VandenBroeck and Keller, Equation (23)

Percentage Accurate: 77.0% → 99.6%

Time: 23.5s

Alternatives: 20

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.0% 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: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ t_1 := \frac{1}{\sin B}\\ \mathbf{if}\;F \leq -1.82 \cdot 10^{+155}:\\ \;\;\;\;\frac{-1}{\sin B} - x \cdot \left(t\_1 \cdot \cos B\right)\\ \mathbf{elif}\;F \leq 200000000:\\ \;\;\;\;\frac{F \cdot {\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1 - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))) (t_1 (/ 1.0 (sin B))))
   (if (<= F -1.82e+155)
     (- (/ -1.0 (sin B)) (* x (* t_1 (cos B))))
     (if (<= F 200000000.0)
       (- (/ (* F (pow (fma F F 2.0) -0.5)) (sin B)) t_0)
       (- t_1 t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double t_1 = 1.0 / sin(B);
	double tmp;
	if (F <= -1.82e+155) {
		tmp = (-1.0 / sin(B)) - (x * (t_1 * cos(B)));
	} else if (F <= 200000000.0) {
		tmp = ((F * pow(fma(F, F, 2.0), -0.5)) / sin(B)) - t_0;
	} else {
		tmp = t_1 - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	t_1 = Float64(1.0 / sin(B))
	tmp = 0.0
	if (F <= -1.82e+155)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x * Float64(t_1 * cos(B))));
	elseif (F <= 200000000.0)
		tmp = Float64(Float64(Float64(F * (fma(F, F, 2.0) ^ -0.5)) / sin(B)) - t_0);
	else
		tmp = Float64(t_1 - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.82e+155], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x * N[(t$95$1 * N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 200000000.0], N[(N[(N[(F * N[Power[N[(F * F + 2.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(t$95$1 - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.81999999999999989e155

   1. Initial program 32.6%

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

   3. Taylor expanded in F around -inf 99.5%

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

      1. tan-quot 99.5%

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

      2. associate-/r/ 99.6%

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

   5. Applied egg-rr 99.6%

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

   if -1.81999999999999989e155 < F < 2e8

   1. Initial program 95.4%

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

   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

   5. Taylor expanded in x around 0 99.6%

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

      1. associate-*l/ 99.6%

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

      2. *-lft-identity 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. fma-undefine 99.6%

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

   7. Simplified 99.6%

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

      1. associate-*r/ 99.7%

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

      2. fma-define 99.7%

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

      3. inv-pow 99.7%

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

      4. sqrt-pow1 99.7%

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

      5. fma-define 99.7%

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

      6. metadata-eval 99.7%

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

   9. Applied egg-rr 99.7%

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

   if 2e8 < F

   1. Initial program 57.2%

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

   3. Simplified 71.1%

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

   5. Taylor expanded in x around 0 71.2%

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

      1. associate-*l/ 71.1%

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

      2. *-lft-identity 71.1%

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

      3. +-commutative 71.1%

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

      4. unpow2 71.1%

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

      5. fma-undefine 71.1%

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

   7. Simplified 71.1%

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

   8. Taylor expanded in F around inf 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.82 \cdot 10^{+155}:\\ \;\;\;\;\frac{-1}{\sin B} - x \cdot \left(\frac{1}{\sin B} \cdot \cos B\right)\\ \mathbf{elif}\;F \leq 200000000:\\ \;\;\;\;\frac{F \cdot {\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2.2 \cdot 10^{+40}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 31000000:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -2.2e+40)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 31000000.0)
       (+
        (* x (/ -1.0 (tan B)))
        (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)))
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -2.2e+40) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 31000000.0) {
		tmp = (x * (-1.0 / tan(B))) + ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5));
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-2.2d+40)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 31000000.0d0) then
        tmp = (x * ((-1.0d0) / tan(b))) + ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)))
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -2.2e+40) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 31000000.0) {
		tmp = (x * (-1.0 / Math.tan(B))) + ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5));
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -2.2e+40:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 31000000.0:
		tmp = (x * (-1.0 / math.tan(B))) + ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5))
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -2.2e+40)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 31000000.0)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -2.2e+40)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 31000000.0)
		tmp = (x * (-1.0 / tan(B))) + ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5));
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.2e+40], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 31000000.0], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 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[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.1999999999999999e40

   1. Initial program 54.1%

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

   3. Simplified 69.0%

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

   5. Taylor expanded in x around 0 69.0%

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

      1. associate-*l/ 69.0%

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

      2. *-lft-identity 69.0%

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

      3. +-commutative 69.0%

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

      4. unpow2 69.0%

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

      5. fma-undefine 69.0%

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

   7. Simplified 69.0%

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

   8. Taylor expanded in F around -inf 99.6%

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

   if -2.1999999999999999e40 < F < 3.1e7

   1. Initial program 99.4%

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

   if 3.1e7 < F

   1. Initial program 57.2%

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

   3. Simplified 71.1%

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

   5. Taylor expanded in x around 0 71.2%

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

      1. associate-*l/ 71.1%

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

      2. *-lft-identity 71.1%

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

      3. +-commutative 71.1%

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

      4. unpow2 71.1%

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

      5. fma-undefine 71.1%

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

   7. Simplified 71.1%

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

   8. Taylor expanded in F around inf 99.9%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.2 \cdot 10^{+40}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 31000000:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.4)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.4)
       (- (* F (/ (sqrt 0.5) (sin B))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.4) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.4) {
		tmp = (F * (sqrt(0.5) / sin(B))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.4d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.4d0) then
        tmp = (f * (sqrt(0.5d0) / sin(b))) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.4) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.4) {
		tmp = (F * (Math.sqrt(0.5) / Math.sin(B))) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.4:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.4:
		tmp = (F * (math.sqrt(0.5) / math.sin(B))) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.4)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.4)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / sin(B))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.4)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.4)
		tmp = (F * (sqrt(0.5) / sin(B))) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.4], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.4], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.3999999999999999

   1. Initial program 57.3%

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

   3. Simplified 71.2%

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

   5. Taylor expanded in x around 0 71.1%

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

      1. associate-*l/ 71.2%

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

      2. *-lft-identity 71.2%

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

      3. +-commutative 71.2%

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

      4. unpow2 71.2%

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

      5. fma-undefine 71.2%

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

   7. Simplified 71.2%

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

   8. Taylor expanded in F around -inf 98.9%

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

   if -1.3999999999999999 < F < 1.3999999999999999

   1. Initial program 99.4%

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

   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

   5. Taylor expanded in x around 0 99.6%

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

      1. associate-*l/ 99.6%

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

      2. *-lft-identity 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. fma-undefine 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in F around 0 98.5%

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

   if 1.3999999999999999 < F

   1. Initial program 57.9%

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

   3. Simplified 71.6%

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

   5. Taylor expanded in x around 0 71.6%

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

      1. associate-*l/ 71.6%

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

      2. *-lft-identity 71.6%

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

      3. +-commutative 71.6%

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

      4. unpow2 71.6%

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

      5. fma-undefine 71.6%

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

   7. Simplified 71.6%

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

   8. Taylor expanded in F around inf 98.8%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;\frac{F}{\frac{\sin B}{\sqrt{0.5}}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.4)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.4)
       (- (/ F (/ (sin B) (sqrt 0.5))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.4) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.4) {
		tmp = (F / (sin(B) / sqrt(0.5))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.4d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.4d0) then
        tmp = (f / (sin(b) / sqrt(0.5d0))) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.4) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.4) {
		tmp = (F / (Math.sin(B) / Math.sqrt(0.5))) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.4:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.4:
		tmp = (F / (math.sin(B) / math.sqrt(0.5))) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.4)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.4)
		tmp = Float64(Float64(F / Float64(sin(B) / sqrt(0.5))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.4)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.4)
		tmp = (F / (sin(B) / sqrt(0.5))) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.4], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.4], N[(N[(F / N[(N[Sin[B], $MachinePrecision] / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.3999999999999999

   1. Initial program 57.3%

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

   3. Simplified 71.2%

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

   5. Taylor expanded in x around 0 71.1%

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

      1. associate-*l/ 71.2%

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

      2. *-lft-identity 71.2%

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

      3. +-commutative 71.2%

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

      4. unpow2 71.2%

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

      5. fma-undefine 71.2%

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

   7. Simplified 71.2%

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

   8. Taylor expanded in F around -inf 98.9%

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

   if -1.3999999999999999 < F < 1.3999999999999999

   1. Initial program 99.4%

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

   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

   5. Taylor expanded in x around 0 99.6%

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

      1. associate-*l/ 99.6%

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

      2. *-lft-identity 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. fma-undefine 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in F around 0 98.5%

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

      1. clear-num 98.5%

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

      2. un-div-inv 98.5%

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

   10. Applied egg-rr 98.5%

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

   if 1.3999999999999999 < F

   1. Initial program 57.9%

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

   3. Simplified 71.6%

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

   5. Taylor expanded in x around 0 71.6%

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

      1. associate-*l/ 71.6%

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

      2. *-lft-identity 71.6%

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

      3. +-commutative 71.6%

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

      4. unpow2 71.6%

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

      5. fma-undefine 71.6%

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

   7. Simplified 71.6%

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

   8. Taylor expanded in F around inf 98.8%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;\frac{F}{\frac{\sin B}{\sqrt{0.5}}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.4)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.4)
       (- (/ (* F (sqrt 0.5)) (sin B)) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.4) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.4) {
		tmp = ((F * sqrt(0.5)) / sin(B)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.4d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.4d0) then
        tmp = ((f * sqrt(0.5d0)) / sin(b)) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.4) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.4) {
		tmp = ((F * Math.sqrt(0.5)) / Math.sin(B)) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.4:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.4:
		tmp = ((F * math.sqrt(0.5)) / math.sin(B)) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.4)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.4)
		tmp = Float64(Float64(Float64(F * sqrt(0.5)) / sin(B)) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.4)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.4)
		tmp = ((F * sqrt(0.5)) / sin(B)) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.4], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.4], N[(N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.3999999999999999

   1. Initial program 57.3%

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

   3. Simplified 71.2%

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

   5. Taylor expanded in x around 0 71.1%

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

      1. associate-*l/ 71.2%

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

      2. *-lft-identity 71.2%

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

      3. +-commutative 71.2%

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

      4. unpow2 71.2%

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

      5. fma-undefine 71.2%

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

   7. Simplified 71.2%

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

   8. Taylor expanded in F around -inf 98.9%

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

   if -1.3999999999999999 < F < 1.3999999999999999

   1. Initial program 99.4%

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

   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

   5. Taylor expanded in x around 0 99.6%

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

      1. associate-*l/ 99.6%

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

      2. *-lft-identity 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. fma-undefine 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in F around 0 98.6%

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

   if 1.3999999999999999 < F

   1. Initial program 57.9%

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

   3. Simplified 71.6%

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

   5. Taylor expanded in x around 0 71.6%

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

      1. associate-*l/ 71.6%

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

      2. *-lft-identity 71.6%

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

      3. +-commutative 71.6%

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

      4. unpow2 71.6%

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

      5. fma-undefine 71.6%

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

   7. Simplified 71.6%

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

   8. Taylor expanded in F around inf 98.8%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3.4 \cdot 10^{-12}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1.9 \cdot 10^{-68}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - t\_0\\ \mathbf{elif}\;F \leq 26000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -3.4e-12)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.9e-68)
       (- (* F (/ (sqrt 0.5) B)) t_0)
       (if (<= F 26000.0)
         (- (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)) (/ x B))
         (- (/ 1.0 (sin B)) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -3.4e-12) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.9e-68) {
		tmp = (F * (sqrt(0.5) / B)) - t_0;
	} else if (F <= 26000.0) {
		tmp = ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B);
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-3.4d-12)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.9d-68) then
        tmp = (f * (sqrt(0.5d0) / b)) - t_0
    else if (f <= 26000.0d0) then
        tmp = ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0))) - (x / b)
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -3.4e-12) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.9e-68) {
		tmp = (F * (Math.sqrt(0.5) / B)) - t_0;
	} else if (F <= 26000.0) {
		tmp = ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B);
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -3.4e-12:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.9e-68:
		tmp = (F * (math.sqrt(0.5) / B)) - t_0
	elif F <= 26000.0:
		tmp = ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B)
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -3.4e-12)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.9e-68)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / B)) - t_0);
	elseif (F <= 26000.0)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -3.4e-12)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.9e-68)
		tmp = (F * (sqrt(0.5) / B)) - t_0;
	elseif (F <= 26000.0)
		tmp = ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5)) - (x / B);
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -3.4e-12], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.9e-68], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 26000.0], 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 / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.4000000000000001e-12

   1. Initial program 58.5%

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

   3. Simplified 71.9%

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

   5. Taylor expanded in x around 0 71.9%

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

      1. associate-*l/ 71.9%

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

      2. *-lft-identity 71.9%

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

      3. +-commutative 71.9%

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

      4. unpow2 71.9%

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

      5. fma-undefine 71.9%

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

   7. Simplified 71.9%

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

   8. Taylor expanded in F around -inf 96.6%

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

   if -3.4000000000000001e-12 < F < 1.90000000000000019e-68

   1. Initial program 99.4%

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

   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

   5. Taylor expanded in x around 0 99.6%

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

      1. associate-*l/ 99.6%

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

      2. *-lft-identity 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. fma-undefine 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in F around 0 99.6%

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

   9. Taylor expanded in B around 0 88.5%

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

   if 1.90000000000000019e-68 < F < 26000

   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 92.5%

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

   if 26000 < F

   1. Initial program 57.2%

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

   3. Simplified 71.1%

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

   5. Taylor expanded in x around 0 71.2%

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

      1. associate-*l/ 71.1%

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

      2. *-lft-identity 71.1%

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

      3. +-commutative 71.1%

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

      4. unpow2 71.1%

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

      5. fma-undefine 71.1%

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

   7. Simplified 71.1%

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

   8. Taylor expanded in F around inf 99.9%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.4 \cdot 10^{-12}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.9 \cdot 10^{-68}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 26000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3.4 \cdot 10^{-12}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1.9 \cdot 10^{-68}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - t\_0\\ \mathbf{elif}\;F \leq 1.3:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -3.4e-12)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.9e-68)
       (- (* F (/ (sqrt 0.5) B)) t_0)
       (if (<= F 1.3)
         (- (* (/ F (sin B)) (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))) (/ x B))
         (- (/ 1.0 (sin B)) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -3.4e-12) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.9e-68) {
		tmp = (F * (sqrt(0.5) / B)) - t_0;
	} else if (F <= 1.3) {
		tmp = ((F / sin(B)) * sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B);
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-3.4d-12)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.9d-68) then
        tmp = (f * (sqrt(0.5d0) / b)) - t_0
    else if (f <= 1.3d0) then
        tmp = ((f / sin(b)) * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))) - (x / b)
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -3.4e-12) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.9e-68) {
		tmp = (F * (Math.sqrt(0.5) / B)) - t_0;
	} else if (F <= 1.3) {
		tmp = ((F / Math.sin(B)) * Math.sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B);
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -3.4e-12:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.9e-68:
		tmp = (F * (math.sqrt(0.5) / B)) - t_0
	elif F <= 1.3:
		tmp = ((F / math.sin(B)) * math.sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B)
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -3.4e-12)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.9e-68)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / B)) - t_0);
	elseif (F <= 1.3)
		tmp = Float64(Float64(Float64(F / sin(B)) * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -3.4e-12)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.9e-68)
		tmp = (F * (sqrt(0.5) / B)) - t_0;
	elseif (F <= 1.3)
		tmp = ((F / sin(B)) * sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B);
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -3.4e-12], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.9e-68], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.3], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.4000000000000001e-12

   1. Initial program 58.5%

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

   3. Simplified 71.9%

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

   5. Taylor expanded in x around 0 71.9%

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

      1. associate-*l/ 71.9%

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

      2. *-lft-identity 71.9%

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

      3. +-commutative 71.9%

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

      4. unpow2 71.9%

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

      5. fma-undefine 71.9%

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

   7. Simplified 71.9%

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

   8. Taylor expanded in F around -inf 96.6%

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

   if -3.4000000000000001e-12 < F < 1.90000000000000019e-68

   1. Initial program 99.4%

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

   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

   5. Taylor expanded in x around 0 99.6%

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

      1. associate-*l/ 99.6%

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

      2. *-lft-identity 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. fma-undefine 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in F around 0 99.6%

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

   9. Taylor expanded in B around 0 88.5%

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

   if 1.90000000000000019e-68 < F < 1.30000000000000004

   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 91.9%

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

   4. Taylor expanded in F around 0 87.0%

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

   if 1.30000000000000004 < F

   1. Initial program 57.9%

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

   3. Simplified 71.6%

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

   5. Taylor expanded in x around 0 71.6%

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

      1. associate-*l/ 71.6%

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

      2. *-lft-identity 71.6%

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

      3. +-commutative 71.6%

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

      4. unpow2 71.6%

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

      5. fma-undefine 71.6%

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

   7. Simplified 71.6%

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

   8. Taylor expanded in F around inf 98.8%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.4 \cdot 10^{-12}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.9 \cdot 10^{-68}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.3:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -6 \cdot 10^{-16}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq -7 \cdot 10^{-77}:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 8.2 \cdot 10^{-54}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -6e-16)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F -7e-77)
       (- (* (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5) (/ F B)) (/ x B))
       (if (<= F 8.2e-54)
         (/ (* x (cos B)) (- (sin B)))
         (- (/ 1.0 (sin B)) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -6e-16) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= -7e-77) {
		tmp = (pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else if (F <= 8.2e-54) {
		tmp = (x * cos(B)) / -sin(B);
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-6d-16)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= (-7d-77)) then
        tmp = ((((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)) * (f / b)) - (x / b)
    else if (f <= 8.2d-54) then
        tmp = (x * cos(b)) / -sin(b)
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -6e-16) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= -7e-77) {
		tmp = (Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else if (F <= 8.2e-54) {
		tmp = (x * Math.cos(B)) / -Math.sin(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -6e-16:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= -7e-77:
		tmp = (math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B)
	elif F <= 8.2e-54:
		tmp = (x * math.cos(B)) / -math.sin(B)
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -6e-16)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= -7e-77)
		tmp = Float64(Float64((Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5) * Float64(F / B)) - Float64(x / B));
	elseif (F <= 8.2e-54)
		tmp = Float64(Float64(x * cos(B)) / Float64(-sin(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -6e-16)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= -7e-77)
		tmp = ((((2.0 + (F * F)) + (x * 2.0)) ^ -0.5) * (F / B)) - (x / B);
	elseif (F <= 8.2e-54)
		tmp = (x * cos(B)) / -sin(B);
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -6e-16], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -7e-77], N[(N[(N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 8.2e-54], N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -5.99999999999999987e-16

   1. Initial program 59.0%

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

   3. Simplified 72.3%

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

   5. Taylor expanded in x around 0 72.3%

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

      1. associate-*l/ 72.3%

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

      2. *-lft-identity 72.3%

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

      3. +-commutative 72.3%

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

      4. unpow2 72.3%

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

      5. fma-undefine 72.3%

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

   7. Simplified 72.3%

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

   8. Taylor expanded in F around -inf 96.6%

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

   if -5.99999999999999987e-16 < F < -7.00000000000000026e-77

   1. Initial program 99.1%

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

   3. Taylor expanded in B around 0 88.3%

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

   4. Taylor expanded in B around 0 72.5%

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

   if -7.00000000000000026e-77 < F < 8.2000000000000001e-54

   1. Initial program 99.5%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 99.7%

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

      1. associate-*l/ 99.7%

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

      2. *-lft-identity 99.7%

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

      3. +-commutative 99.7%

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

      4. unpow2 99.7%

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

      5. fma-undefine 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in F around -inf 34.4%

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

   9. Taylor expanded in x around inf 78.7%

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

       1. mul-1-neg 78.7%

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

   11. Simplified 78.7%

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

   if 8.2000000000000001e-54 < F

   1. Initial program 62.3%

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

   3. Simplified 74.5%

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

   5. Taylor expanded in x around 0 74.6%

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

      1. associate-*l/ 74.5%

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

      2. *-lft-identity 74.5%

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

      3. +-commutative 74.5%

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

      4. unpow2 74.5%

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

      5. fma-undefine 74.5%

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

   7. Simplified 74.5%

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

   8. Taylor expanded in F around inf 90.7%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6 \cdot 10^{-16}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -7 \cdot 10^{-77}:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 8.2 \cdot 10^{-54}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.6 \cdot 10^{-16}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -6.8 \cdot 10^{-77}:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 9.5 \cdot 10^{-53}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.6e-16)
   (- (/ -1.0 B) (/ x (tan B)))
   (if (<= F -6.8e-77)
     (- (* (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5) (/ F B)) (/ x B))
     (if (<= F 9.5e-53)
       (/ (* x (cos B)) (- (sin B)))
       (- (/ 1.0 (sin B)) (/ x B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.6e-16) {
		tmp = (-1.0 / B) - (x / tan(B));
	} else if (F <= -6.8e-77) {
		tmp = (pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else if (F <= 9.5e-53) {
		tmp = (x * cos(B)) / -sin(B);
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-3.6d-16)) then
        tmp = ((-1.0d0) / b) - (x / tan(b))
    else if (f <= (-6.8d-77)) then
        tmp = ((((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)) * (f / b)) - (x / b)
    else if (f <= 9.5d-53) then
        tmp = (x * cos(b)) / -sin(b)
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.6e-16) {
		tmp = (-1.0 / B) - (x / Math.tan(B));
	} else if (F <= -6.8e-77) {
		tmp = (Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else if (F <= 9.5e-53) {
		tmp = (x * Math.cos(B)) / -Math.sin(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.6e-16:
		tmp = (-1.0 / B) - (x / math.tan(B))
	elif F <= -6.8e-77:
		tmp = (math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B)
	elif F <= 9.5e-53:
		tmp = (x * math.cos(B)) / -math.sin(B)
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.6e-16)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / tan(B)));
	elseif (F <= -6.8e-77)
		tmp = Float64(Float64((Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5) * Float64(F / B)) - Float64(x / B));
	elseif (F <= 9.5e-53)
		tmp = Float64(Float64(x * cos(B)) / Float64(-sin(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -3.6e-16)
		tmp = (-1.0 / B) - (x / tan(B));
	elseif (F <= -6.8e-77)
		tmp = ((((2.0 + (F * F)) + (x * 2.0)) ^ -0.5) * (F / B)) - (x / B);
	elseif (F <= 9.5e-53)
		tmp = (x * cos(B)) / -sin(B);
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.6e-16], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -6.8e-77], N[(N[(N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 9.5e-53], N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.59999999999999983e-16

   1. Initial program 59.0%

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

   3. Simplified 72.3%

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

   5. Taylor expanded in x around 0 72.3%

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

      1. associate-*l/ 72.3%

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

      2. *-lft-identity 72.3%

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

      3. +-commutative 72.3%

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

      4. unpow2 72.3%

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

      5. fma-undefine 72.3%

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

   7. Simplified 72.3%

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

   8. Taylor expanded in F around -inf 96.6%

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

   9. Taylor expanded in B around 0 71.0%

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

   if -3.59999999999999983e-16 < F < -6.79999999999999966e-77

   1. Initial program 99.1%

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

   3. Taylor expanded in B around 0 88.3%

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

   4. Taylor expanded in B around 0 72.5%

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

   if -6.79999999999999966e-77 < F < 9.5000000000000008e-53

   1. Initial program 99.5%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 99.7%

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

      1. associate-*l/ 99.7%

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

      2. *-lft-identity 99.7%

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

      3. +-commutative 99.7%

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

      4. unpow2 99.7%

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

      5. fma-undefine 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in F around -inf 34.4%

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

   9. Taylor expanded in x around inf 78.7%

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

       1. mul-1-neg 78.7%

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

   11. Simplified 78.7%

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

   if 9.5000000000000008e-53 < F

   1. Initial program 62.3%

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

   3. Taylor expanded in B around 0 38.2%

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

   4. Taylor expanded in F around inf 66.4%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.6 \cdot 10^{-16}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -6.8 \cdot 10^{-77}:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 9.5 \cdot 10^{-53}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -4.4 \cdot 10^{-16}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.02 \cdot 10^{-76}:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 9.5 \cdot 10^{-53}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4.4e-16)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F -1.02e-76)
     (- (* (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5) (/ F B)) (/ x B))
     (if (<= F 9.5e-53)
       (/ (* x (cos B)) (- (sin B)))
       (- (/ 1.0 (sin B)) (/ x B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.4e-16) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= -1.02e-76) {
		tmp = (pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else if (F <= 9.5e-53) {
		tmp = (x * cos(B)) / -sin(B);
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-4.4d-16)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= (-1.02d-76)) then
        tmp = ((((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)) * (f / b)) - (x / b)
    else if (f <= 9.5d-53) then
        tmp = (x * cos(b)) / -sin(b)
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.4e-16) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= -1.02e-76) {
		tmp = (Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else if (F <= 9.5e-53) {
		tmp = (x * Math.cos(B)) / -Math.sin(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -4.4e-16:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= -1.02e-76:
		tmp = (math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B)
	elif F <= 9.5e-53:
		tmp = (x * math.cos(B)) / -math.sin(B)
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -4.4e-16)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= -1.02e-76)
		tmp = Float64(Float64((Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5) * Float64(F / B)) - Float64(x / B));
	elseif (F <= 9.5e-53)
		tmp = Float64(Float64(x * cos(B)) / Float64(-sin(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -4.4e-16)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= -1.02e-76)
		tmp = ((((2.0 + (F * F)) + (x * 2.0)) ^ -0.5) * (F / B)) - (x / B);
	elseif (F <= 9.5e-53)
		tmp = (x * cos(B)) / -sin(B);
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -4.4e-16], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.02e-76], N[(N[(N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 9.5e-53], N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -4.40000000000000001e-16

   1. Initial program 59.0%

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

   3. Simplified 72.3%

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

   5. Taylor expanded in x around 0 72.3%

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

      1. associate-*l/ 72.3%

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

      2. *-lft-identity 72.3%

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

      3. +-commutative 72.3%

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

      4. unpow2 72.3%

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

      5. fma-undefine 72.3%

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

   7. Simplified 72.3%

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

   8. Taylor expanded in F around -inf 96.6%

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

   if -4.40000000000000001e-16 < F < -1.02000000000000006e-76

   1. Initial program 99.1%

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

   3. Taylor expanded in B around 0 88.3%

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

   4. Taylor expanded in B around 0 72.5%

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

   if -1.02000000000000006e-76 < F < 9.5000000000000008e-53

   1. Initial program 99.5%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 99.7%

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

      1. associate-*l/ 99.7%

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

      2. *-lft-identity 99.7%

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

      3. +-commutative 99.7%

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

      4. unpow2 99.7%

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

      5. fma-undefine 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in F around -inf 34.4%

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

   9. Taylor expanded in x around inf 78.7%

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

       1. mul-1-neg 78.7%

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

   11. Simplified 78.7%

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

   if 9.5000000000000008e-53 < F

   1. Initial program 62.3%

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

   3. Taylor expanded in B around 0 38.2%

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

   4. Taylor expanded in F around inf 66.4%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.4 \cdot 10^{-16}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.02 \cdot 10^{-76}:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 9.5 \cdot 10^{-53}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 90.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3.4 \cdot 10^{-12}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 9.5 \cdot 10^{-53}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -3.4e-12)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 9.5e-53)
       (- (* F (/ (sqrt 0.5) B)) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -3.4e-12) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 9.5e-53) {
		tmp = (F * (sqrt(0.5) / B)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-3.4d-12)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 9.5d-53) then
        tmp = (f * (sqrt(0.5d0) / b)) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -3.4e-12) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 9.5e-53) {
		tmp = (F * (Math.sqrt(0.5) / B)) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -3.4e-12:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 9.5e-53:
		tmp = (F * (math.sqrt(0.5) / B)) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -3.4e-12)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 9.5e-53)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / B)) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -3.4e-12)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 9.5e-53)
		tmp = (F * (sqrt(0.5) / B)) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -3.4e-12], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 9.5e-53], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.4000000000000001e-12

   1. Initial program 58.5%

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

   3. Simplified 71.9%

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

   5. Taylor expanded in x around 0 71.9%

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

      1. associate-*l/ 71.9%

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

      2. *-lft-identity 71.9%

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

      3. +-commutative 71.9%

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

      4. unpow2 71.9%

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

      5. fma-undefine 71.9%

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

   7. Simplified 71.9%

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

   8. Taylor expanded in F around -inf 96.6%

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

   if -3.4000000000000001e-12 < F < 9.5000000000000008e-53

   1. Initial program 99.4%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 99.6%

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

      1. associate-*l/ 99.7%

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

      2. *-lft-identity 99.7%

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

      3. +-commutative 99.7%

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

      4. unpow2 99.7%

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

      5. fma-undefine 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in F around 0 99.7%

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

   9. Taylor expanded in B around 0 88.2%

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

   if 9.5000000000000008e-53 < F

   1. Initial program 62.3%

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

   3. Simplified 74.5%

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

   5. Taylor expanded in x around 0 74.6%

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

      1. associate-*l/ 74.5%

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

      2. *-lft-identity 74.5%

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

      3. +-commutative 74.5%

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

      4. unpow2 74.5%

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

      5. fma-undefine 74.5%

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

   7. Simplified 74.5%

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

   8. Taylor expanded in F around inf 90.7%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.4 \cdot 10^{-12}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 9.5 \cdot 10^{-53}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 45.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 4.8 \cdot 10^{-293}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;B \leq 0.0025:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 4.8e-293)
   (/ (- -1.0 x) B)
   (if (<= B 0.0025)
     (- (* (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5) (/ F B)) (/ x B))
     (- (/ -1.0 B) (/ x (tan B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 4.8e-293) {
		tmp = (-1.0 - x) / B;
	} else if (B <= 0.0025) {
		tmp = (pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else {
		tmp = (-1.0 / B) - (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 (b <= 4.8d-293) then
        tmp = ((-1.0d0) - x) / b
    else if (b <= 0.0025d0) then
        tmp = ((((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)) * (f / b)) - (x / b)
    else
        tmp = ((-1.0d0) / b) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (B <= 4.8e-293) {
		tmp = (-1.0 - x) / B;
	} else if (B <= 0.0025) {
		tmp = (Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else {
		tmp = (-1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if B <= 4.8e-293:
		tmp = (-1.0 - x) / B
	elif B <= 0.0025:
		tmp = (math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B)
	else:
		tmp = (-1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 4.8e-293)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (B <= 0.0025)
		tmp = Float64(Float64((Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5) * Float64(F / B)) - Float64(x / B));
	else
		tmp = Float64(Float64(-1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (B <= 4.8e-293)
		tmp = (-1.0 - x) / B;
	elseif (B <= 0.0025)
		tmp = ((((2.0 + (F * F)) + (x * 2.0)) ^ -0.5) * (F / B)) - (x / B);
	else
		tmp = (-1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 4.8e-293], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[B, 0.0025], N[(N[(N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 4.7999999999999998e-293

   1. Initial program 74.7%

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

   3. Taylor expanded in F around -inf 54.9%

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

   4. Taylor expanded in B around 0 31.7%

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

      1. associate-*r/ 31.7%

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

      2. neg-mul-1 31.7%

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

      3. distribute-neg-in 31.7%

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

      4. metadata-eval 31.7%

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

   6. Simplified 31.7%

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

   if 4.7999999999999998e-293 < B < 0.00250000000000000005

   1. Initial program 78.2%

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

   3. Taylor expanded in B around 0 77.3%

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

   4. Taylor expanded in B around 0 77.3%

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

   if 0.00250000000000000005 < B

   1. Initial program 84.7%

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

   3. Simplified 84.7%

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

   5. Taylor expanded in x around 0 84.7%

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

      1. associate-*l/ 84.7%

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

      2. *-lft-identity 84.7%

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

      3. +-commutative 84.7%

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

      4. unpow2 84.7%

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

      5. fma-undefine 84.7%

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

   7. Simplified 84.7%

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

   8. Taylor expanded in F around -inf 57.1%

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

   9. Taylor expanded in B around 0 49.1%

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

4. Final simplification 45.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4.8 \cdot 10^{-293}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;B \leq 0.0025:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 55.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;x \leq -1.65 \cdot 10^{-176}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-148}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ -1.0 B) (/ x (tan B)))))
   (if (<= x -1.65e-176)
     t_0
     (if (<= x 2.2e-148)
       (- (/ -1.0 (sin B)) (/ x B))
       (if (<= x 2.1e-9) (/ x (- B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 / B) - (x / tan(B));
	double tmp;
	if (x <= -1.65e-176) {
		tmp = t_0;
	} else if (x <= 2.2e-148) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (x <= 2.1e-9) {
		tmp = x / -B;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-1.0d0) / b) - (x / tan(b))
    if (x <= (-1.65d-176)) then
        tmp = t_0
    else if (x <= 2.2d-148) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (x <= 2.1d-9) then
        tmp = x / -b
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (-1.0 / B) - (x / Math.tan(B));
	double tmp;
	if (x <= -1.65e-176) {
		tmp = t_0;
	} else if (x <= 2.2e-148) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (x <= 2.1e-9) {
		tmp = x / -B;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (-1.0 / B) - (x / math.tan(B))
	tmp = 0
	if x <= -1.65e-176:
		tmp = t_0
	elif x <= 2.2e-148:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif x <= 2.1e-9:
		tmp = x / -B
	else:
		tmp = t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 / B) - Float64(x / tan(B)))
	tmp = 0.0
	if (x <= -1.65e-176)
		tmp = t_0;
	elseif (x <= 2.2e-148)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (x <= 2.1e-9)
		tmp = Float64(x / Float64(-B));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (-1.0 / B) - (x / tan(B));
	tmp = 0.0;
	if (x <= -1.65e-176)
		tmp = t_0;
	elseif (x <= 2.2e-148)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (x <= 2.1e-9)
		tmp = x / -B;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.65e-176], t$95$0, If[LessEqual[x, 2.2e-148], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.1e-9], N[(x / (-B)), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.65000000000000006e-176 or 2.10000000000000019e-9 < x

   1. Initial program 79.9%

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

   3. Simplified 90.9%

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

   5. Taylor expanded in x around 0 90.9%

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

      1. associate-*l/ 90.9%

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

      2. *-lft-identity 90.9%

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

      3. +-commutative 90.9%

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

      4. unpow2 90.9%

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

      5. fma-undefine 90.9%

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

   7. Simplified 90.9%

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

   8. Taylor expanded in F around -inf 71.8%

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

   9. Taylor expanded in B around 0 78.9%

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

   if -1.65000000000000006e-176 < x < 2.20000000000000017e-148

   1. Initial program 77.8%

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

   3. Taylor expanded in B around 0 71.3%

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

   4. Taylor expanded in F around -inf 32.8%

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

      1. distribute-lft-in 32.8%

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

      2. associate-*r/ 32.8%

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

      3. metadata-eval 32.8%

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

      4. mul-1-neg 32.8%

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

      5. unsub-neg 32.8%

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

   6. Simplified 32.8%

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

   if 2.20000000000000017e-148 < x < 2.10000000000000019e-9

   1. Initial program 68.5%

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

   3. Taylor expanded in B around 0 61.0%

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

   4. Taylor expanded in x around inf 40.5%

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

      1. associate-*r/ 40.5%

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

      2. neg-mul-1 40.5%

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

   6. Simplified 40.5%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-176}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-148}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2.5 \cdot 10^{-275}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 1.65 \cdot 10^{-219}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;F \leq 4.2 \cdot 10^{-47}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ -1.0 B) (/ x (tan B)))))
   (if (<= F -2.5e-275)
     t_0
     (if (<= F 1.65e-219)
       (/ x (- B))
       (if (<= F 4.2e-47) t_0 (- (/ 1.0 (sin B)) (/ x B)))))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 / B) - (x / tan(B));
	double tmp;
	if (F <= -2.5e-275) {
		tmp = t_0;
	} else if (F <= 1.65e-219) {
		tmp = x / -B;
	} else if (F <= 4.2e-47) {
		tmp = t_0;
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-1.0d0) / b) - (x / tan(b))
    if (f <= (-2.5d-275)) then
        tmp = t_0
    else if (f <= 1.65d-219) then
        tmp = x / -b
    else if (f <= 4.2d-47) then
        tmp = t_0
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (-1.0 / B) - (x / Math.tan(B));
	double tmp;
	if (F <= -2.5e-275) {
		tmp = t_0;
	} else if (F <= 1.65e-219) {
		tmp = x / -B;
	} else if (F <= 4.2e-47) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (-1.0 / B) - (x / math.tan(B))
	tmp = 0
	if F <= -2.5e-275:
		tmp = t_0
	elif F <= 1.65e-219:
		tmp = x / -B
	elif F <= 4.2e-47:
		tmp = t_0
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 / B) - Float64(x / tan(B)))
	tmp = 0.0
	if (F <= -2.5e-275)
		tmp = t_0;
	elseif (F <= 1.65e-219)
		tmp = Float64(x / Float64(-B));
	elseif (F <= 4.2e-47)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (-1.0 / B) - (x / tan(B));
	tmp = 0.0;
	if (F <= -2.5e-275)
		tmp = t_0;
	elseif (F <= 1.65e-219)
		tmp = x / -B;
	elseif (F <= 4.2e-47)
		tmp = t_0;
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.5e-275], t$95$0, If[LessEqual[F, 1.65e-219], N[(x / (-B)), $MachinePrecision], If[LessEqual[F, 4.2e-47], t$95$0, N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.49999999999999992e-275 or 1.6500000000000001e-219 < F < 4.2000000000000001e-47

   1. Initial program 80.6%

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

   3. Simplified 86.9%

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

   5. Taylor expanded in x around 0 86.9%

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

      1. associate-*l/ 86.9%

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

      2. *-lft-identity 86.9%

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

      3. +-commutative 86.9%

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

      4. unpow2 86.9%

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

      5. fma-undefine 86.9%

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

   7. Simplified 86.9%

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

   8. Taylor expanded in F around -inf 64.5%

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

   9. Taylor expanded in B around 0 61.1%

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

   if -2.49999999999999992e-275 < F < 1.6500000000000001e-219

   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 71.9%

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

   4. Taylor expanded in x around inf 52.8%

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

      1. associate-*r/ 52.8%

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

      2. neg-mul-1 52.8%

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

   6. Simplified 52.8%

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

   if 4.2000000000000001e-47 < F

   1. Initial program 61.8%

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

   3. Taylor expanded in B around 0 37.2%

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

   4. Taylor expanded in F around inf 67.3%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.5 \cdot 10^{-275}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.65 \cdot 10^{-219}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;F \leq 4.2 \cdot 10^{-47}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 52.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.8 \cdot 10^{-184} \lor \neg \left(x \leq 3.3 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (or (<= x 3.8e-184) (not (<= x 3.3e-9)))
   (- (/ -1.0 B) (/ x (tan B)))
   (/ x (- B))))

C

double code(double F, double B, double x) {
	double tmp;
	if ((x <= 3.8e-184) || !(x <= 3.3e-9)) {
		tmp = (-1.0 / B) - (x / tan(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 ((x <= 3.8d-184) .or. (.not. (x <= 3.3d-9))) then
        tmp = ((-1.0d0) / b) - (x / tan(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 ((x <= 3.8e-184) || !(x <= 3.3e-9)) {
		tmp = (-1.0 / B) - (x / Math.tan(B));
	} else {
		tmp = x / -B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if (x <= 3.8e-184) or not (x <= 3.3e-9):
		tmp = (-1.0 / B) - (x / math.tan(B))
	else:
		tmp = x / -B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if ((x <= 3.8e-184) || !(x <= 3.3e-9))
		tmp = Float64(Float64(-1.0 / B) - Float64(x / tan(B)));
	else
		tmp = Float64(x / Float64(-B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if ((x <= 3.8e-184) || ~((x <= 3.3e-9)))
		tmp = (-1.0 / B) - (x / tan(B));
	else
		tmp = x / -B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[Or[LessEqual[x, 3.8e-184], N[Not[LessEqual[x, 3.3e-9]], $MachinePrecision]], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / (-B)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.80000000000000017e-184 or 3.30000000000000018e-9 < x

   1. Initial program 80.5%

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

   3. Simplified 88.7%

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

   5. Taylor expanded in x around 0 88.6%

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

      1. associate-*l/ 88.7%

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

      2. *-lft-identity 88.7%

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

      3. +-commutative 88.7%

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

      4. unpow2 88.7%

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

      5. fma-undefine 88.7%

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

   7. Simplified 88.7%

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

   8. Taylor expanded in F around -inf 60.0%

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

   9. Taylor expanded in B around 0 60.9%

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

   if 3.80000000000000017e-184 < x < 3.30000000000000018e-9

   1. Initial program 63.5%

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

   3. Taylor expanded in B around 0 52.6%

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

   4. Taylor expanded in x around inf 35.5%

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

      1. associate-*r/ 35.5%

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

      2. neg-mul-1 35.5%

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

   6. Simplified 35.5%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.8 \cdot 10^{-184} \lor \neg \left(x \leq 3.3 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 43.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 4.15 \cdot 10^{-293}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;B \leq 1.52 \cdot 10^{-44}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 4.15e-293)
   (/ (- -1.0 x) B)
   (if (<= B 1.52e-44)
     (/ (- (* F (sqrt 0.5)) x) B)
     (- (/ -1.0 B) (/ x (tan B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 4.15e-293) {
		tmp = (-1.0 - x) / B;
	} else if (B <= 1.52e-44) {
		tmp = ((F * sqrt(0.5)) - x) / B;
	} else {
		tmp = (-1.0 / B) - (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 (b <= 4.15d-293) then
        tmp = ((-1.0d0) - x) / b
    else if (b <= 1.52d-44) then
        tmp = ((f * sqrt(0.5d0)) - x) / b
    else
        tmp = ((-1.0d0) / b) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (B <= 4.15e-293) {
		tmp = (-1.0 - x) / B;
	} else if (B <= 1.52e-44) {
		tmp = ((F * Math.sqrt(0.5)) - x) / B;
	} else {
		tmp = (-1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if B <= 4.15e-293:
		tmp = (-1.0 - x) / B
	elif B <= 1.52e-44:
		tmp = ((F * math.sqrt(0.5)) - x) / B
	else:
		tmp = (-1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 4.15e-293)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (B <= 1.52e-44)
		tmp = Float64(Float64(Float64(F * sqrt(0.5)) - x) / B);
	else
		tmp = Float64(Float64(-1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (B <= 4.15e-293)
		tmp = (-1.0 - x) / B;
	elseif (B <= 1.52e-44)
		tmp = ((F * sqrt(0.5)) - x) / B;
	else
		tmp = (-1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 4.15e-293], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[B, 1.52e-44], N[(N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 4.14999999999999999e-293

   1. Initial program 74.7%

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

   3. Taylor expanded in F around -inf 54.9%

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

   4. Taylor expanded in B around 0 31.7%

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

      1. associate-*r/ 31.7%

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

      2. neg-mul-1 31.7%

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

      3. distribute-neg-in 31.7%

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

      4. metadata-eval 31.7%

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

   6. Simplified 31.7%

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

   if 4.14999999999999999e-293 < B < 1.51999999999999999e-44

   1. Initial program 77.6%

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

   3. Simplified 86.5%

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

   5. Taylor expanded in x around 0 86.5%

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

      1. associate-*l/ 86.5%

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

      2. *-lft-identity 86.5%

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

      3. +-commutative 86.5%

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

      4. unpow2 86.5%

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

      5. fma-undefine 86.5%

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

   7. Simplified 86.5%

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

   8. Taylor expanded in F around 0 66.8%

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

   9. Taylor expanded in B around 0 66.8%

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

       1. *-commutative 66.8%

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

   11. Simplified 66.8%

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

   if 1.51999999999999999e-44 < B

   1. Initial program 84.6%

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

   3. Simplified 84.6%

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

   5. Taylor expanded in x around 0 84.6%

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

      1. associate-*l/ 84.7%

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

      2. *-lft-identity 84.7%

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

      3. +-commutative 84.7%

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

      4. unpow2 84.7%

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

      5. fma-undefine 84.7%

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

   7. Simplified 84.7%

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

   8. Taylor expanded in F around -inf 55.6%

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

   9. Taylor expanded in B around 0 48.2%

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

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4.15 \cdot 10^{-293}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;B \leq 1.52 \cdot 10^{-44}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 38.1% accurate, 21.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -6 \cdot 10^{-16}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq 6.5 \cdot 10^{-26}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -6e-16) (/ -1.0 B) (if (<= F 6.5e-26) (/ x (- B)) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -6e-16) {
		tmp = -1.0 / B;
	} else if (F <= 6.5e-26) {
		tmp = x / -B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-6d-16)) then
        tmp = (-1.0d0) / b
    else if (f <= 6.5d-26) then
        tmp = x / -b
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -6e-16) {
		tmp = -1.0 / B;
	} else if (F <= 6.5e-26) {
		tmp = x / -B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -6e-16:
		tmp = -1.0 / B
	elif F <= 6.5e-26:
		tmp = x / -B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -6e-16)
		tmp = Float64(-1.0 / B);
	elseif (F <= 6.5e-26)
		tmp = Float64(x / Float64(-B));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -6e-16)
		tmp = -1.0 / B;
	elseif (F <= 6.5e-26)
		tmp = x / -B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -6e-16], N[(-1.0 / B), $MachinePrecision], If[LessEqual[F, 6.5e-26], N[(x / (-B)), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.99999999999999987e-16

   1. Initial program 59.0%

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

   3. Taylor expanded in F around -inf 96.5%

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

      1. log1p-expm1-u 37.6%

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

      2. +-commutative 37.6%

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

      3. add-sqr-sqrt 20.5%

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

      4. sqrt-unprod 32.1%

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

      5. div-inv 32.1%

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

      6. div-inv 32.1%

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

      7. sqr-neg 32.1%

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

      8. sqrt-unprod 11.7%

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

      9. add-sqr-sqrt 27.5%

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

   5. Applied egg-rr 27.5%

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

   6. Taylor expanded in B around 0 29.9%

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

   7. Taylor expanded in x around 0 29.9%

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

   if -5.99999999999999987e-16 < F < 6.5e-26

   1. Initial program 99.4%

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

   3. Taylor expanded in B around 0 64.6%

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

   4. Taylor expanded in x around inf 38.4%

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

      1. associate-*r/ 38.4%

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

      2. neg-mul-1 38.4%

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

   6. Simplified 38.4%

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

   if 6.5e-26 < F

   1. Initial program 59.3%

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

   3. Taylor expanded in B around 0 33.1%

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

   4. Taylor expanded in B around 0 33.2%

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

   5. Taylor expanded in F around inf 48.5%

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

4. Final simplification 38.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6 \cdot 10^{-16}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq 6.5 \cdot 10^{-26}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 45.0% accurate, 21.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.18 \cdot 10^{-66}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 4.1 \cdot 10^{-29}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.18e-66)
   (/ (- -1.0 x) B)
   (if (<= F 4.1e-29) (/ x (- B)) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.18e-66) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 4.1e-29) {
		tmp = x / -B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.18d-66)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 4.1d-29) then
        tmp = x / -b
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.18e-66) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 4.1e-29) {
		tmp = x / -B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.18e-66:
		tmp = (-1.0 - x) / B
	elif F <= 4.1e-29:
		tmp = x / -B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.18e-66)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 4.1e-29)
		tmp = Float64(x / Float64(-B));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.18e-66)
		tmp = (-1.0 - x) / B;
	elseif (F <= 4.1e-29)
		tmp = x / -B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.18e-66], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 4.1e-29], N[(x / (-B)), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.18e-66

   1. Initial program 64.6%

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

   3. Taylor expanded in F around -inf 87.2%

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

   4. Taylor expanded in B around 0 42.6%

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

      1. associate-*r/ 42.6%

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

      2. neg-mul-1 42.6%

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

      3. distribute-neg-in 42.6%

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

      4. metadata-eval 42.6%

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

   6. Simplified 42.6%

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

   if -1.18e-66 < F < 4.0999999999999998e-29

   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 62.2%

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

   4. Taylor expanded in x around inf 39.6%

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

      1. associate-*r/ 39.6%

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

      2. neg-mul-1 39.6%

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

   6. Simplified 39.6%

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

   if 4.0999999999999998e-29 < F

   1. Initial program 59.3%

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

   3. Taylor expanded in B around 0 33.1%

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

   4. Taylor expanded in B around 0 33.2%

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

   5. Taylor expanded in F around inf 48.5%

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

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.18 \cdot 10^{-66}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 4.1 \cdot 10^{-29}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 30.9% accurate, 35.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -6 \cdot 10^{-16}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (if (<= F -6e-16) (/ -1.0 B) (/ x (- B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -6e-16) {
		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 <= (-6d-16)) 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 <= -6e-16) {
		tmp = -1.0 / B;
	} else {
		tmp = x / -B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -6e-16:
		tmp = -1.0 / B
	else:
		tmp = x / -B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -6e-16)
		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 <= -6e-16)
		tmp = -1.0 / B;
	else
		tmp = x / -B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -6e-16], N[(-1.0 / B), $MachinePrecision], N[(x / (-B)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -5.99999999999999987e-16

   1. Initial program 59.0%

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

   3. Taylor expanded in F around -inf 96.5%

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

      1. log1p-expm1-u 37.6%

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

      2. +-commutative 37.6%

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

      3. add-sqr-sqrt 20.5%

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

      4. sqrt-unprod 32.1%

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

      5. div-inv 32.1%

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

      6. div-inv 32.1%

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

      7. sqr-neg 32.1%

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

      8. sqrt-unprod 11.7%

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

      9. add-sqr-sqrt 27.5%

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

   5. Applied egg-rr 27.5%

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

   6. Taylor expanded in B around 0 29.9%

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

   7. Taylor expanded in x around 0 29.9%

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

   if -5.99999999999999987e-16 < F

   1. Initial program 86.0%

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

   3. Taylor expanded in B around 0 54.1%

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

   4. Taylor expanded in x around inf 34.0%

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

      1. associate-*r/ 34.0%

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

      2. neg-mul-1 34.0%

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

   6. Simplified 34.0%

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

4. Final simplification 32.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6 \cdot 10^{-16}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 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 78.2%

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

3. Taylor expanded in F around -inf 54.9%

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

   1. log1p-expm1-u 20.3%

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

   2. +-commutative 20.3%

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

   3. add-sqr-sqrt 10.3%

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

   4. sqrt-unprod 14.5%

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

   5. div-inv 14.5%

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

   6. div-inv 14.5%

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

   7. sqr-neg 14.5%

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

   8. sqrt-unprod 4.1%

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

   9. add-sqr-sqrt 9.2%

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

5. Applied egg-rr 9.2%

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

6. Taylor expanded in B around 0 10.8%

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

7. Taylor expanded in x around 0 11.4%

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

8. Final simplification 11.4%

   \[\leadsto \frac{-1}{B} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(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))))))