VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.7% → 99.7%

Time: 29.3s

Alternatives: 26

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.7% 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: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2 \cdot 10^{+40}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 130000000:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -2e+40)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 130000000.0)
       (- (* F (/ (sqrt (/ 1.0 (fma F F 2.0))) (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 <= -2e+40) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 130000000.0) {
		tmp = (F * (sqrt((1.0 / fma(F, F, 2.0))) / sin(B))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -2e+40)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 130000000.0)
		tmp = Float64(Float64(F * Float64(sqrt(Float64(1.0 / fma(F, F, 2.0))) / sin(B))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2e+40], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 130000000.0], N[(N[(F * N[(N[Sqrt[N[(1.0 / N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $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 < -2.00000000000000006e40

   1. Initial program 56.4%

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

      1. distribute-lft-neg-in 56.4%

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

      2. +-commutative 56.4%

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

      3. cancel-sign-sub-inv 56.4%

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

   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. Taylor expanded in x around 0 74.6%

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

      1. associate-*l/ 74.6%

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

      2. *-lft-identity 74.6%

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

      3. +-commutative 74.6%

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

      4. unpow2 74.6%

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

      5. fma-udef 74.6%

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

   6. Simplified 74.6%

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

   7. Taylor expanded in F around -inf 99.8%

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

   if -2.00000000000000006e40 < F < 1.3e8

   1. Initial program 99.4%

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

      1. distribute-lft-neg-in 99.4%

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

      2. +-commutative 99.4%

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

      3. cancel-sign-sub-inv 99.4%

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

   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. Taylor expanded in x around 0 99.6%

      \[\leadsto F \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)} - \frac{x}{\tan B} \]
   5. 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-udef 99.6%

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

   6. Simplified 99.6%

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

   if 1.3e8 < F

   1. Initial program 56.7%

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

      1. distribute-lft-neg-in 56.7%

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

      2. +-commutative 56.7%

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

      3. cancel-sign-sub-inv 56.7%

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

   3. Simplified 75.2%

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

   4. Taylor expanded in x around 0 75.1%

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

      1. associate-*l/ 75.1%

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

      2. *-lft-identity 75.1%

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

      3. +-commutative 75.1%

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

      4. unpow2 75.1%

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

      5. fma-udef 75.1%

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

   6. Simplified 75.1%

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

   7. Taylor expanded in F around inf 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{+40}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 130000000:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 2: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1150000000:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 100:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} + x \cdot \frac{-1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1150000000.0)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 100.0)
       (+
        (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5))
        (* x (/ -1.0 (tan 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 <= -1150000000.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 100.0) {
		tmp = ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) + (x * (-1.0 / tan(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 <= (-1150000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 100.0d0) then
        tmp = ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0))) + (x * ((-1.0d0) / tan(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 <= -1150000000.0) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 100.0) {
		tmp = ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) + (x * (-1.0 / Math.tan(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 <= -1150000000.0:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 100.0:
		tmp = ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) + (x * (-1.0 / math.tan(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 <= -1150000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 100.0)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)) + Float64(x * Float64(-1.0 / tan(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 <= -1150000000.0)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 100.0)
		tmp = ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5)) + (x * (-1.0 / tan(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, -1150000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 100.0], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] + N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $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 < -1.15e9

   1. Initial program 59.7%

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

      1. distribute-lft-neg-in 59.7%

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

      2. +-commutative 59.7%

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

      3. cancel-sign-sub-inv 59.7%

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

   3. Simplified 76.5%

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

   4. Taylor expanded in x around 0 76.6%

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

      1. associate-*l/ 76.5%

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

      2. *-lft-identity 76.5%

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

      3. +-commutative 76.5%

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

      4. unpow2 76.5%

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

      5. fma-udef 76.5%

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

   6. Simplified 76.5%

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

   7. Taylor expanded in F around -inf 99.8%

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

   if -1.15e9 < F < 100

   1. Initial program 99.4%

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

   if 100 < F

   1. Initial program 56.7%

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

      1. distribute-lft-neg-in 56.7%

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

      2. +-commutative 56.7%

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

      3. cancel-sign-sub-inv 56.7%

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

   3. Simplified 75.2%

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

   4. Taylor expanded in x around 0 75.1%

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

      1. associate-*l/ 75.1%

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

      2. *-lft-identity 75.1%

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

      3. +-commutative 75.1%

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

      4. unpow2 75.1%

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

      5. fma-udef 75.1%

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

   6. Simplified 75.1%

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

   7. Taylor expanded in F around inf 99.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1150000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 100:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} + x \cdot \frac{-1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 3: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1100000000:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 22500000:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1100000000.0)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 22500000.0)
       (+
        (/ -1.0 (/ (tan B) x))
        (* (/ 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 <= -1100000000.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 22500000.0) {
		tmp = (-1.0 / (tan(B) / x)) + ((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 <= (-1100000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 22500000.0d0) then
        tmp = ((-1.0d0) / (tan(b) / x)) + ((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 <= -1100000000.0) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 22500000.0) {
		tmp = (-1.0 / (Math.tan(B) / x)) + ((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 <= -1100000000.0:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 22500000.0:
		tmp = (-1.0 / (math.tan(B) / x)) + ((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 <= -1100000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 22500000.0)
		tmp = Float64(Float64(-1.0 / Float64(tan(B) / x)) + 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 <= -1100000000.0)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 22500000.0)
		tmp = (-1.0 / (tan(B) / x)) + ((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, -1100000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 22500000.0], N[(N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $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 < -1.1e9

   1. Initial program 59.7%

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

      1. distribute-lft-neg-in 59.7%

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

      2. +-commutative 59.7%

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

      3. cancel-sign-sub-inv 59.7%

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

   3. Simplified 76.5%

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

   4. Taylor expanded in x around 0 76.6%

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

      1. associate-*l/ 76.5%

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

      2. *-lft-identity 76.5%

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

      3. +-commutative 76.5%

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

      4. unpow2 76.5%

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

      5. fma-udef 76.5%

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

   6. Simplified 76.5%

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

   7. Taylor expanded in F around -inf 99.8%

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

   if -1.1e9 < F < 2.25e7

   1. Initial program 99.4%

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

      1. div-inv 99.6%

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

      2. clear-num 99.5%

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

   3. Applied egg-rr 99.5%

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

   if 2.25e7 < F

   1. Initial program 56.7%

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

      1. distribute-lft-neg-in 56.7%

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

      2. +-commutative 56.7%

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

      3. cancel-sign-sub-inv 56.7%

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

   3. Simplified 75.2%

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

   4. Taylor expanded in x around 0 75.1%

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

      1. associate-*l/ 75.1%

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

      2. *-lft-identity 75.1%

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

      3. +-commutative 75.1%

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

      4. unpow2 75.1%

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

      5. fma-udef 75.1%

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

   6. Simplified 75.1%

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

   7. Taylor expanded in F around inf 99.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1100000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 22500000:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 4: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1:\\ \;\;\;\;F \cdot \frac{\frac{1}{\frac{-1}{F} - F}}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{-20}:\\ \;\;\;\;F \cdot \left(\frac{1}{\sin B} \cdot \sqrt{0.5 + \left(F \cdot F\right) \cdot -0.25}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{F}{F + \frac{1}{F}}}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.0)
     (- (* F (/ (/ 1.0 (- (/ -1.0 F) F)) (sin B))) t_0)
     (if (<= F 5.2e-20)
       (- (* F (* (/ 1.0 (sin B)) (sqrt (+ 0.5 (* (* F F) -0.25))))) t_0)
       (- (/ (/ F (+ F (/ 1.0 F))) (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.0) {
		tmp = (F * ((1.0 / ((-1.0 / F) - F)) / sin(B))) - t_0;
	} else if (F <= 5.2e-20) {
		tmp = (F * ((1.0 / sin(B)) * sqrt((0.5 + ((F * F) * -0.25))))) - t_0;
	} else {
		tmp = ((F / (F + (1.0 / F))) / 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 <= (-1.0d0)) then
        tmp = (f * ((1.0d0 / (((-1.0d0) / f) - f)) / sin(b))) - t_0
    else if (f <= 5.2d-20) then
        tmp = (f * ((1.0d0 / sin(b)) * sqrt((0.5d0 + ((f * f) * (-0.25d0)))))) - t_0
    else
        tmp = ((f / (f + (1.0d0 / f))) / 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 <= -1.0) {
		tmp = (F * ((1.0 / ((-1.0 / F) - F)) / Math.sin(B))) - t_0;
	} else if (F <= 5.2e-20) {
		tmp = (F * ((1.0 / Math.sin(B)) * Math.sqrt((0.5 + ((F * F) * -0.25))))) - t_0;
	} else {
		tmp = ((F / (F + (1.0 / F))) / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.0:
		tmp = (F * ((1.0 / ((-1.0 / F) - F)) / math.sin(B))) - t_0
	elif F <= 5.2e-20:
		tmp = (F * ((1.0 / math.sin(B)) * math.sqrt((0.5 + ((F * F) * -0.25))))) - t_0
	else:
		tmp = ((F / (F + (1.0 / F))) / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.0)
		tmp = Float64(Float64(F * Float64(Float64(1.0 / Float64(Float64(-1.0 / F) - F)) / sin(B))) - t_0);
	elseif (F <= 5.2e-20)
		tmp = Float64(Float64(F * Float64(Float64(1.0 / sin(B)) * sqrt(Float64(0.5 + Float64(Float64(F * F) * -0.25))))) - t_0);
	else
		tmp = Float64(Float64(Float64(F / Float64(F + Float64(1.0 / F))) / 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 <= -1.0)
		tmp = (F * ((1.0 / ((-1.0 / F) - F)) / sin(B))) - t_0;
	elseif (F <= 5.2e-20)
		tmp = (F * ((1.0 / sin(B)) * sqrt((0.5 + ((F * F) * -0.25))))) - t_0;
	else
		tmp = ((F / (F + (1.0 / F))) / 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, -1.0], N[(N[(F * N[(N[(1.0 / N[(N[(-1.0 / F), $MachinePrecision] - F), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 5.2e-20], N[(N[(F * N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(0.5 + N[(N[(F * F), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[(F / N[(F + N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1

   1. Initial program 60.9%

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

      1. distribute-lft-neg-in 60.9%

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

      2. +-commutative 60.9%

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

      3. cancel-sign-sub-inv 60.9%

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

   3. Simplified 77.2%

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

   4. Taylor expanded in x around 0 77.3%

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

      1. associate-*l/ 77.2%

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

      2. *-lft-identity 77.2%

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

      3. +-commutative 77.2%

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

      4. unpow2 77.2%

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

      5. fma-udef 77.2%

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

   6. Simplified 77.2%

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

      1. sqrt-div 77.2%

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

      2. metadata-eval 77.2%

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

   8. Applied egg-rr 77.2%

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

   9. Taylor expanded in F around -inf 99.6%

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

       1. mul-1-neg 99.6%

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

   11. Simplified 99.6%

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

   if -1 < F < 5.1999999999999999e-20

   1. Initial program 99.4%

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

      1. distribute-lft-neg-in 99.4%

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

      2. +-commutative 99.4%

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

      3. cancel-sign-sub-inv 99.4%

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

   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. Taylor expanded in x around 0 99.6%

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

      1. *-commutative 99.6%

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

      2. +-commutative 99.6%

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

      3. unpow2 99.6%

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

      4. fma-udef 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in F around 0 99.6%

      \[\leadsto F \cdot \left(\sqrt{\color{blue}{0.5 + -0.25 \cdot {F}^{2}}} \cdot \frac{1}{\sin B}\right) - \frac{x}{\tan B} \]
   8. Step-by-step derivation

      1. *-commutative 99.6%

         \[\leadsto F \cdot \left(\sqrt{0.5 + \color{blue}{{F}^{2} \cdot -0.25}} \cdot \frac{1}{\sin B}\right) - \frac{x}{\tan B} \]

      2. unpow2 99.6%

         \[\leadsto F \cdot \left(\sqrt{0.5 + \color{blue}{\left(F \cdot F\right)} \cdot -0.25} \cdot \frac{1}{\sin B}\right) - \frac{x}{\tan B} \]

   9. Simplified 99.6%

      \[\leadsto F \cdot \left(\sqrt{\color{blue}{0.5 + \left(F \cdot F\right) \cdot -0.25}} \cdot \frac{1}{\sin B}\right) - \frac{x}{\tan B} \]

   if 5.1999999999999999e-20 < F

   1. Initial program 60.5%

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

      1. distribute-lft-neg-in 60.5%

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

      2. +-commutative 60.5%

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

      3. cancel-sign-sub-inv 60.5%

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

   3. Simplified 77.4%

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

   4. Taylor expanded in x around 0 77.3%

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

      1. associate-*l/ 77.3%

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

      2. *-lft-identity 77.3%

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

      3. +-commutative 77.3%

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

      4. unpow2 77.3%

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

      5. fma-udef 77.3%

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

   6. Simplified 77.3%

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

      1. sqrt-div 77.3%

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

      2. metadata-eval 77.3%

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

   8. Applied egg-rr 77.3%

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

   9. Taylor expanded in F around inf 99.0%

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

   10. Taylor expanded in B around inf 99.2%

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

       1. associate-/l/ 99.2%

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

   12. Simplified 99.2%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1:\\ \;\;\;\;F \cdot \frac{\frac{1}{\frac{-1}{F} - F}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{-20}:\\ \;\;\;\;F \cdot \left(\frac{1}{\sin B} \cdot \sqrt{0.5 + \left(F \cdot F\right) \cdot -0.25}\right) - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{F}{F + \frac{1}{F}}}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 5: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.91:\\ \;\;\;\;F \cdot \frac{\frac{1}{\frac{-1}{F} - F}}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{-20}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{F}{F + \frac{1}{F}}}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -0.91)
     (- (* F (/ (/ 1.0 (- (/ -1.0 F) F)) (sin B))) t_0)
     (if (<= F 5.2e-20)
       (- (* F (/ (sqrt 0.5) (sin B))) t_0)
       (- (/ (/ F (+ F (/ 1.0 F))) (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -0.91) {
		tmp = (F * ((1.0 / ((-1.0 / F) - F)) / sin(B))) - t_0;
	} else if (F <= 5.2e-20) {
		tmp = (F * (sqrt(0.5) / sin(B))) - t_0;
	} else {
		tmp = ((F / (F + (1.0 / F))) / 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 <= (-0.91d0)) then
        tmp = (f * ((1.0d0 / (((-1.0d0) / f) - f)) / sin(b))) - t_0
    else if (f <= 5.2d-20) then
        tmp = (f * (sqrt(0.5d0) / sin(b))) - t_0
    else
        tmp = ((f / (f + (1.0d0 / f))) / 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 <= -0.91) {
		tmp = (F * ((1.0 / ((-1.0 / F) - F)) / Math.sin(B))) - t_0;
	} else if (F <= 5.2e-20) {
		tmp = (F * (Math.sqrt(0.5) / Math.sin(B))) - t_0;
	} else {
		tmp = ((F / (F + (1.0 / F))) / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -0.91:
		tmp = (F * ((1.0 / ((-1.0 / F) - F)) / math.sin(B))) - t_0
	elif F <= 5.2e-20:
		tmp = (F * (math.sqrt(0.5) / math.sin(B))) - t_0
	else:
		tmp = ((F / (F + (1.0 / F))) / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -0.91)
		tmp = Float64(Float64(F * Float64(Float64(1.0 / Float64(Float64(-1.0 / F) - F)) / sin(B))) - t_0);
	elseif (F <= 5.2e-20)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / sin(B))) - t_0);
	else
		tmp = Float64(Float64(Float64(F / Float64(F + Float64(1.0 / F))) / 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 <= -0.91)
		tmp = (F * ((1.0 / ((-1.0 / F) - F)) / sin(B))) - t_0;
	elseif (F <= 5.2e-20)
		tmp = (F * (sqrt(0.5) / sin(B))) - t_0;
	else
		tmp = ((F / (F + (1.0 / F))) / 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, -0.91], N[(N[(F * N[(N[(1.0 / N[(N[(-1.0 / F), $MachinePrecision] - F), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 5.2e-20], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[(F / N[(F + N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.910000000000000031

   1. Initial program 60.9%

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

      1. distribute-lft-neg-in 60.9%

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

      2. +-commutative 60.9%

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

      3. cancel-sign-sub-inv 60.9%

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

   3. Simplified 77.2%

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

   4. Taylor expanded in x around 0 77.3%

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

      1. associate-*l/ 77.2%

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

      2. *-lft-identity 77.2%

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

      3. +-commutative 77.2%

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

      4. unpow2 77.2%

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

      5. fma-udef 77.2%

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

   6. Simplified 77.2%

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

      1. sqrt-div 77.2%

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

      2. metadata-eval 77.2%

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

   8. Applied egg-rr 77.2%

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

   9. Taylor expanded in F around -inf 99.6%

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

       1. mul-1-neg 99.6%

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

   11. Simplified 99.6%

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

   if -0.910000000000000031 < F < 5.1999999999999999e-20

   1. Initial program 99.4%

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

      1. distribute-lft-neg-in 99.4%

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

      2. +-commutative 99.4%

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

      3. cancel-sign-sub-inv 99.4%

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

   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. Taylor expanded in x around 0 99.6%

      \[\leadsto F \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)} - \frac{x}{\tan B} \]
   5. 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-udef 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in F around 0 99.5%

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

   if 5.1999999999999999e-20 < F

   1. Initial program 60.5%

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

      1. distribute-lft-neg-in 60.5%

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

      2. +-commutative 60.5%

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

      3. cancel-sign-sub-inv 60.5%

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

   3. Simplified 77.4%

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

   4. Taylor expanded in x around 0 77.3%

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

      1. associate-*l/ 77.3%

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

      2. *-lft-identity 77.3%

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

      3. +-commutative 77.3%

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

      4. unpow2 77.3%

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

      5. fma-udef 77.3%

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

   6. Simplified 77.3%

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

      1. sqrt-div 77.3%

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

      2. metadata-eval 77.3%

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

   8. Applied egg-rr 77.3%

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

   9. Taylor expanded in F around inf 99.0%

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

   10. Taylor expanded in B around inf 99.2%

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

       1. associate-/l/ 99.2%

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

   12. Simplified 99.2%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.91:\\ \;\;\;\;F \cdot \frac{\frac{1}{\frac{-1}{F} - F}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{-20}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{F}{F + \frac{1}{F}}}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 6: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.91:\\ \;\;\;\;F \cdot \frac{\frac{1}{\frac{-1}{F} - F}}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{-20}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{F}{F + \frac{1}{F}}}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -0.91)
     (- (* F (/ (/ 1.0 (- (/ -1.0 F) F)) (sin B))) t_0)
     (if (<= F 5.2e-20)
       (- (/ (* F (sqrt 0.5)) (sin B)) t_0)
       (- (/ (/ F (+ F (/ 1.0 F))) (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -0.91) {
		tmp = (F * ((1.0 / ((-1.0 / F) - F)) / sin(B))) - t_0;
	} else if (F <= 5.2e-20) {
		tmp = ((F * sqrt(0.5)) / sin(B)) - t_0;
	} else {
		tmp = ((F / (F + (1.0 / F))) / 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 <= (-0.91d0)) then
        tmp = (f * ((1.0d0 / (((-1.0d0) / f) - f)) / sin(b))) - t_0
    else if (f <= 5.2d-20) then
        tmp = ((f * sqrt(0.5d0)) / sin(b)) - t_0
    else
        tmp = ((f / (f + (1.0d0 / f))) / 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 <= -0.91) {
		tmp = (F * ((1.0 / ((-1.0 / F) - F)) / Math.sin(B))) - t_0;
	} else if (F <= 5.2e-20) {
		tmp = ((F * Math.sqrt(0.5)) / Math.sin(B)) - t_0;
	} else {
		tmp = ((F / (F + (1.0 / F))) / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -0.91:
		tmp = (F * ((1.0 / ((-1.0 / F) - F)) / math.sin(B))) - t_0
	elif F <= 5.2e-20:
		tmp = ((F * math.sqrt(0.5)) / math.sin(B)) - t_0
	else:
		tmp = ((F / (F + (1.0 / F))) / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -0.91)
		tmp = Float64(Float64(F * Float64(Float64(1.0 / Float64(Float64(-1.0 / F) - F)) / sin(B))) - t_0);
	elseif (F <= 5.2e-20)
		tmp = Float64(Float64(Float64(F * sqrt(0.5)) / sin(B)) - t_0);
	else
		tmp = Float64(Float64(Float64(F / Float64(F + Float64(1.0 / F))) / 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 <= -0.91)
		tmp = (F * ((1.0 / ((-1.0 / F) - F)) / sin(B))) - t_0;
	elseif (F <= 5.2e-20)
		tmp = ((F * sqrt(0.5)) / sin(B)) - t_0;
	else
		tmp = ((F / (F + (1.0 / F))) / 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, -0.91], N[(N[(F * N[(N[(1.0 / N[(N[(-1.0 / F), $MachinePrecision] - F), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 5.2e-20], N[(N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[(F / N[(F + N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.910000000000000031

   1. Initial program 60.9%

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

      1. distribute-lft-neg-in 60.9%

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

      2. +-commutative 60.9%

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

      3. cancel-sign-sub-inv 60.9%

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

   3. Simplified 77.2%

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

   4. Taylor expanded in x around 0 77.3%

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

      1. associate-*l/ 77.2%

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

      2. *-lft-identity 77.2%

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

      3. +-commutative 77.2%

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

      4. unpow2 77.2%

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

      5. fma-udef 77.2%

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

   6. Simplified 77.2%

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

      1. sqrt-div 77.2%

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

      2. metadata-eval 77.2%

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

   8. Applied egg-rr 77.2%

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

   9. Taylor expanded in F around -inf 99.6%

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

       1. mul-1-neg 99.6%

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

   11. Simplified 99.6%

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

   if -0.910000000000000031 < F < 5.1999999999999999e-20

   1. Initial program 99.4%

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

      1. distribute-lft-neg-in 99.4%

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

      2. +-commutative 99.4%

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

      3. cancel-sign-sub-inv 99.4%

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

   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. Taylor expanded in x around 0 99.6%

      \[\leadsto F \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)} - \frac{x}{\tan B} \]
   5. 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-udef 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in F around 0 99.5%

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

      1. *-commutative 99.5%

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

   9. Simplified 99.5%

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

   if 5.1999999999999999e-20 < F

   1. Initial program 60.5%

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

      1. distribute-lft-neg-in 60.5%

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

      2. +-commutative 60.5%

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

      3. cancel-sign-sub-inv 60.5%

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

   3. Simplified 77.4%

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

   4. Taylor expanded in x around 0 77.3%

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

      1. associate-*l/ 77.3%

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

      2. *-lft-identity 77.3%

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

      3. +-commutative 77.3%

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

      4. unpow2 77.3%

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

      5. fma-udef 77.3%

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

   6. Simplified 77.3%

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

      1. sqrt-div 77.3%

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

      2. metadata-eval 77.3%

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

   8. Applied egg-rr 77.3%

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

   9. Taylor expanded in F around inf 99.0%

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

   10. Taylor expanded in B around inf 99.2%

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

       1. associate-/l/ 99.2%

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

   12. Simplified 99.2%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.91:\\ \;\;\;\;F \cdot \frac{\frac{1}{\frac{-1}{F} - F}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{-20}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{F}{F + \frac{1}{F}}}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 7: 92.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -9.6 \cdot 10^{-7}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{-20}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{B}{F}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{F}{F + \frac{1}{F}}}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -9.6e-7)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 5.2e-20)
       (- (/ (sqrt 0.5) (/ B F)) t_0)
       (- (/ (/ F (+ F (/ 1.0 F))) (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -9.6e-7) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 5.2e-20) {
		tmp = (sqrt(0.5) / (B / F)) - t_0;
	} else {
		tmp = ((F / (F + (1.0 / F))) / 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 <= (-9.6d-7)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 5.2d-20) then
        tmp = (sqrt(0.5d0) / (b / f)) - t_0
    else
        tmp = ((f / (f + (1.0d0 / f))) / 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 <= -9.6e-7) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 5.2e-20) {
		tmp = (Math.sqrt(0.5) / (B / F)) - t_0;
	} else {
		tmp = ((F / (F + (1.0 / F))) / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -9.6e-7:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 5.2e-20:
		tmp = (math.sqrt(0.5) / (B / F)) - t_0
	else:
		tmp = ((F / (F + (1.0 / F))) / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -9.6e-7)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 5.2e-20)
		tmp = Float64(Float64(sqrt(0.5) / Float64(B / F)) - t_0);
	else
		tmp = Float64(Float64(Float64(F / Float64(F + Float64(1.0 / F))) / 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 <= -9.6e-7)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 5.2e-20)
		tmp = (sqrt(0.5) / (B / F)) - t_0;
	else
		tmp = ((F / (F + (1.0 / F))) / 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, -9.6e-7], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 5.2e-20], N[(N[(N[Sqrt[0.5], $MachinePrecision] / N[(B / F), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[(F / N[(F + N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -9.59999999999999914e-7

   1. Initial program 61.4%

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

      1. distribute-lft-neg-in 61.4%

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

      2. +-commutative 61.4%

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

      3. cancel-sign-sub-inv 61.4%

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

   3. Simplified 77.5%

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

   4. Taylor expanded in x around 0 77.6%

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

      1. associate-*l/ 77.5%

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

      2. *-lft-identity 77.5%

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

      3. +-commutative 77.5%

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

      4. unpow2 77.5%

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

      5. fma-udef 77.5%

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

   6. Simplified 77.5%

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

   7. Taylor expanded in F around -inf 98.0%

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

   if -9.59999999999999914e-7 < F < 5.1999999999999999e-20

   1. Initial program 99.4%

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

      1. distribute-lft-neg-in 99.4%

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

      2. +-commutative 99.4%

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

      3. cancel-sign-sub-inv 99.4%

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

   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. Taylor expanded in x around 0 99.6%

      \[\leadsto F \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)} - \frac{x}{\tan B} \]
   5. 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-udef 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in F around 0 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-/l* 99.6%

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

   9. Simplified 99.6%

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

   10. Taylor expanded in B around 0 87.6%

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

   if 5.1999999999999999e-20 < F

   1. Initial program 60.5%

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

      1. distribute-lft-neg-in 60.5%

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

      2. +-commutative 60.5%

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

      3. cancel-sign-sub-inv 60.5%

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

   3. Simplified 77.4%

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

   4. Taylor expanded in x around 0 77.3%

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

      1. associate-*l/ 77.3%

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

      2. *-lft-identity 77.3%

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

      3. +-commutative 77.3%

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

      4. unpow2 77.3%

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

      5. fma-udef 77.3%

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

   6. Simplified 77.3%

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

      1. sqrt-div 77.3%

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

      2. metadata-eval 77.3%

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

   8. Applied egg-rr 77.3%

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

   9. Taylor expanded in F around inf 99.0%

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

   10. Taylor expanded in B around inf 99.2%

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

       1. associate-/l/ 99.2%

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

   12. Simplified 99.2%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -9.6 \cdot 10^{-7}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{-20}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{B}{F}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{F}{F + \frac{1}{F}}}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 8: 92.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -8.4 \cdot 10^{-7}:\\ \;\;\;\;F \cdot \frac{\frac{1}{\frac{-1}{F} - F}}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{-20}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{B}{F}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{F}{F + \frac{1}{F}}}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -8.4e-7)
     (- (* F (/ (/ 1.0 (- (/ -1.0 F) F)) (sin B))) t_0)
     (if (<= F 5.2e-20)
       (- (/ (sqrt 0.5) (/ B F)) t_0)
       (- (/ (/ F (+ F (/ 1.0 F))) (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -8.4e-7) {
		tmp = (F * ((1.0 / ((-1.0 / F) - F)) / sin(B))) - t_0;
	} else if (F <= 5.2e-20) {
		tmp = (sqrt(0.5) / (B / F)) - t_0;
	} else {
		tmp = ((F / (F + (1.0 / F))) / 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 <= (-8.4d-7)) then
        tmp = (f * ((1.0d0 / (((-1.0d0) / f) - f)) / sin(b))) - t_0
    else if (f <= 5.2d-20) then
        tmp = (sqrt(0.5d0) / (b / f)) - t_0
    else
        tmp = ((f / (f + (1.0d0 / f))) / 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 <= -8.4e-7) {
		tmp = (F * ((1.0 / ((-1.0 / F) - F)) / Math.sin(B))) - t_0;
	} else if (F <= 5.2e-20) {
		tmp = (Math.sqrt(0.5) / (B / F)) - t_0;
	} else {
		tmp = ((F / (F + (1.0 / F))) / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -8.4e-7:
		tmp = (F * ((1.0 / ((-1.0 / F) - F)) / math.sin(B))) - t_0
	elif F <= 5.2e-20:
		tmp = (math.sqrt(0.5) / (B / F)) - t_0
	else:
		tmp = ((F / (F + (1.0 / F))) / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -8.4e-7)
		tmp = Float64(Float64(F * Float64(Float64(1.0 / Float64(Float64(-1.0 / F) - F)) / sin(B))) - t_0);
	elseif (F <= 5.2e-20)
		tmp = Float64(Float64(sqrt(0.5) / Float64(B / F)) - t_0);
	else
		tmp = Float64(Float64(Float64(F / Float64(F + Float64(1.0 / F))) / 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 <= -8.4e-7)
		tmp = (F * ((1.0 / ((-1.0 / F) - F)) / sin(B))) - t_0;
	elseif (F <= 5.2e-20)
		tmp = (sqrt(0.5) / (B / F)) - t_0;
	else
		tmp = ((F / (F + (1.0 / F))) / 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, -8.4e-7], N[(N[(F * N[(N[(1.0 / N[(N[(-1.0 / F), $MachinePrecision] - F), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 5.2e-20], N[(N[(N[Sqrt[0.5], $MachinePrecision] / N[(B / F), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[(F / N[(F + N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -8.4e-7

   1. Initial program 61.4%

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

      1. distribute-lft-neg-in 61.4%

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

      2. +-commutative 61.4%

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

      3. cancel-sign-sub-inv 61.4%

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

   3. Simplified 77.5%

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

   4. Taylor expanded in x around 0 77.6%

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

      1. associate-*l/ 77.5%

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

      2. *-lft-identity 77.5%

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

      3. +-commutative 77.5%

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

      4. unpow2 77.5%

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

      5. fma-udef 77.5%

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

   6. Simplified 77.5%

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

      1. sqrt-div 77.5%

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

      2. metadata-eval 77.5%

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

   8. Applied egg-rr 77.5%

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

   9. Taylor expanded in F around -inf 98.3%

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

       1. mul-1-neg 98.3%

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

   11. Simplified 98.3%

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

   if -8.4e-7 < F < 5.1999999999999999e-20

   1. Initial program 99.4%

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

      1. distribute-lft-neg-in 99.4%

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

      2. +-commutative 99.4%

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

      3. cancel-sign-sub-inv 99.4%

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

   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. Taylor expanded in x around 0 99.6%

      \[\leadsto F \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)} - \frac{x}{\tan B} \]
   5. 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-udef 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in F around 0 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-/l* 99.6%

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

   9. Simplified 99.6%

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

   10. Taylor expanded in B around 0 87.6%

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

   if 5.1999999999999999e-20 < F

   1. Initial program 60.5%

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

      1. distribute-lft-neg-in 60.5%

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

      2. +-commutative 60.5%

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

      3. cancel-sign-sub-inv 60.5%

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

   3. Simplified 77.4%

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

   4. Taylor expanded in x around 0 77.3%

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

      1. associate-*l/ 77.3%

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

      2. *-lft-identity 77.3%

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

      3. +-commutative 77.3%

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

      4. unpow2 77.3%

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

      5. fma-udef 77.3%

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

   6. Simplified 77.3%

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

      1. sqrt-div 77.3%

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

      2. metadata-eval 77.3%

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

   8. Applied egg-rr 77.3%

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

   9. Taylor expanded in F around inf 99.0%

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

   10. Taylor expanded in B around inf 99.2%

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

       1. associate-/l/ 99.2%

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

   12. Simplified 99.2%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -8.4 \cdot 10^{-7}:\\ \;\;\;\;F \cdot \frac{\frac{1}{\frac{-1}{F} - F}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{-20}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{B}{F}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{F}{F + \frac{1}{F}}}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 9: 91.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -9.6 \cdot 10^{-7}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{-19}:\\ \;\;\;\;\frac{F}{\frac{B}{\sqrt{0.5}}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -9.6e-7)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.4e-19)
       (- (/ F (/ 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 <= -9.6e-7) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.4e-19) {
		tmp = (F / (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 <= (-9.6d-7)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.4d-19) then
        tmp = (f / (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 <= -9.6e-7) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.4e-19) {
		tmp = (F / (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 <= -9.6e-7:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.4e-19:
		tmp = (F / (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 <= -9.6e-7)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.4e-19)
		tmp = Float64(Float64(F / Float64(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 <= -9.6e-7)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.4e-19)
		tmp = (F / (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, -9.6e-7], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.4e-19], N[(N[(F / N[(B / 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 < -9.59999999999999914e-7

   1. Initial program 61.4%

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

      1. distribute-lft-neg-in 61.4%

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

      2. +-commutative 61.4%

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

      3. cancel-sign-sub-inv 61.4%

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

   3. Simplified 77.5%

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

   4. Taylor expanded in x around 0 77.6%

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

      1. associate-*l/ 77.5%

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

      2. *-lft-identity 77.5%

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

      3. +-commutative 77.5%

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

      4. unpow2 77.5%

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

      5. fma-udef 77.5%

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

   6. Simplified 77.5%

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

   7. Taylor expanded in F around -inf 98.0%

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

   if -9.59999999999999914e-7 < F < 1.40000000000000001e-19

   1. Initial program 99.4%

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

      1. distribute-lft-neg-in 99.4%

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

      2. +-commutative 99.4%

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

      3. cancel-sign-sub-inv 99.4%

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

   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. Taylor expanded in x around 0 99.6%

      \[\leadsto F \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)} - \frac{x}{\tan B} \]
   5. 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-udef 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in F around 0 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-/l* 99.6%

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

   9. Simplified 99.6%

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

   10. Taylor expanded in B around 0 87.7%

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

       1. associate-/l* 87.7%

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

   12. Simplified 87.7%

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

   if 1.40000000000000001e-19 < F

   1. Initial program 59.9%

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

      1. distribute-lft-neg-in 59.9%

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

      2. +-commutative 59.9%

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

      3. cancel-sign-sub-inv 59.9%

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

   3. Simplified 77.1%

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

   4. Taylor expanded in x around 0 77.0%

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

      1. associate-*l/ 77.0%

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

      2. *-lft-identity 77.0%

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

      3. +-commutative 77.0%

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

      4. unpow2 77.0%

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

      5. fma-udef 77.0%

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

   6. Simplified 77.0%

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

   7. Taylor expanded in F around inf 98.9%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -9.6 \cdot 10^{-7}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{-19}:\\ \;\;\;\;\frac{F}{\frac{B}{\sqrt{0.5}}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 10: 91.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -9.6 \cdot 10^{-7}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{-19}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{B}{F}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -9.6e-7)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.4e-19)
       (- (/ (sqrt 0.5) (/ B F)) 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 <= -9.6e-7) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.4e-19) {
		tmp = (sqrt(0.5) / (B / F)) - 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 <= (-9.6d-7)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.4d-19) then
        tmp = (sqrt(0.5d0) / (b / f)) - 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 <= -9.6e-7) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.4e-19) {
		tmp = (Math.sqrt(0.5) / (B / F)) - 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 <= -9.6e-7:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.4e-19:
		tmp = (math.sqrt(0.5) / (B / F)) - 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 <= -9.6e-7)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.4e-19)
		tmp = Float64(Float64(sqrt(0.5) / Float64(B / F)) - 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 <= -9.6e-7)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.4e-19)
		tmp = (sqrt(0.5) / (B / F)) - 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, -9.6e-7], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.4e-19], N[(N[(N[Sqrt[0.5], $MachinePrecision] / N[(B / F), $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 < -9.59999999999999914e-7

   1. Initial program 61.4%

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

      1. distribute-lft-neg-in 61.4%

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

      2. +-commutative 61.4%

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

      3. cancel-sign-sub-inv 61.4%

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

   3. Simplified 77.5%

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

   4. Taylor expanded in x around 0 77.6%

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

      1. associate-*l/ 77.5%

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

      2. *-lft-identity 77.5%

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

      3. +-commutative 77.5%

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

      4. unpow2 77.5%

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

      5. fma-udef 77.5%

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

   6. Simplified 77.5%

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

   7. Taylor expanded in F around -inf 98.0%

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

   if -9.59999999999999914e-7 < F < 1.40000000000000001e-19

   1. Initial program 99.4%

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

      1. distribute-lft-neg-in 99.4%

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

      2. +-commutative 99.4%

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

      3. cancel-sign-sub-inv 99.4%

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

   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. Taylor expanded in x around 0 99.6%

      \[\leadsto F \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)} - \frac{x}{\tan B} \]
   5. 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-udef 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in F around 0 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-/l* 99.6%

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

   9. Simplified 99.6%

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

   10. Taylor expanded in B around 0 87.7%

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

   if 1.40000000000000001e-19 < F

   1. Initial program 59.9%

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

      1. distribute-lft-neg-in 59.9%

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

      2. +-commutative 59.9%

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

      3. cancel-sign-sub-inv 59.9%

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

   3. Simplified 77.1%

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

   4. Taylor expanded in x around 0 77.0%

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

      1. associate-*l/ 77.0%

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

      2. *-lft-identity 77.0%

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

      3. +-commutative 77.0%

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

      4. unpow2 77.0%

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

      5. fma-udef 77.0%

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

   6. Simplified 77.0%

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

   7. Taylor expanded in F around inf 98.9%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -9.6 \cdot 10^{-7}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{-19}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{B}{F}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 11: 84.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2.6 \cdot 10^{-94}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 400:\\ \;\;\;\;\frac{\frac{F}{B}}{F + \frac{1}{F}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -2.6e-94)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 400.0)
       (- (/ (/ F B) (+ F (/ 1.0 F))) 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 <= -2.6e-94) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 400.0) {
		tmp = ((F / B) / (F + (1.0 / F))) - 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 <= (-2.6d-94)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 400.0d0) then
        tmp = ((f / b) / (f + (1.0d0 / f))) - 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 <= -2.6e-94) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 400.0) {
		tmp = ((F / B) / (F + (1.0 / F))) - 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 <= -2.6e-94:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 400.0:
		tmp = ((F / B) / (F + (1.0 / F))) - 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 <= -2.6e-94)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 400.0)
		tmp = Float64(Float64(Float64(F / B) / Float64(F + Float64(1.0 / F))) - 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 <= -2.6e-94)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 400.0)
		tmp = ((F / B) / (F + (1.0 / F))) - 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, -2.6e-94], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 400.0], N[(N[(N[(F / B), $MachinePrecision] / N[(F + N[(1.0 / F), $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 < -2.59999999999999994e-94

   1. Initial program 65.8%

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

      1. distribute-lft-neg-in 65.8%

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

      2. +-commutative 65.8%

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

      3. cancel-sign-sub-inv 65.8%

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

   3. Simplified 80.1%

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

   4. Taylor expanded in x around 0 80.1%

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

      1. associate-*l/ 80.1%

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

      2. *-lft-identity 80.1%

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

      3. +-commutative 80.1%

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

      4. unpow2 80.1%

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

      5. fma-udef 80.1%

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

   6. Simplified 80.1%

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

   7. Taylor expanded in F around -inf 92.2%

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

   if -2.59999999999999994e-94 < F < 400

   1. Initial program 99.5%

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

      1. distribute-lft-neg-in 99.5%

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

      2. +-commutative 99.5%

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

      3. cancel-sign-sub-inv 99.5%

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

   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. Taylor expanded in x around 0 99.6%

      \[\leadsto F \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)} - \frac{x}{\tan B} \]
   5. 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-udef 99.7%

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

   6. Simplified 99.7%

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

      1. sqrt-div 99.7%

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

      2. metadata-eval 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in F around inf 74.9%

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

   10. Taylor expanded in B around 0 74.8%

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

       1. associate-/r* 74.8%

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

   12. Simplified 74.8%

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

   if 400 < F

   1. Initial program 56.7%

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

      1. distribute-lft-neg-in 56.7%

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

      2. +-commutative 56.7%

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

      3. cancel-sign-sub-inv 56.7%

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

   3. Simplified 75.2%

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

   4. Taylor expanded in x around 0 75.1%

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

      1. associate-*l/ 75.1%

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

      2. *-lft-identity 75.1%

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

      3. +-commutative 75.1%

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

      4. unpow2 75.1%

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

      5. fma-udef 75.1%

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

   6. Simplified 75.1%

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

   7. Taylor expanded in F around inf 99.8%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.6 \cdot 10^{-94}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 400:\\ \;\;\;\;\frac{\frac{F}{B}}{F + \frac{1}{F}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 12: 77.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := F + \frac{1}{F}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2.6 \cdot 10^{-94}:\\ \;\;\;\;\frac{-1}{\sin B} - t_1\\ \mathbf{elif}\;F \leq 6.8 \cdot 10^{+86}:\\ \;\;\;\;\frac{\frac{F}{B}}{t_0} - t_1\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{\frac{1}{t_0}}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (+ F (/ 1.0 F))) (t_1 (/ x (tan B))))
   (if (<= F -2.6e-94)
     (- (/ -1.0 (sin B)) t_1)
     (if (<= F 6.8e+86)
       (- (/ (/ F B) t_0) t_1)
       (- (* F (/ (/ 1.0 t_0) (sin B))) (/ x B))))))

C

double code(double F, double B, double x) {
	double t_0 = F + (1.0 / F);
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -2.6e-94) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= 6.8e+86) {
		tmp = ((F / B) / t_0) - t_1;
	} else {
		tmp = (F * ((1.0 / t_0) / sin(B))) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = f + (1.0d0 / f)
    t_1 = x / tan(b)
    if (f <= (-2.6d-94)) then
        tmp = ((-1.0d0) / sin(b)) - t_1
    else if (f <= 6.8d+86) then
        tmp = ((f / b) / t_0) - t_1
    else
        tmp = (f * ((1.0d0 / t_0) / sin(b))) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = F + (1.0 / F);
	double t_1 = x / Math.tan(B);
	double tmp;
	if (F <= -2.6e-94) {
		tmp = (-1.0 / Math.sin(B)) - t_1;
	} else if (F <= 6.8e+86) {
		tmp = ((F / B) / t_0) - t_1;
	} else {
		tmp = (F * ((1.0 / t_0) / Math.sin(B))) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = F + (1.0 / F)
	t_1 = x / math.tan(B)
	tmp = 0
	if F <= -2.6e-94:
		tmp = (-1.0 / math.sin(B)) - t_1
	elif F <= 6.8e+86:
		tmp = ((F / B) / t_0) - t_1
	else:
		tmp = (F * ((1.0 / t_0) / math.sin(B))) - (x / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(F + Float64(1.0 / F))
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -2.6e-94)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= 6.8e+86)
		tmp = Float64(Float64(Float64(F / B) / t_0) - t_1);
	else
		tmp = Float64(Float64(F * Float64(Float64(1.0 / t_0) / sin(B))) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = F + (1.0 / F);
	t_1 = x / tan(B);
	tmp = 0.0;
	if (F <= -2.6e-94)
		tmp = (-1.0 / sin(B)) - t_1;
	elseif (F <= 6.8e+86)
		tmp = ((F / B) / t_0) - t_1;
	else
		tmp = (F * ((1.0 / t_0) / sin(B))) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(F + N[(1.0 / F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.6e-94], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, 6.8e+86], N[(N[(N[(F / B), $MachinePrecision] / t$95$0), $MachinePrecision] - t$95$1), $MachinePrecision], N[(N[(F * N[(N[(1.0 / t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.59999999999999994e-94

   1. Initial program 65.8%

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

      1. distribute-lft-neg-in 65.8%

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

      2. +-commutative 65.8%

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

      3. cancel-sign-sub-inv 65.8%

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

   3. Simplified 80.1%

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

   4. Taylor expanded in x around 0 80.1%

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

      1. associate-*l/ 80.1%

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

      2. *-lft-identity 80.1%

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

      3. +-commutative 80.1%

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

      4. unpow2 80.1%

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

      5. fma-udef 80.1%

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

   6. Simplified 80.1%

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

   7. Taylor expanded in F around -inf 92.2%

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

   if -2.59999999999999994e-94 < F < 6.7999999999999995e86

   1. Initial program 98.7%

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

      1. distribute-lft-neg-in 98.7%

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

      2. +-commutative 98.7%

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

      3. cancel-sign-sub-inv 98.7%

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

   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. Taylor expanded in x around 0 99.6%

      \[\leadsto F \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)} - \frac{x}{\tan B} \]
   5. 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-udef 99.6%

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

   6. Simplified 99.6%

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

      1. sqrt-div 99.6%

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

      2. metadata-eval 99.6%

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

   8. Applied egg-rr 99.6%

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

   9. Taylor expanded in F around inf 78.0%

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

   10. Taylor expanded in B around 0 76.5%

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

       1. associate-/r* 75.8%

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

   12. Simplified 75.8%

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

   if 6.7999999999999995e86 < F

   1. Initial program 42.7%

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

      1. distribute-lft-neg-in 42.7%

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

      2. +-commutative 42.7%

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

      3. cancel-sign-sub-inv 42.7%

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

   3. Simplified 66.0%

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

   4. Taylor expanded in x around 0 65.9%

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

      1. associate-*l/ 65.9%

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

      2. *-lft-identity 65.9%

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

      3. +-commutative 65.9%

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

      4. unpow2 65.9%

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

      5. fma-udef 65.9%

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

   6. Simplified 65.9%

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

      1. sqrt-div 65.9%

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

      2. metadata-eval 65.9%

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

   8. Applied egg-rr 65.9%

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

   9. Taylor expanded in F around inf 99.5%

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

   10. Taylor expanded in B around 0 82.3%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.6 \cdot 10^{-94}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 6.8 \cdot 10^{+86}:\\ \;\;\;\;\frac{\frac{F}{B}}{F + \frac{1}{F}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{\frac{1}{F + \frac{1}{F}}}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 13: 70.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := F + \frac{1}{F}\\ \mathbf{if}\;F \leq -1.65 \cdot 10^{-111}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq 6.4 \cdot 10^{+86}:\\ \;\;\;\;\frac{\frac{F}{B}}{t_0} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{\frac{1}{t_0}}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (+ F (/ 1.0 F))))
   (if (<= F -1.65e-111)
     (+ (* x (/ -1.0 (tan B))) (/ -1.0 B))
     (if (<= F 6.4e+86)
       (- (/ (/ F B) t_0) (/ x (tan B)))
       (- (* F (/ (/ 1.0 t_0) (sin B))) (/ x B))))))

C

double code(double F, double B, double x) {
	double t_0 = F + (1.0 / F);
	double tmp;
	if (F <= -1.65e-111) {
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / B);
	} else if (F <= 6.4e+86) {
		tmp = ((F / B) / t_0) - (x / tan(B));
	} else {
		tmp = (F * ((1.0 / t_0) / sin(B))) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = f + (1.0d0 / f)
    if (f <= (-1.65d-111)) then
        tmp = (x * ((-1.0d0) / tan(b))) + ((-1.0d0) / b)
    else if (f <= 6.4d+86) then
        tmp = ((f / b) / t_0) - (x / tan(b))
    else
        tmp = (f * ((1.0d0 / t_0) / sin(b))) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = F + (1.0 / F);
	double tmp;
	if (F <= -1.65e-111) {
		tmp = (x * (-1.0 / Math.tan(B))) + (-1.0 / B);
	} else if (F <= 6.4e+86) {
		tmp = ((F / B) / t_0) - (x / Math.tan(B));
	} else {
		tmp = (F * ((1.0 / t_0) / Math.sin(B))) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = F + (1.0 / F)
	tmp = 0
	if F <= -1.65e-111:
		tmp = (x * (-1.0 / math.tan(B))) + (-1.0 / B)
	elif F <= 6.4e+86:
		tmp = ((F / B) / t_0) - (x / math.tan(B))
	else:
		tmp = (F * ((1.0 / t_0) / math.sin(B))) - (x / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(F + Float64(1.0 / F))
	tmp = 0.0
	if (F <= -1.65e-111)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(-1.0 / B));
	elseif (F <= 6.4e+86)
		tmp = Float64(Float64(Float64(F / B) / t_0) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(F * Float64(Float64(1.0 / t_0) / sin(B))) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = F + (1.0 / F);
	tmp = 0.0;
	if (F <= -1.65e-111)
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / B);
	elseif (F <= 6.4e+86)
		tmp = ((F / B) / t_0) - (x / tan(B));
	else
		tmp = (F * ((1.0 / t_0) / sin(B))) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(F + N[(1.0 / F), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.65e-111], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 6.4e+86], N[(N[(N[(F / B), $MachinePrecision] / t$95$0), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(F * N[(N[(1.0 / t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.65e-111

   1. Initial program 67.5%

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

   2. Taylor expanded in F around -inf 89.8%

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

   3. Taylor expanded in B around 0 73.8%

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

   if -1.65e-111 < F < 6.4000000000000001e86

   1. Initial program 98.7%

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

      1. distribute-lft-neg-in 98.7%

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

      2. +-commutative 98.7%

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

      3. cancel-sign-sub-inv 98.7%

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

   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. Taylor expanded in x around 0 99.6%

      \[\leadsto F \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)} - \frac{x}{\tan B} \]
   5. 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-udef 99.6%

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

   6. Simplified 99.6%

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

      1. sqrt-div 99.6%

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

      2. metadata-eval 99.6%

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

   8. Applied egg-rr 99.6%

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

   9. Taylor expanded in F around inf 78.1%

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

   10. Taylor expanded in B around 0 76.5%

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

       1. associate-/r* 75.8%

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

   12. Simplified 75.8%

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

   if 6.4000000000000001e86 < F

   1. Initial program 42.7%

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

      1. distribute-lft-neg-in 42.7%

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

      2. +-commutative 42.7%

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

      3. cancel-sign-sub-inv 42.7%

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

   3. Simplified 66.0%

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

   4. Taylor expanded in x around 0 65.9%

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

      1. associate-*l/ 65.9%

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

      2. *-lft-identity 65.9%

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

      3. +-commutative 65.9%

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

      4. unpow2 65.9%

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

      5. fma-udef 65.9%

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

   6. Simplified 65.9%

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

      1. sqrt-div 65.9%

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

      2. metadata-eval 65.9%

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

   8. Applied egg-rr 65.9%

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

   9. Taylor expanded in F around inf 99.5%

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

   10. Taylor expanded in B around 0 82.3%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.65 \cdot 10^{-111}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq 6.4 \cdot 10^{+86}:\\ \;\;\;\;\frac{\frac{F}{B}}{F + \frac{1}{F}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{\frac{1}{F + \frac{1}{F}}}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 14: 70.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3 \cdot 10^{+38}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq 2.8 \cdot 10^{-29}:\\ \;\;\;\;\frac{\frac{-F}{B}}{F + \frac{1}{F}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -3e+38)
     (+ (* x (/ -1.0 (tan B))) (/ -1.0 B))
     (if (<= F 2.8e-29)
       (- (/ (/ (- F) B) (+ F (/ 1.0 F))) t_0)
       (- (/ 1.0 B) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -3e+38) {
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / B);
	} else if (F <= 2.8e-29) {
		tmp = ((-F / B) / (F + (1.0 / F))) - t_0;
	} else {
		tmp = (1.0 / 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 <= (-3d+38)) then
        tmp = (x * ((-1.0d0) / tan(b))) + ((-1.0d0) / b)
    else if (f <= 2.8d-29) then
        tmp = ((-f / b) / (f + (1.0d0 / f))) - t_0
    else
        tmp = (1.0d0 / 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 <= -3e+38) {
		tmp = (x * (-1.0 / Math.tan(B))) + (-1.0 / B);
	} else if (F <= 2.8e-29) {
		tmp = ((-F / B) / (F + (1.0 / F))) - t_0;
	} else {
		tmp = (1.0 / B) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -3e+38:
		tmp = (x * (-1.0 / math.tan(B))) + (-1.0 / B)
	elif F <= 2.8e-29:
		tmp = ((-F / B) / (F + (1.0 / F))) - t_0
	else:
		tmp = (1.0 / B) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -3e+38)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(-1.0 / B));
	elseif (F <= 2.8e-29)
		tmp = Float64(Float64(Float64(Float64(-F) / B) / Float64(F + Float64(1.0 / F))) - t_0);
	else
		tmp = Float64(Float64(1.0 / 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 <= -3e+38)
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / B);
	elseif (F <= 2.8e-29)
		tmp = ((-F / B) / (F + (1.0 / F))) - t_0;
	else
		tmp = (1.0 / 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, -3e+38], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.8e-29], N[(N[(N[((-F) / B), $MachinePrecision] / N[(F + N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.0000000000000001e38

   1. Initial program 56.4%

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

   2. Taylor expanded in F around -inf 99.8%

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

   3. Taylor expanded in B around 0 79.6%

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

   if -3.0000000000000001e38 < F < 2.8000000000000002e-29

   1. Initial program 99.4%

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

      1. distribute-lft-neg-in 99.4%

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

      2. +-commutative 99.4%

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

      3. cancel-sign-sub-inv 99.4%

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

   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. Taylor expanded in x around 0 99.6%

      \[\leadsto F \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)} - \frac{x}{\tan B} \]
   5. 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-udef 99.6%

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

   6. Simplified 99.6%

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

      1. sqrt-div 99.6%

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

      2. metadata-eval 99.6%

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

   8. Applied egg-rr 99.6%

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

   9. Taylor expanded in F around -inf 73.6%

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

       1. mul-1-neg 73.6%

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

   11. Simplified 73.6%

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

   12. Taylor expanded in B around 0 72.0%

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

       1. mul-1-neg 72.0%

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

       2. associate-/r* 72.0%

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

       3. distribute-neg-frac 72.0%

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

   14. Simplified 72.0%

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

   if 2.8000000000000002e-29 < F

   1. Initial program 62.1%

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

      1. distribute-lft-neg-in 62.1%

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

      2. +-commutative 62.1%

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

      3. cancel-sign-sub-inv 62.1%

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

   3. Simplified 78.3%

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

   4. Taylor expanded in F around inf 96.1%

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

      1. associate-/r* 96.0%

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

   6. Simplified 96.0%

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

   7. Taylor expanded in B around 0 77.4%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3 \cdot 10^{+38}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq 2.8 \cdot 10^{-29}:\\ \;\;\;\;\frac{\frac{-F}{B}}{F + \frac{1}{F}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 15: 70.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.65 \cdot 10^{-111}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq 200000000:\\ \;\;\;\;\frac{\frac{F}{B}}{F + \frac{1}{F}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.65e-111)
     (+ (* x (/ -1.0 (tan B))) (/ -1.0 B))
     (if (<= F 200000000.0)
       (- (/ (/ F B) (+ F (/ 1.0 F))) t_0)
       (- (/ 1.0 B) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.65e-111) {
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / B);
	} else if (F <= 200000000.0) {
		tmp = ((F / B) / (F + (1.0 / F))) - t_0;
	} else {
		tmp = (1.0 / 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 <= (-1.65d-111)) then
        tmp = (x * ((-1.0d0) / tan(b))) + ((-1.0d0) / b)
    else if (f <= 200000000.0d0) then
        tmp = ((f / b) / (f + (1.0d0 / f))) - t_0
    else
        tmp = (1.0d0 / 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 <= -1.65e-111) {
		tmp = (x * (-1.0 / Math.tan(B))) + (-1.0 / B);
	} else if (F <= 200000000.0) {
		tmp = ((F / B) / (F + (1.0 / F))) - t_0;
	} else {
		tmp = (1.0 / B) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.65e-111:
		tmp = (x * (-1.0 / math.tan(B))) + (-1.0 / B)
	elif F <= 200000000.0:
		tmp = ((F / B) / (F + (1.0 / F))) - t_0
	else:
		tmp = (1.0 / B) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.65e-111)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(-1.0 / B));
	elseif (F <= 200000000.0)
		tmp = Float64(Float64(Float64(F / B) / Float64(F + Float64(1.0 / F))) - t_0);
	else
		tmp = Float64(Float64(1.0 / 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 <= -1.65e-111)
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / B);
	elseif (F <= 200000000.0)
		tmp = ((F / B) / (F + (1.0 / F))) - t_0;
	else
		tmp = (1.0 / 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, -1.65e-111], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 200000000.0], N[(N[(N[(F / B), $MachinePrecision] / N[(F + N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.65e-111

   1. Initial program 67.5%

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

   2. Taylor expanded in F around -inf 89.8%

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

   3. Taylor expanded in B around 0 73.8%

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

   if -1.65e-111 < F < 2e8

   1. Initial program 99.5%

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

      1. distribute-lft-neg-in 99.5%

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

      2. +-commutative 99.5%

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

      3. cancel-sign-sub-inv 99.5%

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

   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. Taylor expanded in x around 0 99.6%

      \[\leadsto F \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)} - \frac{x}{\tan B} \]
   5. 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-udef 99.7%

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

   6. Simplified 99.7%

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

      1. sqrt-div 99.7%

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

      2. metadata-eval 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in F around inf 74.9%

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

   10. Taylor expanded in B around 0 74.8%

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

       1. associate-/r* 74.8%

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

   12. Simplified 74.8%

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

   if 2e8 < F

   1. Initial program 56.7%

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

      1. distribute-lft-neg-in 56.7%

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

      2. +-commutative 56.7%

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

      3. cancel-sign-sub-inv 56.7%

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

   3. Simplified 75.2%

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

   4. Taylor expanded in F around inf 99.6%

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

      1. associate-/r* 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in B around 0 78.1%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.65 \cdot 10^{-111}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq 200000000:\\ \;\;\;\;\frac{\frac{F}{B}}{F + \frac{1}{F}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 16: 63.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -9.6 \cdot 10^{-7}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.05 \cdot 10^{-229} \lor \neg \left(F \leq 1.78 \cdot 10^{-152}\right) \land F \leq 10^{-113}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -9.6e-7)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (or (<= F -1.05e-229) (and (not (<= F 1.78e-152)) (<= F 1e-113)))
     (/ (- (* F (sqrt 0.5)) x) B)
     (- (/ 1.0 B) (/ x (tan B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -9.6e-7) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if ((F <= -1.05e-229) || (!(F <= 1.78e-152) && (F <= 1e-113))) {
		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 (f <= (-9.6d-7)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if ((f <= (-1.05d-229)) .or. (.not. (f <= 1.78d-152)) .and. (f <= 1d-113)) 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 (F <= -9.6e-7) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if ((F <= -1.05e-229) || (!(F <= 1.78e-152) && (F <= 1e-113))) {
		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 F <= -9.6e-7:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif (F <= -1.05e-229) or (not (F <= 1.78e-152) and (F <= 1e-113)):
		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 (F <= -9.6e-7)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif ((F <= -1.05e-229) || (!(F <= 1.78e-152) && (F <= 1e-113)))
		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 (F <= -9.6e-7)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif ((F <= -1.05e-229) || (~((F <= 1.78e-152)) && (F <= 1e-113)))
		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[F, -9.6e-7], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[F, -1.05e-229], And[N[Not[LessEqual[F, 1.78e-152]], $MachinePrecision], LessEqual[F, 1e-113]]], 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 F < -9.59999999999999914e-7

   1. Initial program 61.4%

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

   2. Taylor expanded in F around -inf 97.9%

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

   3. Taylor expanded in B around 0 70.1%

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

   if -9.59999999999999914e-7 < F < -1.04999999999999992e-229 or 1.7800000000000001e-152 < F < 9.99999999999999979e-114

   1. Initial program 99.3%

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

      1. distribute-lft-neg-in 99.3%

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

      2. +-commutative 99.3%

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

      3. cancel-sign-sub-inv 99.3%

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

   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. Taylor expanded in x around 0 99.7%

      \[\leadsto F \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)} - \frac{x}{\tan B} \]
   5. 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-udef 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in F around 0 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-/l* 99.6%

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

   9. Simplified 99.6%

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

   10. Taylor expanded in B around 0 64.4%

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

   if -1.04999999999999992e-229 < F < 1.7800000000000001e-152 or 9.99999999999999979e-114 < F

   1. Initial program 80.5%

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

      1. distribute-lft-neg-in 80.5%

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

      2. +-commutative 80.5%

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

      3. cancel-sign-sub-inv 80.5%

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

   3. Simplified 88.8%

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

   4. Taylor expanded in F around inf 63.0%

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

      1. associate-/r* 62.9%

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

   6. Simplified 62.9%

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

   7. Taylor expanded in B around 0 65.7%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -9.6 \cdot 10^{-7}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.05 \cdot 10^{-229} \lor \neg \left(F \leq 1.78 \cdot 10^{-152}\right) \land F \leq 10^{-113}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 17: 55.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F \cdot \sqrt{0.5} - x}{B}\\ t_1 := \frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;B \leq -8.2 \cdot 10^{-114}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 8.8 \cdot 10^{-247}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq 2.1 \cdot 10^{-113}:\\ \;\;\;\;\frac{1 - x}{B}\\ \mathbf{elif}\;B \leq 1.15 \cdot 10^{-49}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- (* F (sqrt 0.5)) x) B)) (t_1 (- (/ 1.0 B) (/ x (tan B)))))
   (if (<= B -8.2e-114)
     t_1
     (if (<= B 8.8e-247)
       t_0
       (if (<= B 2.1e-113) (/ (- 1.0 x) B) (if (<= B 1.15e-49) t_0 t_1))))))

C

double code(double F, double B, double x) {
	double t_0 = ((F * sqrt(0.5)) - x) / B;
	double t_1 = (1.0 / B) - (x / tan(B));
	double tmp;
	if (B <= -8.2e-114) {
		tmp = t_1;
	} else if (B <= 8.8e-247) {
		tmp = t_0;
	} else if (B <= 2.1e-113) {
		tmp = (1.0 - x) / B;
	} else if (B <= 1.15e-49) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((f * sqrt(0.5d0)) - x) / b
    t_1 = (1.0d0 / b) - (x / tan(b))
    if (b <= (-8.2d-114)) then
        tmp = t_1
    else if (b <= 8.8d-247) then
        tmp = t_0
    else if (b <= 2.1d-113) then
        tmp = (1.0d0 - x) / b
    else if (b <= 1.15d-49) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = ((F * Math.sqrt(0.5)) - x) / B;
	double t_1 = (1.0 / B) - (x / Math.tan(B));
	double tmp;
	if (B <= -8.2e-114) {
		tmp = t_1;
	} else if (B <= 8.8e-247) {
		tmp = t_0;
	} else if (B <= 2.1e-113) {
		tmp = (1.0 - x) / B;
	} else if (B <= 1.15e-49) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = ((F * math.sqrt(0.5)) - x) / B
	t_1 = (1.0 / B) - (x / math.tan(B))
	tmp = 0
	if B <= -8.2e-114:
		tmp = t_1
	elif B <= 8.8e-247:
		tmp = t_0
	elif B <= 2.1e-113:
		tmp = (1.0 - x) / B
	elif B <= 1.15e-49:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(Float64(F * sqrt(0.5)) - x) / B)
	t_1 = Float64(Float64(1.0 / B) - Float64(x / tan(B)))
	tmp = 0.0
	if (B <= -8.2e-114)
		tmp = t_1;
	elseif (B <= 8.8e-247)
		tmp = t_0;
	elseif (B <= 2.1e-113)
		tmp = Float64(Float64(1.0 - x) / B);
	elseif (B <= 1.15e-49)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = ((F * sqrt(0.5)) - x) / B;
	t_1 = (1.0 / B) - (x / tan(B));
	tmp = 0.0;
	if (B <= -8.2e-114)
		tmp = t_1;
	elseif (B <= 8.8e-247)
		tmp = t_0;
	elseif (B <= 2.1e-113)
		tmp = (1.0 - x) / B;
	elseif (B <= 1.15e-49)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -8.2e-114], t$95$1, If[LessEqual[B, 8.8e-247], t$95$0, If[LessEqual[B, 2.1e-113], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[B, 1.15e-49], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if B < -8.1999999999999993e-114 or 1.15e-49 < B

   1. Initial program 83.2%

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

      1. distribute-lft-neg-in 83.2%

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

      2. +-commutative 83.2%

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

      3. cancel-sign-sub-inv 83.2%

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

   3. Simplified 85.4%

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

   4. Taylor expanded in F around inf 61.4%

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

      1. associate-/r* 61.4%

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

   6. Simplified 61.4%

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

   7. Taylor expanded in B around 0 61.6%

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

   if -8.1999999999999993e-114 < B < 8.79999999999999966e-247 or 2.1e-113 < B < 1.15e-49

   1. Initial program 77.5%

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

      1. distribute-lft-neg-in 77.5%

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

      2. +-commutative 77.5%

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

      3. cancel-sign-sub-inv 77.5%

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

   3. Simplified 90.8%

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

   4. Taylor expanded in x around 0 90.8%

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

      1. associate-*l/ 90.8%

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

      2. *-lft-identity 90.8%

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

      3. +-commutative 90.8%

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

      4. unpow2 90.8%

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

      5. fma-udef 90.8%

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

   6. Simplified 90.8%

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

   7. Taylor expanded in F around 0 65.5%

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

      1. *-commutative 65.5%

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

      2. associate-/l* 65.5%

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

   9. Simplified 65.5%

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

   10. Taylor expanded in B around 0 69.7%

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

   if 8.79999999999999966e-247 < B < 2.1e-113

   1. Initial program 54.7%

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

      1. distribute-lft-neg-in 54.7%

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

      2. +-commutative 54.7%

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

      3. cancel-sign-sub-inv 54.7%

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

   3. Simplified 95.7%

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

   4. Taylor expanded in F around inf 63.4%

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

      1. associate-/r* 63.3%

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

   6. Simplified 63.3%

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

   7. Taylor expanded in B around 0 71.9%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -8.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;B \leq 8.8 \cdot 10^{-247}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{elif}\;B \leq 2.1 \cdot 10^{-113}:\\ \;\;\;\;\frac{1 - x}{B}\\ \mathbf{elif}\;B \leq 1.15 \cdot 10^{-49}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 18: 61.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F \cdot \sqrt{0.5} - x}{B}\\ t_1 := x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{if}\;F \leq -2.1 \cdot 10^{-119}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq -1.15 \cdot 10^{-227}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 1.2 \cdot 10^{-156}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 1.6 \cdot 10^{-113}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- (* F (sqrt 0.5)) x) B))
        (t_1 (+ (* x (/ -1.0 (tan B))) (/ -1.0 B))))
   (if (<= F -2.1e-119)
     t_1
     (if (<= F -1.15e-227)
       t_0
       (if (<= F 1.2e-156)
         t_1
         (if (<= F 1.6e-113) t_0 (- (/ 1.0 B) (/ x (tan B)))))))))

C

double code(double F, double B, double x) {
	double t_0 = ((F * sqrt(0.5)) - x) / B;
	double t_1 = (x * (-1.0 / tan(B))) + (-1.0 / B);
	double tmp;
	if (F <= -2.1e-119) {
		tmp = t_1;
	} else if (F <= -1.15e-227) {
		tmp = t_0;
	} else if (F <= 1.2e-156) {
		tmp = t_1;
	} else if (F <= 1.6e-113) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((f * sqrt(0.5d0)) - x) / b
    t_1 = (x * ((-1.0d0) / tan(b))) + ((-1.0d0) / b)
    if (f <= (-2.1d-119)) then
        tmp = t_1
    else if (f <= (-1.15d-227)) then
        tmp = t_0
    else if (f <= 1.2d-156) then
        tmp = t_1
    else if (f <= 1.6d-113) then
        tmp = t_0
    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 t_0 = ((F * Math.sqrt(0.5)) - x) / B;
	double t_1 = (x * (-1.0 / Math.tan(B))) + (-1.0 / B);
	double tmp;
	if (F <= -2.1e-119) {
		tmp = t_1;
	} else if (F <= -1.15e-227) {
		tmp = t_0;
	} else if (F <= 1.2e-156) {
		tmp = t_1;
	} else if (F <= 1.6e-113) {
		tmp = t_0;
	} else {
		tmp = (1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = ((F * math.sqrt(0.5)) - x) / B
	t_1 = (x * (-1.0 / math.tan(B))) + (-1.0 / B)
	tmp = 0
	if F <= -2.1e-119:
		tmp = t_1
	elif F <= -1.15e-227:
		tmp = t_0
	elif F <= 1.2e-156:
		tmp = t_1
	elif F <= 1.6e-113:
		tmp = t_0
	else:
		tmp = (1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(Float64(F * sqrt(0.5)) - x) / B)
	t_1 = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(-1.0 / B))
	tmp = 0.0
	if (F <= -2.1e-119)
		tmp = t_1;
	elseif (F <= -1.15e-227)
		tmp = t_0;
	elseif (F <= 1.2e-156)
		tmp = t_1;
	elseif (F <= 1.6e-113)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = ((F * sqrt(0.5)) - x) / B;
	t_1 = (x * (-1.0 / tan(B))) + (-1.0 / B);
	tmp = 0.0;
	if (F <= -2.1e-119)
		tmp = t_1;
	elseif (F <= -1.15e-227)
		tmp = t_0;
	elseif (F <= 1.2e-156)
		tmp = t_1;
	elseif (F <= 1.6e-113)
		tmp = t_0;
	else
		tmp = (1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.1e-119], t$95$1, If[LessEqual[F, -1.15e-227], t$95$0, If[LessEqual[F, 1.2e-156], t$95$1, If[LessEqual[F, 1.6e-113], t$95$0, N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.1e-119 or -1.15000000000000006e-227 < F < 1.2e-156

   1. Initial program 79.6%

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

   2. Taylor expanded in F around -inf 70.0%

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

   3. Taylor expanded in B around 0 67.8%

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

   if -2.1e-119 < F < -1.15000000000000006e-227 or 1.2e-156 < F < 1.6000000000000001e-113

   1. Initial program 99.4%

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

      1. distribute-lft-neg-in 99.4%

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

      2. +-commutative 99.4%

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

      3. cancel-sign-sub-inv 99.4%

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

   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. Taylor expanded in x around 0 99.7%

      \[\leadsto F \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)} - \frac{x}{\tan B} \]
   5. 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-udef 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in F around 0 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-/l* 99.7%

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

   9. Simplified 99.7%

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

   10. Taylor expanded in B around 0 69.6%

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

   if 1.6000000000000001e-113 < F

   1. Initial program 70.6%

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

      1. distribute-lft-neg-in 70.6%

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

      2. +-commutative 70.6%

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

      3. cancel-sign-sub-inv 70.6%

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

   3. Simplified 83.1%

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

   4. Taylor expanded in F around inf 81.5%

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

      1. associate-/r* 81.4%

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

   6. Simplified 81.4%

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

   7. Taylor expanded in B around 0 70.2%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.1 \cdot 10^{-119}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq -1.15 \cdot 10^{-227}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{elif}\;F \leq 1.2 \cdot 10^{-156}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq 1.6 \cdot 10^{-113}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 19: 54.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq -9.5 \cdot 10^{-115} \lor \neg \left(B \leq 9 \cdot 10^{-292}\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (or (<= B -9.5e-115) (not (<= B 9e-292)))
   (- (/ 1.0 B) (/ x (tan B)))
   (/ (- x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if ((B <= -9.5e-115) || !(B <= 9e-292)) {
		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 ((b <= (-9.5d-115)) .or. (.not. (b <= 9d-292))) 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 ((B <= -9.5e-115) || !(B <= 9e-292)) {
		tmp = (1.0 / B) - (x / Math.tan(B));
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if (B <= -9.5e-115) or not (B <= 9e-292):
		tmp = (1.0 / B) - (x / math.tan(B))
	else:
		tmp = -x / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if ((B <= -9.5e-115) || !(B <= 9e-292))
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(-x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if ((B <= -9.5e-115) || ~((B <= 9e-292)))
		tmp = (1.0 / B) - (x / tan(B));
	else
		tmp = -x / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[Or[LessEqual[B, -9.5e-115], N[Not[LessEqual[B, 9e-292]], $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 B < -9.4999999999999996e-115 or 8.99999999999999913e-292 < B

   1. Initial program 80.1%

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

      1. distribute-lft-neg-in 80.1%

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

      2. +-commutative 80.1%

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

      3. cancel-sign-sub-inv 80.1%

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

   3. Simplified 87.8%

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

   4. Taylor expanded in F around inf 57.6%

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

      1. associate-/r* 57.6%

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

   6. Simplified 57.6%

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

   7. Taylor expanded in B around 0 60.1%

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

   if -9.4999999999999996e-115 < B < 8.99999999999999913e-292

   1. Initial program 74.1%

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

      1. distribute-lft-neg-in 74.1%

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

      2. +-commutative 74.1%

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

      3. cancel-sign-sub-inv 74.1%

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

   3. Simplified 88.2%

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

   4. Taylor expanded in F around inf 33.5%

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

      1. associate-/r* 33.5%

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

   6. Simplified 33.5%

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

   7. Taylor expanded in B around 0 45.5%

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

   8. Taylor expanded in x around inf 59.0%

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

      1. associate-*r/ 59.0%

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

      2. neg-mul-1 59.0%

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

   10. Simplified 59.0%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -9.5 \cdot 10^{-115} \lor \neg \left(B \leq 9 \cdot 10^{-292}\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \]

Alternative 20: 44.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -6.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq -1.25 \cdot 10^{-121}:\\ \;\;\;\;\frac{F}{\frac{B}{\sqrt{0.5}}}\\ \mathbf{elif}\;F \leq 2.7 \cdot 10^{-29}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B} + 0.3333333333333333 \cdot \left(B \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -6.2e-5)
   (/ (- -1.0 x) B)
   (if (<= F -1.25e-121)
     (/ F (/ B (sqrt 0.5)))
     (if (<= F 2.7e-29)
       (/ (- x) B)
       (+ (/ (- 1.0 x) B) (* 0.3333333333333333 (* B x)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -6.2e-5) {
		tmp = (-1.0 - x) / B;
	} else if (F <= -1.25e-121) {
		tmp = F / (B / sqrt(0.5));
	} else if (F <= 2.7e-29) {
		tmp = -x / B;
	} else {
		tmp = ((1.0 - x) / B) + (0.3333333333333333 * (B * x));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-6.2d-5)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= (-1.25d-121)) then
        tmp = f / (b / sqrt(0.5d0))
    else if (f <= 2.7d-29) then
        tmp = -x / b
    else
        tmp = ((1.0d0 - x) / b) + (0.3333333333333333d0 * (b * x))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -6.2e-5) {
		tmp = (-1.0 - x) / B;
	} else if (F <= -1.25e-121) {
		tmp = F / (B / Math.sqrt(0.5));
	} else if (F <= 2.7e-29) {
		tmp = -x / B;
	} else {
		tmp = ((1.0 - x) / B) + (0.3333333333333333 * (B * x));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -6.2e-5:
		tmp = (-1.0 - x) / B
	elif F <= -1.25e-121:
		tmp = F / (B / math.sqrt(0.5))
	elif F <= 2.7e-29:
		tmp = -x / B
	else:
		tmp = ((1.0 - x) / B) + (0.3333333333333333 * (B * x))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -6.2e-5)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= -1.25e-121)
		tmp = Float64(F / Float64(B / sqrt(0.5)));
	elseif (F <= 2.7e-29)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(Float64(1.0 - x) / B) + Float64(0.3333333333333333 * Float64(B * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -6.2e-5)
		tmp = (-1.0 - x) / B;
	elseif (F <= -1.25e-121)
		tmp = F / (B / sqrt(0.5));
	elseif (F <= 2.7e-29)
		tmp = -x / B;
	else
		tmp = ((1.0 - x) / B) + (0.3333333333333333 * (B * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -6.2e-5], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, -1.25e-121], N[(F / N[(B / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.7e-29], N[((-x) / B), $MachinePrecision], N[(N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision] + N[(0.3333333333333333 * N[(B * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -6.20000000000000027e-5

   1. Initial program 60.9%

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

   2. Taylor expanded in F around -inf 99.2%

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

   3. Taylor expanded in B around 0 50.7%

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

      1. associate-*r/ 50.7%

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

      2. distribute-lft-in 50.7%

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

      3. metadata-eval 50.7%

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

      4. neg-mul-1 50.7%

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

   5. Simplified 50.7%

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

   if -6.20000000000000027e-5 < F < -1.24999999999999997e-121

   1. Initial program 99.2%

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

      1. distribute-lft-neg-in 99.2%

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

      2. +-commutative 99.2%

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

      3. cancel-sign-sub-inv 99.2%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 99.6%

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

      1. associate-*l/ 99.5%

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

      2. *-lft-identity 99.5%

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

      3. +-commutative 99.5%

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

      4. unpow2 99.5%

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

      5. fma-udef 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in F around 0 98.7%

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

      1. *-commutative 98.7%

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

      2. associate-/l* 98.6%

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

   9. Simplified 98.6%

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

   10. Taylor expanded in B around 0 49.3%

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

   11. Taylor expanded in F around inf 38.1%

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

       1. associate-/l* 38.1%

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

   13. Simplified 38.1%

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

   if -1.24999999999999997e-121 < F < 2.70000000000000023e-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. Step-by-step derivation

      1. distribute-lft-neg-in 99.5%

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

      2. +-commutative 99.5%

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

      3. cancel-sign-sub-inv 99.5%

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

   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. Taylor expanded in F around inf 27.7%

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

      1. associate-/r* 27.7%

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

   6. Simplified 27.7%

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

   7. Taylor expanded in B around 0 18.1%

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

   8. Taylor expanded in x around inf 35.4%

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

      1. associate-*r/ 35.4%

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

      2. neg-mul-1 35.4%

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

   10. Simplified 35.4%

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

   if 2.70000000000000023e-29 < F

   1. Initial program 62.1%

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

      1. distribute-lft-neg-in 62.1%

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

      2. +-commutative 62.1%

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

      3. cancel-sign-sub-inv 62.1%

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

   3. Simplified 78.3%

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

   4. Taylor expanded in F around inf 96.1%

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

      1. associate-/r* 96.0%

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

   6. Simplified 96.0%

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

   7. Taylor expanded in B around 0 77.3%

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

      1. *-commutative 77.3%

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

   9. Simplified 77.3%

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

   10. Taylor expanded in B around 0 58.4%

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

       1. associate--l+ 58.4%

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

       2. div-sub 58.3%

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

   12. Simplified 58.3%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq -1.25 \cdot 10^{-121}:\\ \;\;\;\;\frac{F}{\frac{B}{\sqrt{0.5}}}\\ \mathbf{elif}\;F \leq 2.7 \cdot 10^{-29}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B} + 0.3333333333333333 \cdot \left(B \cdot x\right)\\ \end{array} \]

Alternative 21: 43.9% accurate, 21.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.7 \cdot 10^{-111}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.8 \cdot 10^{-29}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B} + 0.3333333333333333 \cdot \left(B \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.7e-111)
   (/ (- -1.0 x) B)
   (if (<= F 2.8e-29)
     (/ (- x) B)
     (+ (/ (- 1.0 x) B) (* 0.3333333333333333 (* B x))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.7e-111) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 2.8e-29) {
		tmp = -x / B;
	} else {
		tmp = ((1.0 - x) / B) + (0.3333333333333333 * (B * x));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-3.7d-111)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 2.8d-29) then
        tmp = -x / b
    else
        tmp = ((1.0d0 - x) / b) + (0.3333333333333333d0 * (b * x))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.7e-111) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 2.8e-29) {
		tmp = -x / B;
	} else {
		tmp = ((1.0 - x) / B) + (0.3333333333333333 * (B * x));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.7e-111:
		tmp = (-1.0 - x) / B
	elif F <= 2.8e-29:
		tmp = -x / B
	else:
		tmp = ((1.0 - x) / B) + (0.3333333333333333 * (B * x))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.7e-111)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 2.8e-29)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(Float64(1.0 - x) / B) + Float64(0.3333333333333333 * Float64(B * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -3.7e-111)
		tmp = (-1.0 - x) / B;
	elseif (F <= 2.8e-29)
		tmp = -x / B;
	else
		tmp = ((1.0 - x) / B) + (0.3333333333333333 * (B * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.7e-111], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 2.8e-29], N[((-x) / B), $MachinePrecision], N[(N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision] + N[(0.3333333333333333 * N[(B * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.7000000000000002e-111

   1. Initial program 67.1%

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

   2. Taylor expanded in F around -inf 89.7%

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

   3. Taylor expanded in B around 0 45.6%

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

      1. associate-*r/ 45.6%

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

      2. distribute-lft-in 45.6%

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

      3. metadata-eval 45.6%

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

      4. neg-mul-1 45.6%

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

   5. Simplified 45.6%

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

   if -3.7000000000000002e-111 < F < 2.8000000000000002e-29

   1. Initial program 99.4%

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

      1. distribute-lft-neg-in 99.4%

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

      2. +-commutative 99.4%

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

      3. cancel-sign-sub-inv 99.4%

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

   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. Taylor expanded in F around inf 27.2%

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

      1. associate-/r* 27.2%

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

   6. Simplified 27.2%

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

   7. Taylor expanded in B around 0 17.3%

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

   8. Taylor expanded in x around inf 33.7%

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

      1. associate-*r/ 33.7%

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

      2. neg-mul-1 33.7%

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

   10. Simplified 33.7%

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

   if 2.8000000000000002e-29 < F

   1. Initial program 62.1%

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

      1. distribute-lft-neg-in 62.1%

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

      2. +-commutative 62.1%

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

      3. cancel-sign-sub-inv 62.1%

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

   3. Simplified 78.3%

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

   4. Taylor expanded in F around inf 96.1%

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

      1. associate-/r* 96.0%

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

   6. Simplified 96.0%

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

   7. Taylor expanded in B around 0 77.3%

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

      1. *-commutative 77.3%

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

   9. Simplified 77.3%

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

   10. Taylor expanded in B around 0 58.4%

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

       1. associate--l+ 58.4%

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

       2. div-sub 58.3%

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

   12. Simplified 58.3%

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

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.7 \cdot 10^{-111}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.8 \cdot 10^{-29}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B} + 0.3333333333333333 \cdot \left(B \cdot x\right)\\ \end{array} \]

Alternative 22: 43.8% accurate, 29.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -4 \cdot 10^{-111}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.8 \cdot 10^{-29}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4e-111)
   (/ (- -1.0 x) B)
   (if (<= F 2.8e-29) (/ (- x) B) (- (/ 1.0 B) (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4e-111) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 2.8e-29) {
		tmp = -x / B;
	} else {
		tmp = (1.0 / 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 <= (-4d-111)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 2.8d-29) then
        tmp = -x / b
    else
        tmp = (1.0d0 / b) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -4e-111) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 2.8e-29) {
		tmp = -x / B;
	} else {
		tmp = (1.0 / B) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -4e-111:
		tmp = (-1.0 - x) / B
	elif F <= 2.8e-29:
		tmp = -x / B
	else:
		tmp = (1.0 / B) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -4e-111)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 2.8e-29)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -4e-111)
		tmp = (-1.0 - x) / B;
	elseif (F <= 2.8e-29)
		tmp = -x / B;
	else
		tmp = (1.0 / B) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -4e-111], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 2.8e-29], N[((-x) / B), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.00000000000000035e-111

   1. Initial program 67.1%

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

   2. Taylor expanded in F around -inf 89.7%

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

   3. Taylor expanded in B around 0 45.6%

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

      1. associate-*r/ 45.6%

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

      2. distribute-lft-in 45.6%

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

      3. metadata-eval 45.6%

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

      4. neg-mul-1 45.6%

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

   5. Simplified 45.6%

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

   if -4.00000000000000035e-111 < F < 2.8000000000000002e-29

   1. Initial program 99.4%

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

      1. distribute-lft-neg-in 99.4%

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

      2. +-commutative 99.4%

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

      3. cancel-sign-sub-inv 99.4%

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

   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. Taylor expanded in F around inf 27.2%

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

      1. associate-/r* 27.2%

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

   6. Simplified 27.2%

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

   7. Taylor expanded in B around 0 17.3%

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

   8. Taylor expanded in x around inf 33.7%

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

      1. associate-*r/ 33.7%

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

      2. neg-mul-1 33.7%

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

   10. Simplified 33.7%

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

   if 2.8000000000000002e-29 < F

   1. Initial program 62.1%

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

      1. distribute-lft-neg-in 62.1%

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

      2. +-commutative 62.1%

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

      3. cancel-sign-sub-inv 62.1%

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

   3. Simplified 78.3%

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

   4. Taylor expanded in F around inf 96.1%

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

      1. associate-/r* 96.0%

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

   6. Simplified 96.0%

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

   7. Taylor expanded in B around 0 57.5%

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

      1. div-sub 57.6%

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

   9. Applied egg-rr 57.6%

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

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4 \cdot 10^{-111}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.8 \cdot 10^{-29}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{B}\\ \end{array} \]

Alternative 23: 43.8% accurate, 35.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.8 \cdot 10^{-111}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.8 \cdot 10^{-29}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.8e-111)
   (/ (- -1.0 x) B)
   (if (<= F 2.8e-29) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.8e-111) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 2.8e-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 <= (-3.8d-111)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 2.8d-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 <= -3.8e-111) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 2.8e-29) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.8e-111:
		tmp = (-1.0 - x) / B
	elif F <= 2.8e-29:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.8e-111)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 2.8e-29)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -3.8e-111)
		tmp = (-1.0 - x) / B;
	elseif (F <= 2.8e-29)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.8e-111], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 2.8e-29], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.80000000000000022e-111

   1. Initial program 67.1%

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

   2. Taylor expanded in F around -inf 89.7%

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

   3. Taylor expanded in B around 0 45.6%

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

      1. associate-*r/ 45.6%

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

      2. distribute-lft-in 45.6%

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

      3. metadata-eval 45.6%

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

      4. neg-mul-1 45.6%

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

   5. Simplified 45.6%

      \[\leadsto \color{blue}{\frac{-1 + \left(-x\right)}{B}} \]

   if -3.80000000000000022e-111 < F < 2.8000000000000002e-29

   1. Initial program 99.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. distribute-lft-neg-in 99.4%

         \[\leadsto \color{blue}{\left(-x\right) \cdot \frac{1}{\tan B}} + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. +-commutative 99.4%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x\right) \cdot \frac{1}{\tan B}} \]

      3. cancel-sign-sub-inv 99.4%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

   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. Taylor expanded in F around inf 27.2%

      \[\leadsto F \cdot \color{blue}{\frac{1}{F \cdot \sin B}} - \frac{x}{\tan B} \]
   5. Step-by-step derivation

      1. associate-/r* 27.2%

         \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   6. Simplified 27.2%

      \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   7. Taylor expanded in B around 0 17.3%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

   8. Taylor expanded in x around inf 33.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   9. Step-by-step derivation

      1. associate-*r/ 33.7%

         \[\leadsto \color{blue}{\frac{-1 \cdot x}{B}} \]

      2. neg-mul-1 33.7%

         \[\leadsto \frac{\color{blue}{-x}}{B} \]

   10. Simplified 33.7%

       \[\leadsto \color{blue}{\frac{-x}{B}} \]

   if 2.8000000000000002e-29 < F

   1. Initial program 62.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. distribute-lft-neg-in 62.1%

         \[\leadsto \color{blue}{\left(-x\right) \cdot \frac{1}{\tan B}} + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. +-commutative 62.1%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x\right) \cdot \frac{1}{\tan B}} \]

      3. cancel-sign-sub-inv 62.1%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

   3. Simplified 78.3%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]

   4. Taylor expanded in F around inf 96.1%

      \[\leadsto F \cdot \color{blue}{\frac{1}{F \cdot \sin B}} - \frac{x}{\tan B} \]
   5. Step-by-step derivation

      1. associate-/r* 96.0%

         \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   6. Simplified 96.0%

      \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   7. Taylor expanded in B around 0 57.5%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.8 \cdot 10^{-111}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.8 \cdot 10^{-29}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 24: 31.0% accurate, 39.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-223} \lor \neg \left(x \leq 1.22 \cdot 10^{-71}\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (or (<= x -4.2e-223) (not (<= x 1.22e-71))) (/ (- x) B) (/ 1.0 B)))

C

double code(double F, double B, double x) {
	double tmp;
	if ((x <= -4.2e-223) || !(x <= 1.22e-71)) {
		tmp = -x / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-4.2d-223)) .or. (.not. (x <= 1.22d-71))) then
        tmp = -x / b
    else
        tmp = 1.0d0 / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if ((x <= -4.2e-223) || !(x <= 1.22e-71)) {
		tmp = -x / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if (x <= -4.2e-223) or not (x <= 1.22e-71):
		tmp = -x / B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if ((x <= -4.2e-223) || !(x <= 1.22e-71))
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(1.0 / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if ((x <= -4.2e-223) || ~((x <= 1.22e-71)))
		tmp = -x / B;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[Or[LessEqual[x, -4.2e-223], N[Not[LessEqual[x, 1.22e-71]], $MachinePrecision]], N[((-x) / B), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.19999999999999965e-223 or 1.21999999999999999e-71 < x

   1. Initial program 81.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. distribute-lft-neg-in 81.4%

         \[\leadsto \color{blue}{\left(-x\right) \cdot \frac{1}{\tan B}} + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. +-commutative 81.4%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x\right) \cdot \frac{1}{\tan B}} \]

      3. cancel-sign-sub-inv 81.4%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

   3. Simplified 91.4%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]

   4. Taylor expanded in F around inf 62.3%

      \[\leadsto F \cdot \color{blue}{\frac{1}{F \cdot \sin B}} - \frac{x}{\tan B} \]
   5. Step-by-step derivation

      1. associate-/r* 62.2%

         \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   6. Simplified 62.2%

      \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   7. Taylor expanded in B around 0 32.5%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

   8. Taylor expanded in x around inf 36.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   9. Step-by-step derivation

      1. associate-*r/ 36.8%

         \[\leadsto \color{blue}{\frac{-1 \cdot x}{B}} \]

      2. neg-mul-1 36.8%

         \[\leadsto \frac{\color{blue}{-x}}{B} \]

   10. Simplified 36.8%

       \[\leadsto \color{blue}{\frac{-x}{B}} \]

   if -4.19999999999999965e-223 < x < 1.21999999999999999e-71

   1. Initial program 72.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. distribute-lft-neg-in 72.2%

         \[\leadsto \color{blue}{\left(-x\right) \cdot \frac{1}{\tan B}} + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. +-commutative 72.2%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x\right) \cdot \frac{1}{\tan B}} \]

      3. cancel-sign-sub-inv 72.2%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

   3. Simplified 77.8%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]

   4. Taylor expanded in F around inf 27.9%

      \[\leadsto F \cdot \color{blue}{\frac{1}{F \cdot \sin B}} - \frac{x}{\tan B} \]
   5. Step-by-step derivation

      1. associate-/r* 27.8%

         \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   6. Simplified 27.8%

      \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   7. Taylor expanded in B around 0 24.4%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

   8. Taylor expanded in x around 0 24.4%

      \[\leadsto \color{blue}{\frac{1}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-223} \lor \neg \left(x \leq 1.22 \cdot 10^{-71}\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]

Alternative 25: 37.1% accurate, 45.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 2.8 \cdot 10^{-29}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F 2.8e-29) (/ (- x) B) (/ (- 1.0 x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= 2.8e-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 <= 2.8d-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 <= 2.8e-29) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= 2.8e-29:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= 2.8e-29)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= 2.8e-29)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, 2.8e-29], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 2.8000000000000002e-29

   1. Initial program 85.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. distribute-lft-neg-in 85.5%

         \[\leadsto \color{blue}{\left(-x\right) \cdot \frac{1}{\tan B}} + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. +-commutative 85.5%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x\right) \cdot \frac{1}{\tan B}} \]

      3. cancel-sign-sub-inv 85.5%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

   3. Simplified 91.5%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]

   4. Taylor expanded in F around inf 36.8%

      \[\leadsto F \cdot \color{blue}{\frac{1}{F \cdot \sin B}} - \frac{x}{\tan B} \]
   5. Step-by-step derivation

      1. associate-/r* 36.8%

         \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   6. Simplified 36.8%

      \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   7. Taylor expanded in B around 0 20.0%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

   8. Taylor expanded in x around inf 29.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   9. Step-by-step derivation

      1. associate-*r/ 29.4%

         \[\leadsto \color{blue}{\frac{-1 \cdot x}{B}} \]

      2. neg-mul-1 29.4%

         \[\leadsto \frac{\color{blue}{-x}}{B} \]

   10. Simplified 29.4%

       \[\leadsto \color{blue}{\frac{-x}{B}} \]

   if 2.8000000000000002e-29 < F

   1. Initial program 62.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. distribute-lft-neg-in 62.1%

         \[\leadsto \color{blue}{\left(-x\right) \cdot \frac{1}{\tan B}} + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. +-commutative 62.1%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x\right) \cdot \frac{1}{\tan B}} \]

      3. cancel-sign-sub-inv 62.1%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

   3. Simplified 78.3%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]

   4. Taylor expanded in F around inf 96.1%

      \[\leadsto F \cdot \color{blue}{\frac{1}{F \cdot \sin B}} - \frac{x}{\tan B} \]
   5. Step-by-step derivation

      1. associate-/r* 96.0%

         \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   6. Simplified 96.0%

      \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   7. Taylor expanded in B around 0 57.5%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 2.8 \cdot 10^{-29}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 26: 10.0% accurate, 108.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (/ 1.0 B))

C

double code(double F, double B, double x) {
	return 1.0 / B;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = 1.0d0 / b
end function

Java

public static double code(double F, double B, double x) {
	return 1.0 / B;
}

Python

def code(F, B, x):
	return 1.0 / B

Julia

function code(F, B, x)
	return Float64(1.0 / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = 1.0 / B;
end

Wolfram

code[F_, B_, x_] := N[(1.0 / B), $MachinePrecision]

Derivation

1. Initial program 79.0%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
2. Step-by-step derivation

   1. distribute-lft-neg-in 79.0%

      \[\leadsto \color{blue}{\left(-x\right) \cdot \frac{1}{\tan B}} + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. +-commutative 79.0%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x\right) \cdot \frac{1}{\tan B}} \]

   3. cancel-sign-sub-inv 79.0%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

3. Simplified 87.8%

   \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]

4. Taylor expanded in F around inf 53.3%

   \[\leadsto F \cdot \color{blue}{\frac{1}{F \cdot \sin B}} - \frac{x}{\tan B} \]
5. Step-by-step derivation

   1. associate-/r* 53.2%

      \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

6. Simplified 53.2%

   \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

7. Taylor expanded in B around 0 30.4%

   \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

8. Taylor expanded in x around 0 10.2%

   \[\leadsto \color{blue}{\frac{1}{B}} \]

9. Final simplification 10.2%

   \[\leadsto \frac{1}{B} \]

Reproduce

herbie shell --seed 2023287 
(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))))))