else if (F <= 3.6e-35) {
		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 <= (-0.0054d0)) then
        tmp = (-1.0d0) / b
    else if (f <= 3.6d-35) 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 <= -0.0054) {
		tmp = -1.0 / B;
	} else if (F <= 3.6e-35) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -0.0054:
		tmp = -1.0 / B
	elif F <= 3.6e-35:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.0054)
		tmp = Float64(-1.0 / B);
	elseif (F <= 3.6e-35)
		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 <= -0.0054)
		tmp = -1.0 / B;
	elseif (F <= 3.6e-35)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.0054], N[(-1.0 / B), $MachinePrecision], If[LessEqual[F, 3.6e-35], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.0054000000000000003

   1. Initial program 69.3%

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

   2. Taylor expanded in F around -inf 98.0%

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

   3. Taylor expanded in B around 0 50.9%

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

      1. associate-*r/ 50.9%

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

      2. mul-1-neg 50.9%

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

   5. Simplified 50.9%

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

   6. Taylor expanded in x around 0 32.2%

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

   if -0.0054000000000000003 < F < 3.60000000000000019e-35

   1. Initial program 99.4%

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

   2. Taylor expanded in F around -inf 37.6%

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

   3. Taylor expanded in B around 0 17.7%

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

      1. associate-*r/ 17.7%

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

      2. mul-1-neg 17.7%

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

   5. Simplified 17.7%

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

   6. Taylor expanded in x around inf 31.8%

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

      1. associate-*r/ 31.8%

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

      2. neg-mul-1 31.8%

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

   8. Simplified 31.8%

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

   if 3.60000000000000019e-35 < F

   1. Initial program 55.3%

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

   2. Taylor expanded in F around inf 96.1%

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

   3. Taylor expanded in B around 0 55.5%

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

4. Final simplification 37.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.0054:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq 3.6 \cdot 10^{-35}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 23: 43.6% accurate, 35.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.15 \cdot 10^{-24}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 3.4 \cdot 10^{-36}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.15e-24)
   (/ (- -1.0 x) B)
   (if (<= F 3.4e-36) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.15e-24) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 3.4e-36) {
		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 <= (-1.15d-24)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 3.4d-36) 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 <= -1.15e-24) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 3.4e-36) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.15e-24:
		tmp = (-1.0 - x) / B
	elif F <= 3.4e-36:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.15e-24)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 3.4e-36)
		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 <= -1.15e-24)
		tmp = (-1.0 - x) / B;
	elseif (F <= 3.4e-36)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.15e-24], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 3.4e-36], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.1500000000000001e-24

   1. Initial program 70.3%

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

   2. Taylor expanded in F around -inf 96.0%

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

   3. Taylor expanded in B around 0 50.4%

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

      1. associate-*r/ 50.4%

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

      2. mul-1-neg 50.4%

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

   5. Simplified 50.4%

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

   if -1.1500000000000001e-24 < F < 3.4000000000000003e-36

   1. Initial program 99.4%

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

   2. Taylor expanded in F around -inf 37.6%

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

   3. Taylor expanded in B around 0 17.1%

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

      1. associate-*r/ 17.1%

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

      2. mul-1-neg 17.1%

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

   5. Simplified 17.1%

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

   6. Taylor expanded in x around inf 31.7%

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

      1. associate-*r/ 31.7%

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

      2. neg-mul-1 31.7%

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

   8. Simplified 31.7%

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

   if 3.4000000000000003e-36 < F

   1. Initial program 55.3%

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

   2. Taylor expanded in F around inf 96.1%

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

   3. Taylor expanded in B around 0 55.5%

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

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.15 \cdot 10^{-24}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 3.4 \cdot 10^{-36}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 24: 30.4% accurate, 39.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-68} \lor \neg \left(x \leq 1.4 \cdot 10^{-72}\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (or (<= x -4.4e-68) (not (<= x 1.4e-72))) (/ (- x) B) (/ -1.0 B)))

C

double code(double F, double B, double x) {
	double tmp;
	if ((x <= -4.4e-68) || !(x <= 1.4e-72)) {
		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 <= (-4.4d-68)) .or. (.not. (x <= 1.4d-72))) 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 <= -4.4e-68) || !(x <= 1.4e-72)) {
		tmp = -x / B;
	} else {
		tmp = -1.0 / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if (x <= -4.4e-68) or not (x <= 1.4e-72):
		tmp = -x / B
	else:
		tmp = -1.0 / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if ((x <= -4.4e-68) || !(x <= 1.4e-72))
		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 <= -4.4e-68) || ~((x <= 1.4e-72)))
		tmp = -x / B;
	else
		tmp = -1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[Or[LessEqual[x, -4.4e-68], N[Not[LessEqual[x, 1.4e-72]], $MachinePrecision]], N[((-x) / B), $MachinePrecision], N[(-1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.40000000000000005e-68 or 1.3999999999999999e-72 < x

   1. Initial program 87.0%

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

   2. Taylor expanded in F around -inf 79.1%

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

   3. Taylor expanded in B around 0 37.5%

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

      1. associate-*r/ 37.5%

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

      2. mul-1-neg 37.5%

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

   5. Simplified 37.5%

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

   6. Taylor expanded in x around inf 43.0%

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

      1. associate-*r/ 43.0%

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

      2. neg-mul-1 43.0%

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

   8. Simplified 43.0%

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

   if -4.40000000000000005e-68 < x < 1.3999999999999999e-72

   1. Initial program 67.4%

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

   2. Taylor expanded in F around -inf 36.2%

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

   3. Taylor expanded in B around 0 22.3%

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

      1. associate-*r/ 22.3%

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

      2. mul-1-neg 22.3%

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

   5. Simplified 22.3%

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

   6. Taylor expanded in x around 0 22.3%

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

4. Final simplification 33.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-68} \lor \neg \left(x \leq 1.4 \cdot 10^{-72}\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B}\\ \end{array} \]

Alternative 25: 10.1% 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 78.0%

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

2. Taylor expanded in F around -inf 59.3%

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

3. Taylor expanded in B around 0 30.5%

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

   1. associate-*r/ 30.5%

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

   2. mul-1-neg 30.5%

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

5. Simplified 30.5%

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

6. Taylor expanded in x around 0 13.4%

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

7. Final simplification 13.4%

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

Reproduce

herbie shell --seed 2023321 
(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))))))