VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.0% → 98.1%

Time: 21.7s

Alternatives: 25

Speedup: 1.4×

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: 76.0% 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: 98.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -1.05 \cdot 10^{+16}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.75 \cdot 10^{-34}:\\ \;\;\;\;t\_0 + \frac{F \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B)))))
   (if (<= F -1.05e+16)
     (- (/ -1.0 (sin B)) (/ x (tan B)))
     (if (<= F 1.75e-34)
       (+ t_0 (/ (* F (pow (fma 2.0 x (fma F F 2.0)) -0.5)) (sin B)))
       (+ t_0 (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double tmp;
	if (F <= -1.05e+16) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 1.75e-34) {
		tmp = t_0 + ((F * pow(fma(2.0, x, fma(F, F, 2.0)), -0.5)) / sin(B));
	} else {
		tmp = t_0 + (1.0 / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	tmp = 0.0
	if (F <= -1.05e+16)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 1.75e-34)
		tmp = Float64(t_0 + Float64(Float64(F * (fma(2.0, x, fma(F, F, 2.0)) ^ -0.5)) / sin(B)));
	else
		tmp = Float64(t_0 + Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.05e+16], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.75e-34], N[(t$95$0 + N[(N[(F * N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.05e16

   1. Initial program 55.8%

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

   3. Taylor expanded in F around -inf 99.8%

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

      1. +-commutative 99.8%

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

      2. unsub-neg 99.8%

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

      3. un-div-inv 99.9%

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

   5. Applied egg-rr 99.9%

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

   if -1.05e16 < F < 1.75e-34

   1. Initial program 99.6%

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

      1. associate-*l/ 99.7%

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

      2. +-commutative 99.7%

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

      3. fma-define 99.7%

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

      4. fma-define 99.7%

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

      5. metadata-eval 99.7%

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

      6. metadata-eval 99.7%

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

   4. Applied egg-rr 99.7%

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

   if 1.75e-34 < F

   1. Initial program 59.9%

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

   3. Taylor expanded in F around inf 99.6%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.05 \cdot 10^{+16}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.75 \cdot 10^{-34}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -7 \cdot 10^{+52}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 50000:\\ \;\;\;\;F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -7e+52)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 50000.0)
       (- (* F (/ (pow (fma x 2.0 (fma F F 2.0)) -0.5) (sin B))) t_0)
       (+ (* x (/ -1.0 (tan B))) (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -7e+52) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 50000.0) {
		tmp = (F * (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) / sin(B))) - t_0;
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -7e+52)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 50000.0)
		tmp = Float64(Float64(F * Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) / sin(B))) - t_0);
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -7e+52], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 50000.0], N[(N[(F * N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -7e52

   1. Initial program 47.4%

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

   3. Taylor expanded in F around -inf 99.9%

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

      1. +-commutative 99.9%

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

      2. unsub-neg 99.9%

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

      3. un-div-inv 99.9%

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

   5. Applied egg-rr 99.9%

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

   if -7e52 < F < 5e4

   1. Initial program 98.8%

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

   3. Simplified 99.6%

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

   if 5e4 < F

   1. Initial program 58.3%

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

   3. Taylor expanded in F around inf 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7 \cdot 10^{+52}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 50000:\\ \;\;\;\;F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -6 \cdot 10^{+15}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 50000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - x \cdot \frac{\cos B}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -6e+15)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 50000.0)
     (-
      (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5))
      (* x (/ (cos B) (sin B))))
     (+ (* x (/ -1.0 (tan B))) (/ 1.0 (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -6e+15) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 50000.0) {
		tmp = ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x * (cos(B) / sin(B)));
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-6d+15)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 50000.0d0) then
        tmp = ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0))) - (x * (cos(b) / sin(b)))
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -6e+15) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 50000.0) {
		tmp = ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x * (Math.cos(B) / Math.sin(B)));
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -6e+15:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 50000.0:
		tmp = ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x * (math.cos(B) / math.sin(B)))
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -6e+15)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 50000.0)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)) - Float64(x * Float64(cos(B) / sin(B))));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -6e+15)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 50000.0)
		tmp = ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5)) - (x * (cos(B) / sin(B)));
	else
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -6e+15], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 50000.0], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - N[(x * N[(N[Cos[B], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -6e15

   1. Initial program 55.8%

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

   3. Taylor expanded in F around -inf 99.8%

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

      1. +-commutative 99.8%

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

      2. unsub-neg 99.8%

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

      3. un-div-inv 99.9%

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

   5. Applied egg-rr 99.9%

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

   if -6e15 < F < 5e4

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 99.6%

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

      1. associate-/l* 99.5%

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

   5. Simplified 99.5%

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

   if 5e4 < F

   1. Initial program 58.3%

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

   3. Taylor expanded in F around inf 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6 \cdot 10^{+15}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 50000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - x \cdot \frac{\cos B}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -1 \cdot 10^{+16}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.75 \cdot 10^{-34}:\\ \;\;\;\;t\_0 + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B)))))
   (if (<= F -1e+16)
     (- (/ -1.0 (sin B)) (/ x (tan B)))
     (if (<= F 1.75e-34)
       (+ t_0 (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)))
       (+ t_0 (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double tmp;
	if (F <= -1e+16) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 1.75e-34) {
		tmp = t_0 + ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5));
	} else {
		tmp = t_0 + (1.0 / sin(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * ((-1.0d0) / tan(b))
    if (f <= (-1d+16)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 1.75d-34) then
        tmp = t_0 + ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)))
    else
        tmp = t_0 + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / Math.tan(B));
	double tmp;
	if (F <= -1e+16) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 1.75e-34) {
		tmp = t_0 + ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5));
	} else {
		tmp = t_0 + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * (-1.0 / math.tan(B))
	tmp = 0
	if F <= -1e+16:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 1.75e-34:
		tmp = t_0 + ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5))
	else:
		tmp = t_0 + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	tmp = 0.0
	if (F <= -1e+16)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 1.75e-34)
		tmp = Float64(t_0 + Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)));
	else
		tmp = Float64(t_0 + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x * (-1.0 / tan(B));
	tmp = 0.0;
	if (F <= -1e+16)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 1.75e-34)
		tmp = t_0 + ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5));
	else
		tmp = t_0 + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1e+16], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.75e-34], N[(t$95$0 + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1e16

   1. Initial program 55.8%

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

   3. Taylor expanded in F around -inf 99.8%

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

      1. +-commutative 99.8%

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

      2. unsub-neg 99.8%

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

      3. un-div-inv 99.9%

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

   5. Applied egg-rr 99.9%

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

   if -1e16 < F < 1.75e-34

   1. Initial program 99.6%

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

   if 1.75e-34 < F

   1. Initial program 59.9%

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

   3. Taylor expanded in F around inf 99.6%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{+16}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.75 \cdot 10^{-34}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.42:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1.5:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.42)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.5)
       (- (* F (/ (sqrt 0.5) (sin B))) t_0)
       (+ (* x (/ -1.0 (tan B))) (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.42) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.5) {
		tmp = (F * (sqrt(0.5) / sin(B))) - t_0;
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.42d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.5d0) then
        tmp = (f * (sqrt(0.5d0) / sin(b))) - t_0
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.42) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.5) {
		tmp = (F * (Math.sqrt(0.5) / Math.sin(B))) - t_0;
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.42:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.5:
		tmp = (F * (math.sqrt(0.5) / math.sin(B))) - t_0
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.42)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.5)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / sin(B))) - t_0);
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.42)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.5)
		tmp = (F * (sqrt(0.5) / sin(B))) - t_0;
	else
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.42], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.5], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.4199999999999999

   1. Initial program 57.2%

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

   3. Taylor expanded in F around -inf 99.8%

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

      1. +-commutative 99.8%

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

      2. unsub-neg 99.8%

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

      3. un-div-inv 99.9%

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

   5. Applied egg-rr 99.9%

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

   if -1.4199999999999999 < F < 1.5

   1. Initial program 99.5%

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

   3. Taylor expanded in F around 0 98.2%

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

   4. Taylor expanded in x around 0 98.3%

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

      1. +-commutative 98.3%

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

      2. unsub-neg 98.3%

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{0.5}}{\sin B} - x \cdot \frac{1}{\tan B}} \]

      3. associate-/l* 98.2%

         \[\leadsto \color{blue}{F \cdot \frac{\sqrt{0.5}}{\sin B}} - x \cdot \frac{1}{\tan B} \]

      4. un-div-inv 98.3%

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

   6. Applied egg-rr 98.3%

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

   if 1.5 < F

   1. Initial program 58.3%

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

   3. Taylor expanded in F around inf 99.7%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.42:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.5:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.42:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1.5:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.42)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.5)
       (- (/ (* F (sqrt 0.5)) (sin B)) t_0)
       (+ (* x (/ -1.0 (tan B))) (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.42) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.5) {
		tmp = ((F * sqrt(0.5)) / sin(B)) - t_0;
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.42d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.5d0) then
        tmp = ((f * sqrt(0.5d0)) / sin(b)) - t_0
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.42) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.5) {
		tmp = ((F * Math.sqrt(0.5)) / Math.sin(B)) - t_0;
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.42:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.5:
		tmp = ((F * math.sqrt(0.5)) / math.sin(B)) - t_0
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.42)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.5)
		tmp = Float64(Float64(Float64(F * sqrt(0.5)) / sin(B)) - t_0);
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.42)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.5)
		tmp = ((F * sqrt(0.5)) / sin(B)) - t_0;
	else
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.42], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.5], N[(N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.4199999999999999

   1. Initial program 57.2%

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

   3. Taylor expanded in F around -inf 99.8%

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

      1. +-commutative 99.8%

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

      2. unsub-neg 99.8%

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

      3. un-div-inv 99.9%

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

   5. Applied egg-rr 99.9%

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

   if -1.4199999999999999 < F < 1.5

   1. Initial program 99.5%

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

   3. Taylor expanded in F around 0 98.2%

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

   4. Taylor expanded in x around 0 98.3%

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

      1. div-inv 98.4%

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

   6. Applied egg-rr 98.4%

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

   if 1.5 < F

   1. Initial program 58.3%

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

   3. Taylor expanded in F around inf 99.7%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.42:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.5:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ t_1 := {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ t_2 := \frac{1}{\sin B}\\ \mathbf{if}\;F \leq -5.7 \cdot 10^{+15}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -2.35 \cdot 10^{-66}:\\ \;\;\;\;\frac{F}{\sin B} \cdot t\_1 - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.2 \cdot 10^{-87}:\\ \;\;\;\;t\_0 + \frac{F \cdot \sqrt{0.5}}{B}\\ \mathbf{elif}\;F \leq 1.75 \cdot 10^{-34}:\\ \;\;\;\;t\_1 \cdot \left(F \cdot t\_2\right) - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B))))
        (t_1 (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5))
        (t_2 (/ 1.0 (sin B))))
   (if (<= F -5.7e+15)
     (- (/ -1.0 (sin B)) (/ x (tan B)))
     (if (<= F -2.35e-66)
       (- (* (/ F (sin B)) t_1) (/ x B))
       (if (<= F 1.2e-87)
         (+ t_0 (/ (* F (sqrt 0.5)) B))
         (if (<= F 1.75e-34) (- (* t_1 (* F t_2)) (/ x B)) (+ t_0 t_2)))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double t_1 = pow(((2.0 + (F * F)) + (x * 2.0)), -0.5);
	double t_2 = 1.0 / sin(B);
	double tmp;
	if (F <= -5.7e+15) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= -2.35e-66) {
		tmp = ((F / sin(B)) * t_1) - (x / B);
	} else if (F <= 1.2e-87) {
		tmp = t_0 + ((F * sqrt(0.5)) / B);
	} else if (F <= 1.75e-34) {
		tmp = (t_1 * (F * t_2)) - (x / B);
	} else {
		tmp = t_0 + t_2;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = x * ((-1.0d0) / tan(b))
    t_1 = ((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)
    t_2 = 1.0d0 / sin(b)
    if (f <= (-5.7d+15)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= (-2.35d-66)) then
        tmp = ((f / sin(b)) * t_1) - (x / b)
    else if (f <= 1.2d-87) then
        tmp = t_0 + ((f * sqrt(0.5d0)) / b)
    else if (f <= 1.75d-34) then
        tmp = (t_1 * (f * t_2)) - (x / b)
    else
        tmp = t_0 + t_2
    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(((2.0 + (F * F)) + (x * 2.0)), -0.5);
	double t_2 = 1.0 / Math.sin(B);
	double tmp;
	if (F <= -5.7e+15) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= -2.35e-66) {
		tmp = ((F / Math.sin(B)) * t_1) - (x / B);
	} else if (F <= 1.2e-87) {
		tmp = t_0 + ((F * Math.sqrt(0.5)) / B);
	} else if (F <= 1.75e-34) {
		tmp = (t_1 * (F * t_2)) - (x / B);
	} else {
		tmp = t_0 + t_2;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * (-1.0 / math.tan(B))
	t_1 = math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)
	t_2 = 1.0 / math.sin(B)
	tmp = 0
	if F <= -5.7e+15:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= -2.35e-66:
		tmp = ((F / math.sin(B)) * t_1) - (x / B)
	elif F <= 1.2e-87:
		tmp = t_0 + ((F * math.sqrt(0.5)) / B)
	elif F <= 1.75e-34:
		tmp = (t_1 * (F * t_2)) - (x / B)
	else:
		tmp = t_0 + t_2
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	t_1 = Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5
	t_2 = Float64(1.0 / sin(B))
	tmp = 0.0
	if (F <= -5.7e+15)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= -2.35e-66)
		tmp = Float64(Float64(Float64(F / sin(B)) * t_1) - Float64(x / B));
	elseif (F <= 1.2e-87)
		tmp = Float64(t_0 + Float64(Float64(F * sqrt(0.5)) / B));
	elseif (F <= 1.75e-34)
		tmp = Float64(Float64(t_1 * Float64(F * t_2)) - Float64(x / B));
	else
		tmp = Float64(t_0 + t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x * (-1.0 / tan(B));
	t_1 = ((2.0 + (F * F)) + (x * 2.0)) ^ -0.5;
	t_2 = 1.0 / sin(B);
	tmp = 0.0;
	if (F <= -5.7e+15)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= -2.35e-66)
		tmp = ((F / sin(B)) * t_1) - (x / B);
	elseif (F <= 1.2e-87)
		tmp = t_0 + ((F * sqrt(0.5)) / B);
	elseif (F <= 1.75e-34)
		tmp = (t_1 * (F * t_2)) - (x / B);
	else
		tmp = t_0 + t_2;
	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[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -5.7e+15], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -2.35e-66], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.2e-87], N[(t$95$0 + N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.75e-34], N[(N[(t$95$1 * N[(F * t$95$2), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + t$95$2), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -5.7e15

   1. Initial program 55.8%

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

   3. Taylor expanded in F around -inf 99.8%

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

      1. +-commutative 99.8%

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

      2. unsub-neg 99.8%

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

      3. un-div-inv 99.9%

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

   5. Applied egg-rr 99.9%

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

   if -5.7e15 < F < -2.35e-66

   1. Initial program 99.6%

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

   3. Taylor expanded in B around 0 94.0%

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

   if -2.35e-66 < F < 1.2e-87

   1. Initial program 99.6%

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

   3. Taylor expanded in F around 0 99.6%

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

   4. Taylor expanded in x around 0 99.7%

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

   5. Taylor expanded in B around 0 93.2%

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

   if 1.2e-87 < F < 1.75e-34

   1. Initial program 99.5%

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

   3. Taylor expanded in B around 0 81.3%

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

      1. div-inv 81.4%

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

   5. Applied egg-rr 81.4%

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

   if 1.75e-34 < F

   1. Initial program 59.9%

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

   3. Taylor expanded in F around inf 99.6%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5.7 \cdot 10^{+15}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -2.35 \cdot 10^{-66}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.2 \cdot 10^{-87}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F \cdot \sqrt{0.5}}{B}\\ \mathbf{elif}\;F \leq 1.75 \cdot 10^{-34}:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \left(F \cdot \frac{1}{\sin B}\right) - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{\sin B} - \frac{x}{B}\\ t_1 := F \cdot \frac{\sqrt{0.5}}{\sin B}\\ \mathbf{if}\;F \leq -2.05 \cdot 10^{+156}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq -4.5 \cdot 10^{+81}:\\ \;\;\;\;F \cdot \frac{-1}{F \cdot B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -0.305:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq -1.02 \cdot 10^{-66}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;F \leq 1.3 \cdot 10^{-87}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{-30}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ -1.0 (sin B)) (/ x B))) (t_1 (* F (/ (sqrt 0.5) (sin B)))))
   (if (<= F -2.05e+156)
     t_0
     (if (<= F -4.5e+81)
       (- (* F (/ -1.0 (* F B))) (/ x (tan B)))
       (if (<= F -0.305)
         t_0
         (if (<= F -1.02e-66)
           t_1
           (if (<= F 1.3e-87)
             (/ x (- (tan B)))
             (if (<= F 2.3e-30) t_1 (- (/ 1.0 (sin B)) (/ x B))))))))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 / sin(B)) - (x / B);
	double t_1 = F * (sqrt(0.5) / sin(B));
	double tmp;
	if (F <= -2.05e+156) {
		tmp = t_0;
	} else if (F <= -4.5e+81) {
		tmp = (F * (-1.0 / (F * B))) - (x / tan(B));
	} else if (F <= -0.305) {
		tmp = t_0;
	} else if (F <= -1.02e-66) {
		tmp = t_1;
	} else if (F <= 1.3e-87) {
		tmp = x / -tan(B);
	} else if (F <= 2.3e-30) {
		tmp = t_1;
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((-1.0d0) / sin(b)) - (x / b)
    t_1 = f * (sqrt(0.5d0) / sin(b))
    if (f <= (-2.05d+156)) then
        tmp = t_0
    else if (f <= (-4.5d+81)) then
        tmp = (f * ((-1.0d0) / (f * b))) - (x / tan(b))
    else if (f <= (-0.305d0)) then
        tmp = t_0
    else if (f <= (-1.02d-66)) then
        tmp = t_1
    else if (f <= 1.3d-87) then
        tmp = x / -tan(b)
    else if (f <= 2.3d-30) then
        tmp = t_1
    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 / Math.sin(B)) - (x / B);
	double t_1 = F * (Math.sqrt(0.5) / Math.sin(B));
	double tmp;
	if (F <= -2.05e+156) {
		tmp = t_0;
	} else if (F <= -4.5e+81) {
		tmp = (F * (-1.0 / (F * B))) - (x / Math.tan(B));
	} else if (F <= -0.305) {
		tmp = t_0;
	} else if (F <= -1.02e-66) {
		tmp = t_1;
	} else if (F <= 1.3e-87) {
		tmp = x / -Math.tan(B);
	} else if (F <= 2.3e-30) {
		tmp = t_1;
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (-1.0 / math.sin(B)) - (x / B)
	t_1 = F * (math.sqrt(0.5) / math.sin(B))
	tmp = 0
	if F <= -2.05e+156:
		tmp = t_0
	elif F <= -4.5e+81:
		tmp = (F * (-1.0 / (F * B))) - (x / math.tan(B))
	elif F <= -0.305:
		tmp = t_0
	elif F <= -1.02e-66:
		tmp = t_1
	elif F <= 1.3e-87:
		tmp = x / -math.tan(B)
	elif F <= 2.3e-30:
		tmp = t_1
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 / sin(B)) - Float64(x / B))
	t_1 = Float64(F * Float64(sqrt(0.5) / sin(B)))
	tmp = 0.0
	if (F <= -2.05e+156)
		tmp = t_0;
	elseif (F <= -4.5e+81)
		tmp = Float64(Float64(F * Float64(-1.0 / Float64(F * B))) - Float64(x / tan(B)));
	elseif (F <= -0.305)
		tmp = t_0;
	elseif (F <= -1.02e-66)
		tmp = t_1;
	elseif (F <= 1.3e-87)
		tmp = Float64(x / Float64(-tan(B)));
	elseif (F <= 2.3e-30)
		tmp = t_1;
	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 / sin(B)) - (x / B);
	t_1 = F * (sqrt(0.5) / sin(B));
	tmp = 0.0;
	if (F <= -2.05e+156)
		tmp = t_0;
	elseif (F <= -4.5e+81)
		tmp = (F * (-1.0 / (F * B))) - (x / tan(B));
	elseif (F <= -0.305)
		tmp = t_0;
	elseif (F <= -1.02e-66)
		tmp = t_1;
	elseif (F <= 1.3e-87)
		tmp = x / -tan(B);
	elseif (F <= 2.3e-30)
		tmp = t_1;
	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 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.05e+156], t$95$0, If[LessEqual[F, -4.5e+81], N[(N[(F * N[(-1.0 / N[(F * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -0.305], t$95$0, If[LessEqual[F, -1.02e-66], t$95$1, If[LessEqual[F, 1.3e-87], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 2.3e-30], t$95$1, N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -2.0500000000000001e156 or -4.50000000000000017e81 < F < -0.304999999999999993

   1. Initial program 51.0%

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

   3. Taylor expanded in B around 0 35.9%

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

   4. Taylor expanded in F around -inf 83.2%

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

      1. distribute-lft-in 83.2%

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

      2. mul-1-neg 83.2%

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

      3. unsub-neg 83.2%

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

      4. associate-*r/ 83.2%

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

      5. metadata-eval 83.2%

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

   6. Simplified 83.2%

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

   if -2.0500000000000001e156 < F < -4.50000000000000017e81

   1. Initial program 73.3%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in F around -inf 99.8%

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

   6. Taylor expanded in B around 0 94.2%

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

      1. *-commutative 94.2%

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

   8. Simplified 94.2%

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

   if -0.304999999999999993 < F < -1.01999999999999996e-66 or 1.30000000000000001e-87 < F < 2.29999999999999984e-30

   1. Initial program 99.5%

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

   3. Taylor expanded in F around 0 94.4%

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

   4. Taylor expanded in x around 0 94.5%

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

   5. Taylor expanded in x around 0 70.2%

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

      1. associate-*r/ 70.1%

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

   7. Simplified 70.1%

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

   if -1.01999999999999996e-66 < F < 1.30000000000000001e-87

   1. Initial program 99.6%

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

   3. Taylor expanded in F around -inf 41.4%

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

   4. Taylor expanded in x around inf 81.2%

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

      1. mul-1-neg 81.2%

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

      2. associate-/l* 81.2%

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

      3. distribute-rgt-neg-in 81.2%

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

      4. distribute-neg-frac2 81.2%

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

   6. Simplified 81.2%

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

      1. add-sqr-sqrt 45.0%

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

      2. sqrt-unprod 40.7%

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

      3. sqr-neg 40.7%

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

      4. sqrt-unprod 0.7%

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

      5. add-sqr-sqrt 2.0%

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

      6. clear-num 2.0%

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

      7. tan-quot 2.0%

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

      8. add-sqr-sqrt 1.4%

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

      9. sqrt-unprod 26.4%

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

      10. sqr-neg 26.4%

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

      11. sqrt-unprod 32.2%

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

      12. add-sqr-sqrt 81.3%

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

      13. neg-sub0 81.3%

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

      14. un-div-inv 81.4%

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

   8. Applied egg-rr 81.4%

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

      1. neg-sub0 81.4%

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

      2. distribute-frac-neg 81.4%

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

   10. Simplified 81.4%

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

   if 2.29999999999999984e-30 < F

   1. Initial program 59.4%

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

   3. Taylor expanded in B around 0 44.2%

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

   4. Taylor expanded in F around inf 84.3%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.05 \cdot 10^{+156}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -4.5 \cdot 10^{+81}:\\ \;\;\;\;F \cdot \frac{-1}{F \cdot B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -0.305:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.02 \cdot 10^{-66}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 1.3 \cdot 10^{-87}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{-30}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{\sin B} - \frac{x}{B}\\ t_1 := \frac{F \cdot \sqrt{0.5}}{\sin B}\\ \mathbf{if}\;F \leq -1.95 \cdot 10^{+156}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq -6.5 \cdot 10^{+81}:\\ \;\;\;\;F \cdot \frac{-1}{F \cdot B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -0.27:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq -9.2 \cdot 10^{-66}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;F \leq 1.25 \cdot 10^{-87}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{-30}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ -1.0 (sin B)) (/ x B))) (t_1 (/ (* F (sqrt 0.5)) (sin B))))
   (if (<= F -1.95e+156)
     t_0
     (if (<= F -6.5e+81)
       (- (* F (/ -1.0 (* F B))) (/ x (tan B)))
       (if (<= F -0.27)
         t_0
         (if (<= F -9.2e-66)
           t_1
           (if (<= F 1.25e-87)
             (/ x (- (tan B)))
             (if (<= F 2.3e-30) t_1 (- (/ 1.0 (sin B)) (/ x B))))))))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 / sin(B)) - (x / B);
	double t_1 = (F * sqrt(0.5)) / sin(B);
	double tmp;
	if (F <= -1.95e+156) {
		tmp = t_0;
	} else if (F <= -6.5e+81) {
		tmp = (F * (-1.0 / (F * B))) - (x / tan(B));
	} else if (F <= -0.27) {
		tmp = t_0;
	} else if (F <= -9.2e-66) {
		tmp = t_1;
	} else if (F <= 1.25e-87) {
		tmp = x / -tan(B);
	} else if (F <= 2.3e-30) {
		tmp = t_1;
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((-1.0d0) / sin(b)) - (x / b)
    t_1 = (f * sqrt(0.5d0)) / sin(b)
    if (f <= (-1.95d+156)) then
        tmp = t_0
    else if (f <= (-6.5d+81)) then
        tmp = (f * ((-1.0d0) / (f * b))) - (x / tan(b))
    else if (f <= (-0.27d0)) then
        tmp = t_0
    else if (f <= (-9.2d-66)) then
        tmp = t_1
    else if (f <= 1.25d-87) then
        tmp = x / -tan(b)
    else if (f <= 2.3d-30) then
        tmp = t_1
    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 / Math.sin(B)) - (x / B);
	double t_1 = (F * Math.sqrt(0.5)) / Math.sin(B);
	double tmp;
	if (F <= -1.95e+156) {
		tmp = t_0;
	} else if (F <= -6.5e+81) {
		tmp = (F * (-1.0 / (F * B))) - (x / Math.tan(B));
	} else if (F <= -0.27) {
		tmp = t_0;
	} else if (F <= -9.2e-66) {
		tmp = t_1;
	} else if (F <= 1.25e-87) {
		tmp = x / -Math.tan(B);
	} else if (F <= 2.3e-30) {
		tmp = t_1;
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (-1.0 / math.sin(B)) - (x / B)
	t_1 = (F * math.sqrt(0.5)) / math.sin(B)
	tmp = 0
	if F <= -1.95e+156:
		tmp = t_0
	elif F <= -6.5e+81:
		tmp = (F * (-1.0 / (F * B))) - (x / math.tan(B))
	elif F <= -0.27:
		tmp = t_0
	elif F <= -9.2e-66:
		tmp = t_1
	elif F <= 1.25e-87:
		tmp = x / -math.tan(B)
	elif F <= 2.3e-30:
		tmp = t_1
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 / sin(B)) - Float64(x / B))
	t_1 = Float64(Float64(F * sqrt(0.5)) / sin(B))
	tmp = 0.0
	if (F <= -1.95e+156)
		tmp = t_0;
	elseif (F <= -6.5e+81)
		tmp = Float64(Float64(F * Float64(-1.0 / Float64(F * B))) - Float64(x / tan(B)));
	elseif (F <= -0.27)
		tmp = t_0;
	elseif (F <= -9.2e-66)
		tmp = t_1;
	elseif (F <= 1.25e-87)
		tmp = Float64(x / Float64(-tan(B)));
	elseif (F <= 2.3e-30)
		tmp = t_1;
	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 / sin(B)) - (x / B);
	t_1 = (F * sqrt(0.5)) / sin(B);
	tmp = 0.0;
	if (F <= -1.95e+156)
		tmp = t_0;
	elseif (F <= -6.5e+81)
		tmp = (F * (-1.0 / (F * B))) - (x / tan(B));
	elseif (F <= -0.27)
		tmp = t_0;
	elseif (F <= -9.2e-66)
		tmp = t_1;
	elseif (F <= 1.25e-87)
		tmp = x / -tan(B);
	elseif (F <= 2.3e-30)
		tmp = t_1;
	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 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.95e+156], t$95$0, If[LessEqual[F, -6.5e+81], N[(N[(F * N[(-1.0 / N[(F * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -0.27], t$95$0, If[LessEqual[F, -9.2e-66], t$95$1, If[LessEqual[F, 1.25e-87], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 2.3e-30], t$95$1, N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -1.9499999999999999e156 or -6.4999999999999996e81 < F < -0.27000000000000002

   1. Initial program 51.0%

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

   3. Taylor expanded in B around 0 35.9%

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

   4. Taylor expanded in F around -inf 83.2%

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

      1. distribute-lft-in 83.2%

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

      2. mul-1-neg 83.2%

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

      3. unsub-neg 83.2%

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

      4. associate-*r/ 83.2%

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

      5. metadata-eval 83.2%

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

   6. Simplified 83.2%

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

   if -1.9499999999999999e156 < F < -6.4999999999999996e81

   1. Initial program 73.3%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in F around -inf 99.8%

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

   6. Taylor expanded in B around 0 94.2%

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

      1. *-commutative 94.2%

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

   8. Simplified 94.2%

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

   if -0.27000000000000002 < F < -9.19999999999999967e-66 or 1.25000000000000011e-87 < F < 2.29999999999999984e-30

   1. Initial program 99.5%

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

   3. Taylor expanded in F around 0 94.4%

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

   4. Taylor expanded in x around 0 94.5%

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

   5. Taylor expanded in x around 0 70.2%

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

   if -9.19999999999999967e-66 < F < 1.25000000000000011e-87

   1. Initial program 99.6%

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

   3. Taylor expanded in F around -inf 41.4%

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

   4. Taylor expanded in x around inf 81.2%

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

      1. mul-1-neg 81.2%

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

      2. associate-/l* 81.2%

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

      3. distribute-rgt-neg-in 81.2%

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

      4. distribute-neg-frac2 81.2%

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

   6. Simplified 81.2%

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

      1. add-sqr-sqrt 45.0%

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

      2. sqrt-unprod 40.7%

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

      3. sqr-neg 40.7%

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

      4. sqrt-unprod 0.7%

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

      5. add-sqr-sqrt 2.0%

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

      6. clear-num 2.0%

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

      7. tan-quot 2.0%

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

      8. add-sqr-sqrt 1.4%

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

      9. sqrt-unprod 26.4%

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

      10. sqr-neg 26.4%

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

      11. sqrt-unprod 32.2%

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

      12. add-sqr-sqrt 81.3%

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

      13. neg-sub0 81.3%

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

      14. un-div-inv 81.4%

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

   8. Applied egg-rr 81.4%

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

      1. neg-sub0 81.4%

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

      2. distribute-frac-neg 81.4%

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

   10. Simplified 81.4%

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

   if 2.29999999999999984e-30 < F

   1. Initial program 59.4%

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

   3. Taylor expanded in B around 0 44.2%

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

   4. Taylor expanded in F around inf 84.3%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.95 \cdot 10^{+156}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -6.5 \cdot 10^{+81}:\\ \;\;\;\;F \cdot \frac{-1}{F \cdot B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -0.27:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -9.2 \cdot 10^{-66}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 1.25 \cdot 10^{-87}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{-30}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 82.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F}{\sin B} \cdot \sqrt{0.5} - \frac{x}{B}\\ \mathbf{if}\;F \leq -0.55:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -6 \cdot 10^{-139}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 1.1 \cdot 10^{-121}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{-30}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (* (/ F (sin B)) (sqrt 0.5)) (/ x B))))
   (if (<= F -0.55)
     (- (/ -1.0 (sin B)) (/ x (tan B)))
     (if (<= F -6e-139)
       t_0
       (if (<= F 1.1e-121)
         (/ x (- (tan B)))
         (if (<= F 2.3e-30) t_0 (- (/ 1.0 (sin B)) (/ x 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 <= -0.55) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= -6e-139) {
		tmp = t_0;
	} else if (F <= 1.1e-121) {
		tmp = x / -tan(B);
	} else if (F <= 2.3e-30) {
		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 = ((f / sin(b)) * sqrt(0.5d0)) - (x / b)
    if (f <= (-0.55d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= (-6d-139)) then
        tmp = t_0
    else if (f <= 1.1d-121) then
        tmp = x / -tan(b)
    else if (f <= 2.3d-30) 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 = ((F / Math.sin(B)) * Math.sqrt(0.5)) - (x / B);
	double tmp;
	if (F <= -0.55) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= -6e-139) {
		tmp = t_0;
	} else if (F <= 1.1e-121) {
		tmp = x / -Math.tan(B);
	} else if (F <= 2.3e-30) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / 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 <= -0.55:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= -6e-139:
		tmp = t_0
	elif F <= 1.1e-121:
		tmp = x / -math.tan(B)
	elif F <= 2.3e-30:
		tmp = t_0
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(Float64(F / sin(B)) * sqrt(0.5)) - Float64(x / B))
	tmp = 0.0
	if (F <= -0.55)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= -6e-139)
		tmp = t_0;
	elseif (F <= 1.1e-121)
		tmp = Float64(x / Float64(-tan(B)));
	elseif (F <= 2.3e-30)
		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 = ((F / sin(B)) * sqrt(0.5)) - (x / B);
	tmp = 0.0;
	if (F <= -0.55)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= -6e-139)
		tmp = t_0;
	elseif (F <= 1.1e-121)
		tmp = x / -tan(B);
	elseif (F <= 2.3e-30)
		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[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.55], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -6e-139], t$95$0, If[LessEqual[F, 1.1e-121], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 2.3e-30], t$95$0, N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.55000000000000004

   1. Initial program 57.2%

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

   3. Taylor expanded in F around -inf 99.8%

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

      1. +-commutative 99.8%

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

      2. unsub-neg 99.8%

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

      3. un-div-inv 99.9%

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

   5. Applied egg-rr 99.9%

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

   if -0.55000000000000004 < F < -5.9999999999999998e-139 or 1.10000000000000011e-121 < F < 2.29999999999999984e-30

   1. Initial program 99.6%

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

   3. Taylor expanded in B around 0 77.2%

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

   4. Taylor expanded in F around 0 74.1%

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

   5. Taylor expanded in x around 0 74.0%

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

   if -5.9999999999999998e-139 < F < 1.10000000000000011e-121

   1. Initial program 99.5%

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

   3. Taylor expanded in F around -inf 44.1%

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

   4. Taylor expanded in x around inf 87.3%

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

      1. mul-1-neg 87.3%

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

      2. associate-/l* 87.2%

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

      3. distribute-rgt-neg-in 87.2%

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

      4. distribute-neg-frac2 87.2%

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

   6. Simplified 87.2%

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

      1. add-sqr-sqrt 43.2%

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

      2. sqrt-unprod 38.9%

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

      3. sqr-neg 38.9%

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

      4. sqrt-unprod 0.7%

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

      5. add-sqr-sqrt 1.8%

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

      6. clear-num 1.8%

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

      7. tan-quot 1.8%

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

      8. add-sqr-sqrt 1.3%

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

      9. sqrt-unprod 26.0%

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

      10. sqr-neg 26.0%

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

      11. sqrt-unprod 31.3%

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

      12. add-sqr-sqrt 87.3%

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

      13. neg-sub0 87.3%

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

      14. un-div-inv 87.4%

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

   8. Applied egg-rr 87.4%

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

      1. neg-sub0 87.4%

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

      2. distribute-frac-neg 87.4%

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

   10. Simplified 87.4%

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

   if 2.29999999999999984e-30 < F

   1. Initial program 59.4%

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

   3. Taylor expanded in B around 0 44.2%

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

   4. Taylor expanded in F around inf 84.3%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.55:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -6 \cdot 10^{-139}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.1 \cdot 10^{-121}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{-30}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 88.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F}{\sin B} \cdot \sqrt{0.5} - \frac{x}{B}\\ \mathbf{if}\;F \leq -0.29:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -5.8 \cdot 10^{-143}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 10^{-123}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 1.75 \cdot 10^{-34}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (* (/ F (sin B)) (sqrt 0.5)) (/ x B))))
   (if (<= F -0.29)
     (- (/ -1.0 (sin B)) (/ x (tan B)))
     (if (<= F -5.8e-143)
       t_0
       (if (<= F 1e-123)
         (/ x (- (tan B)))
         (if (<= F 1.75e-34)
           t_0
           (+ (* x (/ -1.0 (tan B))) (/ 1.0 (sin B)))))))))

C

double code(double F, double B, double x) {
	double t_0 = ((F / sin(B)) * sqrt(0.5)) - (x / B);
	double tmp;
	if (F <= -0.29) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= -5.8e-143) {
		tmp = t_0;
	} else if (F <= 1e-123) {
		tmp = x / -tan(B);
	} else if (F <= 1.75e-34) {
		tmp = t_0;
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((f / sin(b)) * sqrt(0.5d0)) - (x / b)
    if (f <= (-0.29d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= (-5.8d-143)) then
        tmp = t_0
    else if (f <= 1d-123) then
        tmp = x / -tan(b)
    else if (f <= 1.75d-34) then
        tmp = t_0
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = ((F / Math.sin(B)) * Math.sqrt(0.5)) - (x / B);
	double tmp;
	if (F <= -0.29) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= -5.8e-143) {
		tmp = t_0;
	} else if (F <= 1e-123) {
		tmp = x / -Math.tan(B);
	} else if (F <= 1.75e-34) {
		tmp = t_0;
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = ((F / math.sin(B)) * math.sqrt(0.5)) - (x / B)
	tmp = 0
	if F <= -0.29:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= -5.8e-143:
		tmp = t_0
	elif F <= 1e-123:
		tmp = x / -math.tan(B)
	elif F <= 1.75e-34:
		tmp = t_0
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(Float64(F / sin(B)) * sqrt(0.5)) - Float64(x / B))
	tmp = 0.0
	if (F <= -0.29)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= -5.8e-143)
		tmp = t_0;
	elseif (F <= 1e-123)
		tmp = Float64(x / Float64(-tan(B)));
	elseif (F <= 1.75e-34)
		tmp = t_0;
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = ((F / sin(B)) * sqrt(0.5)) - (x / B);
	tmp = 0.0;
	if (F <= -0.29)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= -5.8e-143)
		tmp = t_0;
	elseif (F <= 1e-123)
		tmp = x / -tan(B);
	elseif (F <= 1.75e-34)
		tmp = t_0;
	else
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.29], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -5.8e-143], t$95$0, If[LessEqual[F, 1e-123], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 1.75e-34], t$95$0, N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.28999999999999998

   1. Initial program 57.2%

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

   3. Taylor expanded in F around -inf 99.8%

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

      1. +-commutative 99.8%

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

      2. unsub-neg 99.8%

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

      3. un-div-inv 99.9%

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

   5. Applied egg-rr 99.9%

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

   if -0.28999999999999998 < F < -5.8000000000000002e-143 or 1.0000000000000001e-123 < F < 1.75e-34

   1. Initial program 99.6%

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

   3. Taylor expanded in B around 0 78.7%

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

   4. Taylor expanded in F around 0 75.5%

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

   5. Taylor expanded in x around 0 75.5%

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

   if -5.8000000000000002e-143 < F < 1.0000000000000001e-123

   1. Initial program 99.5%

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

   3. Taylor expanded in F around -inf 44.1%

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

   4. Taylor expanded in x around inf 87.3%

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

      1. mul-1-neg 87.3%

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

      2. associate-/l* 87.2%

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

      3. distribute-rgt-neg-in 87.2%

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

      4. distribute-neg-frac2 87.2%

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

   6. Simplified 87.2%

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

      1. add-sqr-sqrt 43.2%

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

      2. sqrt-unprod 38.9%

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

      3. sqr-neg 38.9%

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

      4. sqrt-unprod 0.7%

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

      5. add-sqr-sqrt 1.8%

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

      6. clear-num 1.8%

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

      7. tan-quot 1.8%

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

      8. add-sqr-sqrt 1.3%

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

      9. sqrt-unprod 26.0%

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

      10. sqr-neg 26.0%

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

      11. sqrt-unprod 31.3%

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

      12. add-sqr-sqrt 87.3%

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

      13. neg-sub0 87.3%

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

      14. un-div-inv 87.4%

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

   8. Applied egg-rr 87.4%

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

      1. neg-sub0 87.4%

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

      2. distribute-frac-neg 87.4%

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

   10. Simplified 87.4%

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

   if 1.75e-34 < F

   1. Initial program 59.9%

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

   3. Taylor expanded in F around inf 99.6%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.29:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -5.8 \cdot 10^{-143}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 10^{-123}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 1.75 \cdot 10^{-34}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 91.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F}{\sin B} \cdot \sqrt{0.5} - \frac{x}{B}\\ t_1 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -0.3:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -5.4 \cdot 10^{-67}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 1.1 \cdot 10^{-87}:\\ \;\;\;\;t\_1 + \frac{F \cdot \sqrt{0.5}}{B}\\ \mathbf{elif}\;F \leq 1.75 \cdot 10^{-34}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (* (/ F (sin B)) (sqrt 0.5)) (/ x B)))
        (t_1 (* x (/ -1.0 (tan B)))))
   (if (<= F -0.3)
     (- (/ -1.0 (sin B)) (/ x (tan B)))
     (if (<= F -5.4e-67)
       t_0
       (if (<= F 1.1e-87)
         (+ t_1 (/ (* F (sqrt 0.5)) B))
         (if (<= F 1.75e-34) t_0 (+ t_1 (/ 1.0 (sin B)))))))))

C

double code(double F, double B, double x) {
	double t_0 = ((F / sin(B)) * sqrt(0.5)) - (x / B);
	double t_1 = x * (-1.0 / tan(B));
	double tmp;
	if (F <= -0.3) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= -5.4e-67) {
		tmp = t_0;
	} else if (F <= 1.1e-87) {
		tmp = t_1 + ((F * sqrt(0.5)) / B);
	} else if (F <= 1.75e-34) {
		tmp = t_0;
	} else {
		tmp = t_1 + (1.0 / sin(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((f / sin(b)) * sqrt(0.5d0)) - (x / b)
    t_1 = x * ((-1.0d0) / tan(b))
    if (f <= (-0.3d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= (-5.4d-67)) then
        tmp = t_0
    else if (f <= 1.1d-87) then
        tmp = t_1 + ((f * sqrt(0.5d0)) / b)
    else if (f <= 1.75d-34) then
        tmp = t_0
    else
        tmp = t_1 + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = ((F / Math.sin(B)) * Math.sqrt(0.5)) - (x / B);
	double t_1 = x * (-1.0 / Math.tan(B));
	double tmp;
	if (F <= -0.3) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= -5.4e-67) {
		tmp = t_0;
	} else if (F <= 1.1e-87) {
		tmp = t_1 + ((F * Math.sqrt(0.5)) / B);
	} else if (F <= 1.75e-34) {
		tmp = t_0;
	} else {
		tmp = t_1 + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = ((F / math.sin(B)) * math.sqrt(0.5)) - (x / B)
	t_1 = x * (-1.0 / math.tan(B))
	tmp = 0
	if F <= -0.3:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= -5.4e-67:
		tmp = t_0
	elif F <= 1.1e-87:
		tmp = t_1 + ((F * math.sqrt(0.5)) / B)
	elif F <= 1.75e-34:
		tmp = t_0
	else:
		tmp = t_1 + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(Float64(F / sin(B)) * sqrt(0.5)) - Float64(x / B))
	t_1 = Float64(x * Float64(-1.0 / tan(B)))
	tmp = 0.0
	if (F <= -0.3)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= -5.4e-67)
		tmp = t_0;
	elseif (F <= 1.1e-87)
		tmp = Float64(t_1 + Float64(Float64(F * sqrt(0.5)) / B));
	elseif (F <= 1.75e-34)
		tmp = t_0;
	else
		tmp = Float64(t_1 + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = ((F / sin(B)) * sqrt(0.5)) - (x / B);
	t_1 = x * (-1.0 / tan(B));
	tmp = 0.0;
	if (F <= -0.3)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= -5.4e-67)
		tmp = t_0;
	elseif (F <= 1.1e-87)
		tmp = t_1 + ((F * sqrt(0.5)) / B);
	elseif (F <= 1.75e-34)
		tmp = t_0;
	else
		tmp = t_1 + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.3], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -5.4e-67], t$95$0, If[LessEqual[F, 1.1e-87], N[(t$95$1 + N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.75e-34], t$95$0, N[(t$95$1 + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.299999999999999989

   1. Initial program 57.2%

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

   3. Taylor expanded in F around -inf 99.8%

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

      1. +-commutative 99.8%

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

      2. unsub-neg 99.8%

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

      3. un-div-inv 99.9%

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

   5. Applied egg-rr 99.9%

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

   if -0.299999999999999989 < F < -5.40000000000000032e-67 or 1.09999999999999994e-87 < F < 1.75e-34

   1. Initial program 99.5%

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

   3. Taylor expanded in B around 0 87.2%

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

   4. Taylor expanded in F around 0 81.9%

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

   5. Taylor expanded in x around 0 81.9%

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

   if -5.40000000000000032e-67 < F < 1.09999999999999994e-87

   1. Initial program 99.6%

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

   3. Taylor expanded in F around 0 99.6%

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

   4. Taylor expanded in x around 0 99.7%

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

   5. Taylor expanded in B around 0 93.2%

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

   if 1.75e-34 < F

   1. Initial program 59.9%

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

   3. Taylor expanded in F around inf 99.6%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.3:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -5.4 \cdot 10^{-67}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.1 \cdot 10^{-87}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F \cdot \sqrt{0.5}}{B}\\ \mathbf{elif}\;F \leq 1.75 \cdot 10^{-34}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 91.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ t_1 := \frac{F}{\sin B}\\ \mathbf{if}\;F \leq -5.7 \cdot 10^{+15}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -7.8 \cdot 10^{-65}:\\ \;\;\;\;t\_1 \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 10^{-87}:\\ \;\;\;\;t\_0 + \frac{F \cdot \sqrt{0.5}}{B}\\ \mathbf{elif}\;F \leq 1.75 \cdot 10^{-34}:\\ \;\;\;\;t\_1 \cdot \sqrt{0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B)))) (t_1 (/ F (sin B))))
   (if (<= F -5.7e+15)
     (- (/ -1.0 (sin B)) (/ x (tan B)))
     (if (<= F -7.8e-65)
       (- (* t_1 (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)) (/ x B))
       (if (<= F 1e-87)
         (+ t_0 (/ (* F (sqrt 0.5)) B))
         (if (<= F 1.75e-34)
           (- (* t_1 (sqrt 0.5)) (/ x B))
           (+ t_0 (/ 1.0 (sin B)))))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double t_1 = F / sin(B);
	double tmp;
	if (F <= -5.7e+15) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= -7.8e-65) {
		tmp = (t_1 * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B);
	} else if (F <= 1e-87) {
		tmp = t_0 + ((F * sqrt(0.5)) / B);
	} else if (F <= 1.75e-34) {
		tmp = (t_1 * sqrt(0.5)) - (x / B);
	} else {
		tmp = t_0 + (1.0 / sin(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x * ((-1.0d0) / tan(b))
    t_1 = f / sin(b)
    if (f <= (-5.7d+15)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= (-7.8d-65)) then
        tmp = (t_1 * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0))) - (x / b)
    else if (f <= 1d-87) then
        tmp = t_0 + ((f * sqrt(0.5d0)) / b)
    else if (f <= 1.75d-34) then
        tmp = (t_1 * sqrt(0.5d0)) - (x / b)
    else
        tmp = t_0 + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / Math.tan(B));
	double t_1 = F / Math.sin(B);
	double tmp;
	if (F <= -5.7e+15) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= -7.8e-65) {
		tmp = (t_1 * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B);
	} else if (F <= 1e-87) {
		tmp = t_0 + ((F * Math.sqrt(0.5)) / B);
	} else if (F <= 1.75e-34) {
		tmp = (t_1 * Math.sqrt(0.5)) - (x / B);
	} else {
		tmp = t_0 + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * (-1.0 / math.tan(B))
	t_1 = F / math.sin(B)
	tmp = 0
	if F <= -5.7e+15:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= -7.8e-65:
		tmp = (t_1 * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B)
	elif F <= 1e-87:
		tmp = t_0 + ((F * math.sqrt(0.5)) / B)
	elif F <= 1.75e-34:
		tmp = (t_1 * math.sqrt(0.5)) - (x / B)
	else:
		tmp = t_0 + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	t_1 = Float64(F / sin(B))
	tmp = 0.0
	if (F <= -5.7e+15)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= -7.8e-65)
		tmp = Float64(Float64(t_1 * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)) - Float64(x / B));
	elseif (F <= 1e-87)
		tmp = Float64(t_0 + Float64(Float64(F * sqrt(0.5)) / B));
	elseif (F <= 1.75e-34)
		tmp = Float64(Float64(t_1 * sqrt(0.5)) - Float64(x / B));
	else
		tmp = Float64(t_0 + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x * (-1.0 / tan(B));
	t_1 = F / sin(B);
	tmp = 0.0;
	if (F <= -5.7e+15)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= -7.8e-65)
		tmp = (t_1 * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5)) - (x / B);
	elseif (F <= 1e-87)
		tmp = t_0 + ((F * sqrt(0.5)) / B);
	elseif (F <= 1.75e-34)
		tmp = (t_1 * sqrt(0.5)) - (x / B);
	else
		tmp = t_0 + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -5.7e+15], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -7.8e-65], N[(N[(t$95$1 * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1e-87], N[(t$95$0 + N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.75e-34], N[(N[(t$95$1 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -5.7e15

   1. Initial program 55.8%

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

   3. Taylor expanded in F around -inf 99.8%

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

      1. +-commutative 99.8%

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

      2. unsub-neg 99.8%

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

      3. un-div-inv 99.9%

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

   5. Applied egg-rr 99.9%

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

   if -5.7e15 < F < -7.8000000000000007e-65

   1. Initial program 99.6%

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

   3. Taylor expanded in B around 0 94.0%

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

   if -7.8000000000000007e-65 < F < 1.00000000000000002e-87

   1. Initial program 99.6%

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

   3. Taylor expanded in F around 0 99.6%

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

   4. Taylor expanded in x around 0 99.7%

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

   5. Taylor expanded in B around 0 93.2%

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

   if 1.00000000000000002e-87 < F < 1.75e-34

   1. Initial program 99.5%

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

   3. Taylor expanded in B around 0 81.3%

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

   4. Taylor expanded in F around 0 81.3%

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

   5. Taylor expanded in x around 0 81.3%

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

   if 1.75e-34 < F

   1. Initial program 59.9%

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

   3. Taylor expanded in F around inf 99.6%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5.7 \cdot 10^{+15}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -7.8 \cdot 10^{-65}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 10^{-87}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F \cdot \sqrt{0.5}}{B}\\ \mathbf{elif}\;F \leq 1.75 \cdot 10^{-34}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 79.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F \cdot \sqrt{0.5}}{\sin B}\\ \mathbf{if}\;F \leq -0.27:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -3 \cdot 10^{-65}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 1.15 \cdot 10^{-87}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{-30}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (* F (sqrt 0.5)) (sin B))))
   (if (<= F -0.27)
     (- (/ -1.0 (sin B)) (/ x (tan B)))
     (if (<= F -3e-65)
       t_0
       (if (<= F 1.15e-87)
         (/ x (- (tan B)))
         (if (<= F 2.3e-30) t_0 (- (/ 1.0 (sin B)) (/ x B))))))))

C

double code(double F, double B, double x) {
	double t_0 = (F * sqrt(0.5)) / sin(B);
	double tmp;
	if (F <= -0.27) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= -3e-65) {
		tmp = t_0;
	} else if (F <= 1.15e-87) {
		tmp = x / -tan(B);
	} else if (F <= 2.3e-30) {
		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 = (f * sqrt(0.5d0)) / sin(b)
    if (f <= (-0.27d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= (-3d-65)) then
        tmp = t_0
    else if (f <= 1.15d-87) then
        tmp = x / -tan(b)
    else if (f <= 2.3d-30) 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 = (F * Math.sqrt(0.5)) / Math.sin(B);
	double tmp;
	if (F <= -0.27) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= -3e-65) {
		tmp = t_0;
	} else if (F <= 1.15e-87) {
		tmp = x / -Math.tan(B);
	} else if (F <= 2.3e-30) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (F * math.sqrt(0.5)) / math.sin(B)
	tmp = 0
	if F <= -0.27:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= -3e-65:
		tmp = t_0
	elif F <= 1.15e-87:
		tmp = x / -math.tan(B)
	elif F <= 2.3e-30:
		tmp = t_0
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(F * sqrt(0.5)) / sin(B))
	tmp = 0.0
	if (F <= -0.27)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= -3e-65)
		tmp = t_0;
	elseif (F <= 1.15e-87)
		tmp = Float64(x / Float64(-tan(B)));
	elseif (F <= 2.3e-30)
		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 = (F * sqrt(0.5)) / sin(B);
	tmp = 0.0;
	if (F <= -0.27)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= -3e-65)
		tmp = t_0;
	elseif (F <= 1.15e-87)
		tmp = x / -tan(B);
	elseif (F <= 2.3e-30)
		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[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.27], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -3e-65], t$95$0, If[LessEqual[F, 1.15e-87], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 2.3e-30], t$95$0, N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.27000000000000002

   1. Initial program 57.2%

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

   3. Taylor expanded in F around -inf 99.8%

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

      1. +-commutative 99.8%

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

      2. unsub-neg 99.8%

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

      3. un-div-inv 99.9%

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

   5. Applied egg-rr 99.9%

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

   if -0.27000000000000002 < F < -2.99999999999999998e-65 or 1.1500000000000001e-87 < F < 2.29999999999999984e-30

   1. Initial program 99.5%

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

   3. Taylor expanded in F around 0 94.4%

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

   4. Taylor expanded in x around 0 94.5%

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

   5. Taylor expanded in x around 0 70.2%

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

   if -2.99999999999999998e-65 < F < 1.1500000000000001e-87

   1. Initial program 99.6%

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

   3. Taylor expanded in F around -inf 41.4%

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

   4. Taylor expanded in x around inf 81.2%

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

      1. mul-1-neg 81.2%

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

      2. associate-/l* 81.2%

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

      3. distribute-rgt-neg-in 81.2%

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

      4. distribute-neg-frac2 81.2%

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

   6. Simplified 81.2%

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

      1. add-sqr-sqrt 45.0%

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

      2. sqrt-unprod 40.7%

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

      3. sqr-neg 40.7%

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

      4. sqrt-unprod 0.7%

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

      5. add-sqr-sqrt 2.0%

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

      6. clear-num 2.0%

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

      7. tan-quot 2.0%

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

      8. add-sqr-sqrt 1.4%

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

      9. sqrt-unprod 26.4%

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

      10. sqr-neg 26.4%

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

      11. sqrt-unprod 32.2%

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

      12. add-sqr-sqrt 81.3%

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

      13. neg-sub0 81.3%

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

      14. un-div-inv 81.4%

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

   8. Applied egg-rr 81.4%

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

      1. neg-sub0 81.4%

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

      2. distribute-frac-neg 81.4%

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

   10. Simplified 81.4%

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

   if 2.29999999999999984e-30 < F

   1. Initial program 59.4%

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

   3. Taylor expanded in B around 0 44.2%

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

   4. Taylor expanded in F around inf 84.3%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.27:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -3 \cdot 10^{-65}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 1.15 \cdot 10^{-87}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{-30}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 71.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{if}\;F \leq -1.95 \cdot 10^{+156}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq -6.8 \cdot 10^{+81}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} - \frac{1}{B}\\ \mathbf{elif}\;F \leq -1.1 \cdot 10^{+27}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 7200000000:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ -1.0 (sin B)) (/ x B))))
   (if (<= F -1.95e+156)
     t_0
     (if (<= F -6.8e+81)
       (- (* x (/ -1.0 (tan B))) (/ 1.0 B))
       (if (<= F -1.1e+27)
         t_0
         (if (<= F 7200000000.0)
           (/ x (- (tan B)))
           (- (/ 1.0 (sin B)) (/ x B))))))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 / sin(B)) - (x / B);
	double tmp;
	if (F <= -1.95e+156) {
		tmp = t_0;
	} else if (F <= -6.8e+81) {
		tmp = (x * (-1.0 / tan(B))) - (1.0 / B);
	} else if (F <= -1.1e+27) {
		tmp = t_0;
	} else if (F <= 7200000000.0) {
		tmp = x / -tan(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) / sin(b)) - (x / b)
    if (f <= (-1.95d+156)) then
        tmp = t_0
    else if (f <= (-6.8d+81)) then
        tmp = (x * ((-1.0d0) / tan(b))) - (1.0d0 / b)
    else if (f <= (-1.1d+27)) then
        tmp = t_0
    else if (f <= 7200000000.0d0) then
        tmp = x / -tan(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 / Math.sin(B)) - (x / B);
	double tmp;
	if (F <= -1.95e+156) {
		tmp = t_0;
	} else if (F <= -6.8e+81) {
		tmp = (x * (-1.0 / Math.tan(B))) - (1.0 / B);
	} else if (F <= -1.1e+27) {
		tmp = t_0;
	} else if (F <= 7200000000.0) {
		tmp = x / -Math.tan(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (-1.0 / math.sin(B)) - (x / B)
	tmp = 0
	if F <= -1.95e+156:
		tmp = t_0
	elif F <= -6.8e+81:
		tmp = (x * (-1.0 / math.tan(B))) - (1.0 / B)
	elif F <= -1.1e+27:
		tmp = t_0
	elif F <= 7200000000.0:
		tmp = x / -math.tan(B)
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 / sin(B)) - Float64(x / B))
	tmp = 0.0
	if (F <= -1.95e+156)
		tmp = t_0;
	elseif (F <= -6.8e+81)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) - Float64(1.0 / B));
	elseif (F <= -1.1e+27)
		tmp = t_0;
	elseif (F <= 7200000000.0)
		tmp = Float64(x / Float64(-tan(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 / sin(B)) - (x / B);
	tmp = 0.0;
	if (F <= -1.95e+156)
		tmp = t_0;
	elseif (F <= -6.8e+81)
		tmp = (x * (-1.0 / tan(B))) - (1.0 / B);
	elseif (F <= -1.1e+27)
		tmp = t_0;
	elseif (F <= 7200000000.0)
		tmp = x / -tan(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 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.95e+156], t$95$0, If[LessEqual[F, -6.8e+81], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.1e+27], t$95$0, If[LessEqual[F, 7200000000.0], N[(x / (-N[Tan[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 < -1.9499999999999999e156 or -6.80000000000000005e81 < F < -1.0999999999999999e27

   1. Initial program 44.0%

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

   3. Taylor expanded in B around 0 35.6%

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

   4. Taylor expanded in F around -inf 89.7%

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

      1. distribute-lft-in 89.7%

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

      2. mul-1-neg 89.7%

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

      3. unsub-neg 89.7%

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

      4. associate-*r/ 89.7%

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

      5. metadata-eval 89.7%

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

   6. Simplified 89.7%

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

   if -1.9499999999999999e156 < F < -6.80000000000000005e81

   1. Initial program 73.3%

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

   3. Taylor expanded in F around -inf 99.8%

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

   4. Taylor expanded in B around 0 94.1%

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

   if -1.0999999999999999e27 < F < 7.2e9

   1. Initial program 99.5%

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

   3. Taylor expanded in F around -inf 39.4%

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

   4. Taylor expanded in x around inf 66.8%

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

      1. mul-1-neg 66.8%

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

      2. associate-/l* 66.8%

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

      3. distribute-rgt-neg-in 66.8%

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

      4. distribute-neg-frac2 66.8%

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

   6. Simplified 66.8%

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

      1. add-sqr-sqrt 36.1%

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

      2. sqrt-unprod 33.8%

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

      3. sqr-neg 33.8%

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

      4. sqrt-unprod 1.0%

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

      5. add-sqr-sqrt 2.2%

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

      6. clear-num 2.2%

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

      7. tan-quot 2.2%

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

      8. add-sqr-sqrt 1.3%

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

      9. sqrt-unprod 20.6%

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

      10. sqr-neg 20.6%

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

      11. sqrt-unprod 26.5%

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

      12. add-sqr-sqrt 66.8%

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

      13. neg-sub0 66.8%

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

      14. un-div-inv 66.9%

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

   8. Applied egg-rr 66.9%

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

      1. neg-sub0 66.9%

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

      2. distribute-frac-neg 66.9%

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

   10. Simplified 66.9%

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

   if 7.2e9 < F

   1. Initial program 57.1%

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

   3. Taylor expanded in B around 0 43.8%

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

   4. Taylor expanded in F around inf 86.1%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.95 \cdot 10^{+156}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -6.8 \cdot 10^{+81}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} - \frac{1}{B}\\ \mathbf{elif}\;F \leq -1.1 \cdot 10^{+27}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 7200000000:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 71.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{if}\;F \leq -2.15 \cdot 10^{+156}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq -7 \cdot 10^{+80}:\\ \;\;\;\;F \cdot \frac{-1}{F \cdot B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.1 \cdot 10^{+27}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 7200000000:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ -1.0 (sin B)) (/ x B))))
   (if (<= F -2.15e+156)
     t_0
     (if (<= F -7e+80)
       (- (* F (/ -1.0 (* F B))) (/ x (tan B)))
       (if (<= F -1.1e+27)
         t_0
         (if (<= F 7200000000.0)
           (/ x (- (tan B)))
           (- (/ 1.0 (sin B)) (/ x B))))))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 / sin(B)) - (x / B);
	double tmp;
	if (F <= -2.15e+156) {
		tmp = t_0;
	} else if (F <= -7e+80) {
		tmp = (F * (-1.0 / (F * B))) - (x / tan(B));
	} else if (F <= -1.1e+27) {
		tmp = t_0;
	} else if (F <= 7200000000.0) {
		tmp = x / -tan(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) / sin(b)) - (x / b)
    if (f <= (-2.15d+156)) then
        tmp = t_0
    else if (f <= (-7d+80)) then
        tmp = (f * ((-1.0d0) / (f * b))) - (x / tan(b))
    else if (f <= (-1.1d+27)) then
        tmp = t_0
    else if (f <= 7200000000.0d0) then
        tmp = x / -tan(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 / Math.sin(B)) - (x / B);
	double tmp;
	if (F <= -2.15e+156) {
		tmp = t_0;
	} else if (F <= -7e+80) {
		tmp = (F * (-1.0 / (F * B))) - (x / Math.tan(B));
	} else if (F <= -1.1e+27) {
		tmp = t_0;
	} else if (F <= 7200000000.0) {
		tmp = x / -Math.tan(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (-1.0 / math.sin(B)) - (x / B)
	tmp = 0
	if F <= -2.15e+156:
		tmp = t_0
	elif F <= -7e+80:
		tmp = (F * (-1.0 / (F * B))) - (x / math.tan(B))
	elif F <= -1.1e+27:
		tmp = t_0
	elif F <= 7200000000.0:
		tmp = x / -math.tan(B)
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 / sin(B)) - Float64(x / B))
	tmp = 0.0
	if (F <= -2.15e+156)
		tmp = t_0;
	elseif (F <= -7e+80)
		tmp = Float64(Float64(F * Float64(-1.0 / Float64(F * B))) - Float64(x / tan(B)));
	elseif (F <= -1.1e+27)
		tmp = t_0;
	elseif (F <= 7200000000.0)
		tmp = Float64(x / Float64(-tan(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 / sin(B)) - (x / B);
	tmp = 0.0;
	if (F <= -2.15e+156)
		tmp = t_0;
	elseif (F <= -7e+80)
		tmp = (F * (-1.0 / (F * B))) - (x / tan(B));
	elseif (F <= -1.1e+27)
		tmp = t_0;
	elseif (F <= 7200000000.0)
		tmp = x / -tan(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 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.15e+156], t$95$0, If[LessEqual[F, -7e+80], N[(N[(F * N[(-1.0 / N[(F * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.1e+27], t$95$0, If[LessEqual[F, 7200000000.0], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.14999999999999993e156 or -6.99999999999999987e80 < F < -1.0999999999999999e27

   1. Initial program 44.0%

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

   3. Taylor expanded in B around 0 35.6%

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

   4. Taylor expanded in F around -inf 89.7%

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

      1. distribute-lft-in 89.7%

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

      2. mul-1-neg 89.7%

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

      3. unsub-neg 89.7%

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

      4. associate-*r/ 89.7%

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

      5. metadata-eval 89.7%

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

   6. Simplified 89.7%

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

   if -2.14999999999999993e156 < F < -6.99999999999999987e80

   1. Initial program 73.3%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in F around -inf 99.8%

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

   6. Taylor expanded in B around 0 94.2%

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

      1. *-commutative 94.2%

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

   8. Simplified 94.2%

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

   if -1.0999999999999999e27 < F < 7.2e9

   1. Initial program 99.5%

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

   3. Taylor expanded in F around -inf 39.4%

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

   4. Taylor expanded in x around inf 66.8%

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

      1. mul-1-neg 66.8%

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

      2. associate-/l* 66.8%

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

      3. distribute-rgt-neg-in 66.8%

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

      4. distribute-neg-frac2 66.8%

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

   6. Simplified 66.8%

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

      1. add-sqr-sqrt 36.1%

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

      2. sqrt-unprod 33.8%

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

      3. sqr-neg 33.8%

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

      4. sqrt-unprod 1.0%

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

      5. add-sqr-sqrt 2.2%

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

      6. clear-num 2.2%

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

      7. tan-quot 2.2%

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

      8. add-sqr-sqrt 1.3%

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

      9. sqrt-unprod 20.6%

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

      10. sqr-neg 20.6%

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

      11. sqrt-unprod 26.5%

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

      12. add-sqr-sqrt 66.8%

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

      13. neg-sub0 66.8%

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

      14. un-div-inv 66.9%

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

   8. Applied egg-rr 66.9%

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

      1. neg-sub0 66.9%

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

      2. distribute-frac-neg 66.9%

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

   10. Simplified 66.9%

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

   if 7.2e9 < F

   1. Initial program 57.1%

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

   3. Taylor expanded in B around 0 43.8%

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

   4. Taylor expanded in F around inf 86.1%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.15 \cdot 10^{+156}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -7 \cdot 10^{+80}:\\ \;\;\;\;F \cdot \frac{-1}{F \cdot B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.1 \cdot 10^{+27}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 7200000000:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 56.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.8 \cdot 10^{+113} \lor \neg \left(F \leq -6.8 \cdot 10^{+81}\right) \land F \leq -1.12 \cdot 10^{+27}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (or (<= F -3.8e+113) (and (not (<= F -6.8e+81)) (<= F -1.12e+27)))
   (/ -1.0 (sin B))
   (/ x (- (tan B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if ((F <= -3.8e+113) || (!(F <= -6.8e+81) && (F <= -1.12e+27))) {
		tmp = -1.0 / sin(B);
	} else {
		tmp = x / -tan(B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((f <= (-3.8d+113)) .or. (.not. (f <= (-6.8d+81))) .and. (f <= (-1.12d+27))) then
        tmp = (-1.0d0) / sin(b)
    else
        tmp = x / -tan(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if ((F <= -3.8e+113) || (!(F <= -6.8e+81) && (F <= -1.12e+27))) {
		tmp = -1.0 / Math.sin(B);
	} else {
		tmp = x / -Math.tan(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if (F <= -3.8e+113) or (not (F <= -6.8e+81) and (F <= -1.12e+27)):
		tmp = -1.0 / math.sin(B)
	else:
		tmp = x / -math.tan(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if ((F <= -3.8e+113) || (!(F <= -6.8e+81) && (F <= -1.12e+27)))
		tmp = Float64(-1.0 / sin(B));
	else
		tmp = Float64(x / Float64(-tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if ((F <= -3.8e+113) || (~((F <= -6.8e+81)) && (F <= -1.12e+27)))
		tmp = -1.0 / sin(B);
	else
		tmp = x / -tan(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[Or[LessEqual[F, -3.8e+113], And[N[Not[LessEqual[F, -6.8e+81]], $MachinePrecision], LessEqual[F, -1.12e+27]]], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -3.8000000000000003e113 or -6.80000000000000005e81 < F < -1.12e27

   1. Initial program 49.5%

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

   3. Taylor expanded in B around 0 39.3%

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

   4. Taylor expanded in F around -inf 88.3%

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

      1. distribute-lft-in 88.3%

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

      2. mul-1-neg 88.3%

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

      3. unsub-neg 88.3%

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

      4. associate-*r/ 88.3%

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

      5. metadata-eval 88.3%

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

   6. Simplified 88.3%

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

   7. Taylor expanded in B around inf 75.8%

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

   if -3.8000000000000003e113 < F < -6.80000000000000005e81 or -1.12e27 < F

   1. Initial program 84.4%

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

   3. Taylor expanded in F around -inf 39.1%

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

   4. Taylor expanded in x around inf 56.9%

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

      1. mul-1-neg 56.9%

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

      2. associate-/l* 56.8%

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

      3. distribute-rgt-neg-in 56.8%

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

      4. distribute-neg-frac2 56.8%

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

   6. Simplified 56.8%

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

      1. add-sqr-sqrt 30.5%

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

      2. sqrt-unprod 28.9%

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

      3. sqr-neg 28.9%

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

      4. sqrt-unprod 0.9%

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

      5. add-sqr-sqrt 2.3%

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

      6. clear-num 2.3%

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

      7. tan-quot 2.3%

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

      8. add-sqr-sqrt 1.2%

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

      9. sqrt-unprod 17.2%

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

      10. sqr-neg 17.2%

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

      11. sqrt-unprod 22.8%

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

      12. add-sqr-sqrt 56.8%

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

      13. neg-sub0 56.8%

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

      14. un-div-inv 56.9%

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

   8. Applied egg-rr 56.9%

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

      1. neg-sub0 56.9%

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

      2. distribute-frac-neg 56.9%

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

   10. Simplified 56.9%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.8 \cdot 10^{+113} \lor \neg \left(F \leq -6.8 \cdot 10^{+81}\right) \land F \leq -1.12 \cdot 10^{+27}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 37.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{\sin B}\\ \mathbf{if}\;F \leq -3.8 \cdot 10^{+152}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq -3.5 \cdot 10^{+81}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq -3.2 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 3.2 \cdot 10^{+46}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ -1.0 (sin B))))
   (if (<= F -3.8e+152)
     t_0
     (if (<= F -3.5e+81)
       (/ (- -1.0 x) B)
       (if (<= F -3.2e-5)
         t_0
         (if (<= F 3.2e+46) (/ x (- B)) (/ (+ x 1.0) B)))))))

C

double code(double F, double B, double x) {
	double t_0 = -1.0 / sin(B);
	double tmp;
	if (F <= -3.8e+152) {
		tmp = t_0;
	} else if (F <= -3.5e+81) {
		tmp = (-1.0 - x) / B;
	} else if (F <= -3.2e-5) {
		tmp = t_0;
	} else if (F <= 3.2e+46) {
		tmp = x / -B;
	} else {
		tmp = (x + 1.0) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-1.0d0) / sin(b)
    if (f <= (-3.8d+152)) then
        tmp = t_0
    else if (f <= (-3.5d+81)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= (-3.2d-5)) then
        tmp = t_0
    else if (f <= 3.2d+46) then
        tmp = x / -b
    else
        tmp = (x + 1.0d0) / 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 tmp;
	if (F <= -3.8e+152) {
		tmp = t_0;
	} else if (F <= -3.5e+81) {
		tmp = (-1.0 - x) / B;
	} else if (F <= -3.2e-5) {
		tmp = t_0;
	} else if (F <= 3.2e+46) {
		tmp = x / -B;
	} else {
		tmp = (x + 1.0) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -1.0 / math.sin(B)
	tmp = 0
	if F <= -3.8e+152:
		tmp = t_0
	elif F <= -3.5e+81:
		tmp = (-1.0 - x) / B
	elif F <= -3.2e-5:
		tmp = t_0
	elif F <= 3.2e+46:
		tmp = x / -B
	else:
		tmp = (x + 1.0) / B
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(-1.0 / sin(B))
	tmp = 0.0
	if (F <= -3.8e+152)
		tmp = t_0;
	elseif (F <= -3.5e+81)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= -3.2e-5)
		tmp = t_0;
	elseif (F <= 3.2e+46)
		tmp = Float64(x / Float64(-B));
	else
		tmp = Float64(Float64(x + 1.0) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = -1.0 / sin(B);
	tmp = 0.0;
	if (F <= -3.8e+152)
		tmp = t_0;
	elseif (F <= -3.5e+81)
		tmp = (-1.0 - x) / B;
	elseif (F <= -3.2e-5)
		tmp = t_0;
	elseif (F <= 3.2e+46)
		tmp = x / -B;
	else
		tmp = (x + 1.0) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -3.8e+152], t$95$0, If[LessEqual[F, -3.5e+81], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, -3.2e-5], t$95$0, If[LessEqual[F, 3.2e+46], N[(x / (-B)), $MachinePrecision], N[(N[(x + 1.0), $MachinePrecision] / B), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.8e152 or -3.5e81 < F < -3.19999999999999986e-5

   1. Initial program 54.8%

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

   3. Taylor expanded in B around 0 39.0%

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

   4. Taylor expanded in F around -inf 77.5%

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

      1. distribute-lft-in 77.5%

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

      2. mul-1-neg 77.5%

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

      3. unsub-neg 77.5%

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

      4. associate-*r/ 77.5%

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

      5. metadata-eval 77.5%

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

   6. Simplified 77.5%

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

   7. Taylor expanded in B around inf 71.5%

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

   if -3.8e152 < F < -3.5e81

   1. Initial program 71.8%

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

   3. Taylor expanded in F around -inf 99.9%

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

   4. Taylor expanded in B around 0 66.8%

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

      1. mul-1-neg 66.8%

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

      2. distribute-neg-frac2 66.8%

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

   6. Simplified 66.8%

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

   7. Taylor expanded in B around 0 66.8%

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

      1. associate-*r/ 66.8%

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

      2. distribute-lft-in 66.8%

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

      3. metadata-eval 66.8%

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

      4. neg-mul-1 66.8%

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

      5. unsub-neg 66.8%

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

   9. Simplified 66.8%

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

   if -3.19999999999999986e-5 < F < 3.1999999999999998e46

   1. Initial program 98.8%

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

   3. Taylor expanded in F around -inf 38.2%

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

   4. Taylor expanded in B around 0 26.0%

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

      1. mul-1-neg 26.0%

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

      2. distribute-neg-frac2 26.0%

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

   6. Simplified 26.0%

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

   7. Taylor expanded in x around inf 39.5%

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

      1. mul-1-neg 39.5%

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

      2. distribute-neg-frac2 39.5%

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

   9. Simplified 39.5%

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

   if 3.1999999999999998e46 < F

   1. Initial program 49.9%

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

   3. Taylor expanded in F around -inf 30.1%

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

   4. Taylor expanded in B around 0 17.3%

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

      1. mul-1-neg 17.3%

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

      2. distribute-neg-frac2 17.3%

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

   6. Simplified 17.3%

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

      1. *-un-lft-identity 17.3%

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

      2. +-commutative 17.3%

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

      3. add-sqr-sqrt 9.0%

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

      4. sqrt-unprod 18.2%

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

      5. sqr-neg 18.2%

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

      6. sqrt-prod 11.6%

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

      7. add-sqr-sqrt 26.7%

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

   8. Applied egg-rr 26.7%

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

      1. *-lft-identity 26.7%

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

      2. +-commutative 26.7%

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

   10. Simplified 26.7%

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

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.8 \cdot 10^{+152}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -3.5 \cdot 10^{+81}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq -3.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 3.2 \cdot 10^{+46}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 71.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.1 \cdot 10^{+27}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 7200000000:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.1e+27)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 7200000000.0) (/ x (- (tan B))) (- (/ 1.0 (sin B)) (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.1e+27) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 7200000000.0) {
		tmp = x / -tan(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 <= (-1.1d+27)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 7200000000.0d0) then
        tmp = x / -tan(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 <= -1.1e+27) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 7200000000.0) {
		tmp = x / -Math.tan(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.1e+27:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 7200000000.0:
		tmp = x / -math.tan(B)
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.1e+27)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 7200000000.0)
		tmp = Float64(x / Float64(-tan(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 <= -1.1e+27)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 7200000000.0)
		tmp = x / -tan(B);
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.1e+27], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7200000000.0], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.0999999999999999e27

   1. Initial program 52.9%

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

   3. Taylor expanded in B around 0 37.4%

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

   4. Taylor expanded in F around -inf 83.1%

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

      1. distribute-lft-in 83.1%

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

      2. mul-1-neg 83.1%

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

      3. unsub-neg 83.1%

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

      4. associate-*r/ 83.1%

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

      5. metadata-eval 83.1%

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

   6. Simplified 83.1%

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

   if -1.0999999999999999e27 < F < 7.2e9

   1. Initial program 99.5%

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

   3. Taylor expanded in F around -inf 39.4%

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

   4. Taylor expanded in x around inf 66.8%

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

      1. mul-1-neg 66.8%

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

      2. associate-/l* 66.8%

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

      3. distribute-rgt-neg-in 66.8%

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

      4. distribute-neg-frac2 66.8%

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

   6. Simplified 66.8%

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

      1. add-sqr-sqrt 36.1%

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

      2. sqrt-unprod 33.8%

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

      3. sqr-neg 33.8%

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

      4. sqrt-unprod 1.0%

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

      5. add-sqr-sqrt 2.2%

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

      6. clear-num 2.2%

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

      7. tan-quot 2.2%

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

      8. add-sqr-sqrt 1.3%

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

      9. sqrt-unprod 20.6%

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

      10. sqr-neg 20.6%

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

      11. sqrt-unprod 26.5%

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

      12. add-sqr-sqrt 66.8%

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

      13. neg-sub0 66.8%

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

      14. un-div-inv 66.9%

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

   8. Applied egg-rr 66.9%

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

      1. neg-sub0 66.9%

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

      2. distribute-frac-neg 66.9%

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

   10. Simplified 66.9%

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

   if 7.2e9 < F

   1. Initial program 57.1%

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

   3. Taylor expanded in B around 0 43.8%

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

   4. Taylor expanded in F around inf 86.1%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.1 \cdot 10^{+27}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 7200000000:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 62.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.1 \cdot 10^{+27}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.1e+27) (- (/ -1.0 (sin B)) (/ x B)) (/ x (- (tan B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.1e+27) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else {
		tmp = x / -tan(B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.1d+27)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else
        tmp = x / -tan(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.1e+27) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else {
		tmp = x / -Math.tan(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.1e+27:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	else:
		tmp = x / -math.tan(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.1e+27)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	else
		tmp = Float64(x / Float64(-tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.1e+27)
		tmp = (-1.0 / sin(B)) - (x / B);
	else
		tmp = x / -tan(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.1e+27], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -1.0999999999999999e27

   1. Initial program 52.9%

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

   3. Taylor expanded in B around 0 37.4%

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

   4. Taylor expanded in F around -inf 83.1%

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

      1. distribute-lft-in 83.1%

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

      2. mul-1-neg 83.1%

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

      3. unsub-neg 83.1%

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

      4. associate-*r/ 83.1%

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

      5. metadata-eval 83.1%

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

   6. Simplified 83.1%

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

   if -1.0999999999999999e27 < F

   1. Initial program 84.4%

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

   3. Taylor expanded in F around -inf 37.3%

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

   4. Taylor expanded in x around inf 55.6%

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

      1. mul-1-neg 55.6%

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

      2. associate-/l* 55.6%

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

      3. distribute-rgt-neg-in 55.6%

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

      4. distribute-neg-frac2 55.6%

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

   6. Simplified 55.6%

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

      1. add-sqr-sqrt 29.4%

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

      2. sqrt-unprod 27.8%

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

      3. sqr-neg 27.8%

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

      4. sqrt-unprod 0.9%

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

      5. add-sqr-sqrt 2.4%

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

      6. clear-num 2.4%

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

      7. tan-quot 2.4%

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

      8. add-sqr-sqrt 1.2%

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

      9. sqrt-unprod 16.7%

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

      10. sqr-neg 16.7%

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

      11. sqrt-unprod 22.4%

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

      12. add-sqr-sqrt 55.6%

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

      13. neg-sub0 55.6%

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

      14. un-div-inv 55.7%

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

   8. Applied egg-rr 55.7%

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

      1. neg-sub0 55.7%

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

      2. distribute-frac-neg 55.7%

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

   10. Simplified 55.7%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.1 \cdot 10^{+27}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 37.1% accurate, 14.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -5.8 \cdot 10^{-32}:\\ \;\;\;\;\left(B \cdot -0.16666666666666666 + x \cdot \left(B \cdot 0.3333333333333333 + \frac{-1}{B}\right)\right) + \frac{-1}{B}\\ \mathbf{elif}\;F \leq 4.8 \cdot 10^{+48}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -5.8e-32)
   (+
    (+
     (* B -0.16666666666666666)
     (* x (+ (* B 0.3333333333333333) (/ -1.0 B))))
    (/ -1.0 B))
   (if (<= F 4.8e+48) (/ x (- B)) (/ (+ x 1.0) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -5.8e-32) {
		tmp = ((B * -0.16666666666666666) + (x * ((B * 0.3333333333333333) + (-1.0 / B)))) + (-1.0 / B);
	} else if (F <= 4.8e+48) {
		tmp = x / -B;
	} else {
		tmp = (x + 1.0) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-5.8d-32)) then
        tmp = ((b * (-0.16666666666666666d0)) + (x * ((b * 0.3333333333333333d0) + ((-1.0d0) / b)))) + ((-1.0d0) / b)
    else if (f <= 4.8d+48) then
        tmp = x / -b
    else
        tmp = (x + 1.0d0) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -5.8e-32) {
		tmp = ((B * -0.16666666666666666) + (x * ((B * 0.3333333333333333) + (-1.0 / B)))) + (-1.0 / B);
	} else if (F <= 4.8e+48) {
		tmp = x / -B;
	} else {
		tmp = (x + 1.0) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -5.8e-32:
		tmp = ((B * -0.16666666666666666) + (x * ((B * 0.3333333333333333) + (-1.0 / B)))) + (-1.0 / B)
	elif F <= 4.8e+48:
		tmp = x / -B
	else:
		tmp = (x + 1.0) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -5.8e-32)
		tmp = Float64(Float64(Float64(B * -0.16666666666666666) + Float64(x * Float64(Float64(B * 0.3333333333333333) + Float64(-1.0 / B)))) + Float64(-1.0 / B));
	elseif (F <= 4.8e+48)
		tmp = Float64(x / Float64(-B));
	else
		tmp = Float64(Float64(x + 1.0) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -5.8e-32)
		tmp = ((B * -0.16666666666666666) + (x * ((B * 0.3333333333333333) + (-1.0 / B)))) + (-1.0 / B);
	elseif (F <= 4.8e+48)
		tmp = x / -B;
	else
		tmp = (x + 1.0) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -5.8e-32], N[(N[(N[(B * -0.16666666666666666), $MachinePrecision] + N[(x * N[(N[(B * 0.3333333333333333), $MachinePrecision] + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.8e+48], N[(x / (-B)), $MachinePrecision], N[(N[(x + 1.0), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.79999999999999991e-32

   1. Initial program 61.3%

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

   3. Taylor expanded in F around -inf 92.5%

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

   4. Taylor expanded in B around 0 50.0%

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

   5. Taylor expanded in x around 0 50.2%

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

   if -5.79999999999999991e-32 < F < 4.8000000000000002e48

   1. Initial program 98.8%

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

   3. Taylor expanded in F around -inf 38.4%

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

   4. Taylor expanded in B around 0 26.1%

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

      1. mul-1-neg 26.1%

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

      2. distribute-neg-frac2 26.1%

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

   6. Simplified 26.1%

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

   7. Taylor expanded in x around inf 40.1%

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

      1. mul-1-neg 40.1%

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

      2. distribute-neg-frac2 40.1%

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

   9. Simplified 40.1%

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

   if 4.8000000000000002e48 < F

   1. Initial program 49.9%

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

   3. Taylor expanded in F around -inf 30.1%

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

   4. Taylor expanded in B around 0 17.3%

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

      1. mul-1-neg 17.3%

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

      2. distribute-neg-frac2 17.3%

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

   6. Simplified 17.3%

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

      1. *-un-lft-identity 17.3%

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

      2. +-commutative 17.3%

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

      3. add-sqr-sqrt 9.0%

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

      4. sqrt-unprod 18.2%

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

      5. sqr-neg 18.2%

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

      6. sqrt-prod 11.6%

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

      7. add-sqr-sqrt 26.7%

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

   8. Applied egg-rr 26.7%

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

      1. *-lft-identity 26.7%

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

      2. +-commutative 26.7%

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

   10. Simplified 26.7%

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

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5.8 \cdot 10^{-32}:\\ \;\;\;\;\left(B \cdot -0.16666666666666666 + x \cdot \left(B \cdot 0.3333333333333333 + \frac{-1}{B}\right)\right) + \frac{-1}{B}\\ \mathbf{elif}\;F \leq 4.8 \cdot 10^{+48}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 30.6% accurate, 21.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{+44}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.2e-5)
   (/ -1.0 B)
   (if (<= F 5.2e+44) (/ x (- B)) (/ (+ x 1.0) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.2e-5) {
		tmp = -1.0 / B;
	} else if (F <= 5.2e+44) {
		tmp = x / -B;
	} else {
		tmp = (x + 1.0) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-3.2d-5)) then
        tmp = (-1.0d0) / b
    else if (f <= 5.2d+44) then
        tmp = x / -b
    else
        tmp = (x + 1.0d0) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.2e-5) {
		tmp = -1.0 / B;
	} else if (F <= 5.2e+44) {
		tmp = x / -B;
	} else {
		tmp = (x + 1.0) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.2e-5:
		tmp = -1.0 / B
	elif F <= 5.2e+44:
		tmp = x / -B
	else:
		tmp = (x + 1.0) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.2e-5)
		tmp = Float64(-1.0 / B);
	elseif (F <= 5.2e+44)
		tmp = Float64(x / Float64(-B));
	else
		tmp = Float64(Float64(x + 1.0) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -3.2e-5)
		tmp = -1.0 / B;
	elseif (F <= 5.2e+44)
		tmp = x / -B;
	else
		tmp = (x + 1.0) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.2e-5], N[(-1.0 / B), $MachinePrecision], If[LessEqual[F, 5.2e+44], N[(x / (-B)), $MachinePrecision], N[(N[(x + 1.0), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.19999999999999986e-5

   1. Initial program 59.1%

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

   3. Taylor expanded in F around -inf 96.0%

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

   4. Taylor expanded in B around 0 51.5%

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

      1. mul-1-neg 51.5%

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

      2. distribute-neg-frac2 51.5%

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

   6. Simplified 51.5%

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

   7. Taylor expanded in x around 0 38.4%

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

   if -3.19999999999999986e-5 < F < 5.1999999999999998e44

   1. Initial program 98.8%

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

   3. Taylor expanded in F around -inf 38.2%

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

   4. Taylor expanded in B around 0 26.0%

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

      1. mul-1-neg 26.0%

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

      2. distribute-neg-frac2 26.0%

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

   6. Simplified 26.0%

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

   7. Taylor expanded in x around inf 39.5%

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

      1. mul-1-neg 39.5%

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

      2. distribute-neg-frac2 39.5%

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

   9. Simplified 39.5%

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

   if 5.1999999999999998e44 < F

   1. Initial program 49.9%

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

   3. Taylor expanded in F around -inf 30.1%

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

   4. Taylor expanded in B around 0 17.3%

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

      1. mul-1-neg 17.3%

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

      2. distribute-neg-frac2 17.3%

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

   6. Simplified 17.3%

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

      1. *-un-lft-identity 17.3%

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

      2. +-commutative 17.3%

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

      3. add-sqr-sqrt 9.0%

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

      4. sqrt-unprod 18.2%

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

      5. sqr-neg 18.2%

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

      6. sqrt-prod 11.6%

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

      7. add-sqr-sqrt 26.7%

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

   8. Applied egg-rr 26.7%

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

      1. *-lft-identity 26.7%

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

      2. +-commutative 26.7%

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

   10. Simplified 26.7%

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

4. Final simplification 36.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{+44}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 37.0% accurate, 21.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.55 \cdot 10^{-28}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 3.6 \cdot 10^{+45}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.55e-28)
   (/ (- -1.0 x) B)
   (if (<= F 3.6e+45) (/ x (- B)) (/ (+ x 1.0) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.55e-28) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 3.6e+45) {
		tmp = x / -B;
	} else {
		tmp = (x + 1.0) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.55d-28)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 3.6d+45) then
        tmp = x / -b
    else
        tmp = (x + 1.0d0) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.55e-28) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 3.6e+45) {
		tmp = x / -B;
	} else {
		tmp = (x + 1.0) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.55e-28:
		tmp = (-1.0 - x) / B
	elif F <= 3.6e+45:
		tmp = x / -B
	else:
		tmp = (x + 1.0) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.55e-28)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 3.6e+45)
		tmp = Float64(x / Float64(-B));
	else
		tmp = Float64(Float64(x + 1.0) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.55e-28)
		tmp = (-1.0 - x) / B;
	elseif (F <= 3.6e+45)
		tmp = x / -B;
	else
		tmp = (x + 1.0) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.55e-28], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 3.6e+45], N[(x / (-B)), $MachinePrecision], N[(N[(x + 1.0), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.54999999999999996e-28

   1. Initial program 61.3%

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

   3. Taylor expanded in F around -inf 92.5%

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

   4. Taylor expanded in B around 0 49.9%

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

      1. mul-1-neg 49.9%

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

      2. distribute-neg-frac2 49.9%

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

   6. Simplified 49.9%

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

   7. Taylor expanded in B around 0 49.9%

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

      1. associate-*r/ 49.9%

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

      2. distribute-lft-in 49.9%

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

      3. metadata-eval 49.9%

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

      4. neg-mul-1 49.9%

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

      5. unsub-neg 49.9%

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

   9. Simplified 49.9%

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

   if -1.54999999999999996e-28 < F < 3.6e45

   1. Initial program 98.8%

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

   3. Taylor expanded in F around -inf 38.4%

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

   4. Taylor expanded in B around 0 26.1%

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

      1. mul-1-neg 26.1%

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

      2. distribute-neg-frac2 26.1%

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

   6. Simplified 26.1%

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

   7. Taylor expanded in x around inf 40.1%

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

      1. mul-1-neg 40.1%

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

      2. distribute-neg-frac2 40.1%

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

   9. Simplified 40.1%

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

   if 3.6e45 < F

   1. Initial program 49.9%

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

   3. Taylor expanded in F around -inf 30.1%

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

   4. Taylor expanded in B around 0 17.3%

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

      1. mul-1-neg 17.3%

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

      2. distribute-neg-frac2 17.3%

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

   6. Simplified 17.3%

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

      1. *-un-lft-identity 17.3%

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

      2. +-commutative 17.3%

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

      3. add-sqr-sqrt 9.0%

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

      4. sqrt-unprod 18.2%

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

      5. sqr-neg 18.2%

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

      6. sqrt-prod 11.6%

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

      7. add-sqr-sqrt 26.7%

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

   8. Applied egg-rr 26.7%

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

      1. *-lft-identity 26.7%

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

      2. +-commutative 26.7%

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

   10. Simplified 26.7%

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

4. Final simplification 39.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.55 \cdot 10^{-28}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 3.6 \cdot 10^{+45}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 30.0% accurate, 35.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (if (<= F -3.2e-5) (/ -1.0 B) (/ x (- B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.2e-5) {
		tmp = -1.0 / B;
	} else {
		tmp = x / -B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-3.2d-5)) then
        tmp = (-1.0d0) / b
    else
        tmp = x / -b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.2e-5) {
		tmp = -1.0 / B;
	} else {
		tmp = x / -B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.2e-5:
		tmp = -1.0 / B
	else:
		tmp = x / -B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.2e-5)
		tmp = Float64(-1.0 / B);
	else
		tmp = Float64(x / Float64(-B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -3.2e-5)
		tmp = -1.0 / B;
	else
		tmp = x / -B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.2e-5], N[(-1.0 / B), $MachinePrecision], N[(x / (-B)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -3.19999999999999986e-5

   1. Initial program 59.1%

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

   3. Taylor expanded in F around -inf 96.0%

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

   4. Taylor expanded in B around 0 51.5%

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

      1. mul-1-neg 51.5%

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

      2. distribute-neg-frac2 51.5%

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

   6. Simplified 51.5%

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

   7. Taylor expanded in x around 0 38.4%

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

   if -3.19999999999999986e-5 < F

   1. Initial program 83.7%

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

   3. Taylor expanded in F around -inf 35.7%

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

   4. Taylor expanded in B around 0 23.3%

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

      1. mul-1-neg 23.3%

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

      2. distribute-neg-frac2 23.3%

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

   6. Simplified 23.3%

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

   7. Taylor expanded in x around inf 32.6%

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

      1. mul-1-neg 32.6%

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

      2. distribute-neg-frac2 32.6%

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

   9. Simplified 32.6%

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

4. Final simplification 34.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 10.6% accurate, 108.0× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (/ -1.0 B))

C

double code(double F, double B, double x) {
	return -1.0 / B;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = (-1.0d0) / b
end function

Java

public static double code(double F, double B, double x) {
	return -1.0 / B;
}

Python

def code(F, B, x):
	return -1.0 / B

Julia

function code(F, B, x)
	return Float64(-1.0 / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = -1.0 / B;
end

Wolfram

code[F_, B_, x_] := N[(-1.0 / B), $MachinePrecision]

Derivation

1. Initial program 77.2%

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

3. Taylor expanded in F around -inf 51.7%

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

4. Taylor expanded in B around 0 30.8%

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

   1. mul-1-neg 30.8%

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

   2. distribute-neg-frac2 30.8%

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

6. Simplified 30.8%

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

7. Taylor expanded in x around 0 13.2%

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

8. Final simplification 13.2%

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

Reproduce

herbie shell --seed 2024112 
(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))))))