VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.4% → 99.7%

Time: 19.4s

Alternatives: 23

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -14000:\\ \;\;\;\;\frac{\frac{1}{{F}^{2}} + -1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 3350000:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -14000.0)
     (- (/ (+ (/ 1.0 (pow F 2.0)) -1.0) (sin B)) t_0)
     (if (<= F 3350000.0)
       (- (* 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 <= -14000.0) {
		tmp = (((1.0 / pow(F, 2.0)) + -1.0) / sin(B)) - t_0;
	} else if (F <= 3350000.0) {
		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 <= -14000.0)
		tmp = Float64(Float64(Float64(Float64(1.0 / (F ^ 2.0)) + -1.0) / sin(B)) - t_0);
	elseif (F <= 3350000.0)
		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, -14000.0], N[(N[(N[(N[(1.0 / N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 3350000.0], 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 < -14000

   1. Initial program 57.6%

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

   3. Simplified 70.1%

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

   5. Taylor expanded in x around 0 70.1%

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

      1. associate-*l/ 70.1%

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

      2. *-lft-identity 70.1%

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

      3. +-commutative 70.1%

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

      4. unpow2 70.1%

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

      5. fma-undefine 70.1%

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

   7. Simplified 70.1%

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

      1. associate-*r/ 70.1%

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

      2. sqrt-div 70.1%

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

      3. metadata-eval 70.1%

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

      4. un-div-inv 70.2%

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

   9. Applied egg-rr 70.2%

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

   10. Taylor expanded in F around -inf 99.7%

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

   if -14000 < F < 3.35e6

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 99.6%

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

      1. associate-*l/ 99.6%

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

      2. *-lft-identity 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. fma-undefine 99.6%

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

   7. Simplified 99.6%

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

   if 3.35e6 < F

   1. Initial program 63.3%

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

   3. Simplified 77.6%

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

   5. Taylor expanded in x around 0 77.6%

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

      1. associate-*l/ 77.6%

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

      2. *-lft-identity 77.6%

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

      3. +-commutative 77.6%

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

      4. unpow2 77.6%

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

      5. fma-undefine 77.6%

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

   7. Simplified 77.6%

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

   8. Taylor expanded in F around inf 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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -14000:\\ \;\;\;\;\frac{\frac{1}{{F}^{2}} + -1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 3350000:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.9:\\ \;\;\;\;\frac{\frac{F}{\frac{-1}{F} - F}}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 3.6 \cdot 10^{-13}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{F}{F + \frac{1}{F}}}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -0.9)
     (- (/ (/ F (- (/ -1.0 F) F)) (sin B)) t_0)
     (if (<= F 3.6e-13)
       (- (* 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 <= -0.9) {
		tmp = ((F / ((-1.0 / F) - F)) / sin(B)) - t_0;
	} else if (F <= 3.6e-13) {
		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 <= (-0.9d0)) then
        tmp = ((f / (((-1.0d0) / f) - f)) / sin(b)) - t_0
    else if (f <= 3.6d-13) 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 <= -0.9) {
		tmp = ((F / ((-1.0 / F) - F)) / Math.sin(B)) - t_0;
	} else if (F <= 3.6e-13) {
		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 <= -0.9:
		tmp = ((F / ((-1.0 / F) - F)) / math.sin(B)) - t_0
	elif F <= 3.6e-13:
		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 <= -0.9)
		tmp = Float64(Float64(Float64(F / Float64(Float64(-1.0 / F) - F)) / sin(B)) - t_0);
	elseif (F <= 3.6e-13)
		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 <= -0.9)
		tmp = ((F / ((-1.0 / F) - F)) / sin(B)) - t_0;
	elseif (F <= 3.6e-13)
		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, -0.9], N[(N[(N[(F / N[(N[(-1.0 / F), $MachinePrecision] - F), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 3.6e-13], 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 < -0.900000000000000022

   1. Initial program 57.6%

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

   3. Simplified 70.1%

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

   5. Taylor expanded in x around 0 70.1%

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

      1. associate-*l/ 70.1%

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

      2. *-lft-identity 70.1%

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

      3. +-commutative 70.1%

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

      4. unpow2 70.1%

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

      5. fma-undefine 70.1%

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

   7. Simplified 70.1%

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

      1. associate-*r/ 70.1%

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

      2. sqrt-div 70.1%

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

      3. metadata-eval 70.1%

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

      4. un-div-inv 70.2%

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

   9. Applied egg-rr 70.2%

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

   10. Taylor expanded in F around -inf 99.7%

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

       1. neg-mul-1 99.7%

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

   12. Simplified 99.7%

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

   if -0.900000000000000022 < F < 3.5999999999999998e-13

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 99.6%

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

      1. associate-*l/ 99.6%

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

      2. *-lft-identity 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. fma-undefine 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in F around 0 99.5%

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

      1. *-commutative 99.5%

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

      2. *-rgt-identity 99.5%

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

      3. times-frac 99.6%

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

      4. /-rgt-identity 99.6%

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

   10. Simplified 99.6%

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

   if 3.5999999999999998e-13 < F

   1. Initial program 64.7%

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

   3. Simplified 78.4%

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

   5. Taylor expanded in x around 0 78.5%

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

      1. associate-*l/ 78.4%

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

      2. *-lft-identity 78.4%

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

      3. +-commutative 78.4%

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

      4. unpow2 78.4%

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

      5. fma-undefine 78.4%

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

   7. Simplified 78.4%

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

      1. associate-*r/ 78.6%

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

      2. sqrt-div 78.6%

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

      3. metadata-eval 78.6%

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

      4. un-div-inv 78.6%

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

   9. Applied egg-rr 78.6%

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

   10. Taylor expanded in F around inf 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.9:\\ \;\;\;\;\frac{\frac{F}{\frac{-1}{F} - F}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 3.6 \cdot 10^{-13}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{F}{F + \frac{1}{F}}}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{\frac{1}{{F}^{2}} + -1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 3.6 \cdot 10^{-13}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{F}{F + \frac{1}{F}}}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.4)
     (- (/ (+ (/ 1.0 (pow F 2.0)) -1.0) (sin B)) t_0)
     (if (<= F 3.6e-13)
       (- (* 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 <= -1.4) {
		tmp = (((1.0 / pow(F, 2.0)) + -1.0) / sin(B)) - t_0;
	} else if (F <= 3.6e-13) {
		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 <= (-1.4d0)) then
        tmp = (((1.0d0 / (f ** 2.0d0)) + (-1.0d0)) / sin(b)) - t_0
    else if (f <= 3.6d-13) 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 <= -1.4) {
		tmp = (((1.0 / Math.pow(F, 2.0)) + -1.0) / Math.sin(B)) - t_0;
	} else if (F <= 3.6e-13) {
		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 <= -1.4:
		tmp = (((1.0 / math.pow(F, 2.0)) + -1.0) / math.sin(B)) - t_0
	elif F <= 3.6e-13:
		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 <= -1.4)
		tmp = Float64(Float64(Float64(Float64(1.0 / (F ^ 2.0)) + -1.0) / sin(B)) - t_0);
	elseif (F <= 3.6e-13)
		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 <= -1.4)
		tmp = (((1.0 / (F ^ 2.0)) + -1.0) / sin(B)) - t_0;
	elseif (F <= 3.6e-13)
		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, -1.4], N[(N[(N[(N[(1.0 / N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 3.6e-13], 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 < -1.3999999999999999

   1. Initial program 57.6%

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

   3. Simplified 70.1%

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

   5. Taylor expanded in x around 0 70.1%

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

      1. associate-*l/ 70.1%

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

      2. *-lft-identity 70.1%

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

      3. +-commutative 70.1%

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

      4. unpow2 70.1%

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

      5. fma-undefine 70.1%

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

   7. Simplified 70.1%

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

      1. associate-*r/ 70.1%

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

      2. sqrt-div 70.1%

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

      3. metadata-eval 70.1%

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

      4. un-div-inv 70.2%

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

   9. Applied egg-rr 70.2%

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

   10. Taylor expanded in F around -inf 99.7%

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

   if -1.3999999999999999 < F < 3.5999999999999998e-13

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 99.6%

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

      1. associate-*l/ 99.6%

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

      2. *-lft-identity 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. fma-undefine 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in F around 0 99.5%

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

      1. *-commutative 99.5%

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

      2. *-rgt-identity 99.5%

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

      3. times-frac 99.6%

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

      4. /-rgt-identity 99.6%

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

   10. Simplified 99.6%

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

   if 3.5999999999999998e-13 < F

   1. Initial program 64.7%

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

   3. Simplified 78.4%

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

   5. Taylor expanded in x around 0 78.5%

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

      1. associate-*l/ 78.4%

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

      2. *-lft-identity 78.4%

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

      3. +-commutative 78.4%

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

      4. unpow2 78.4%

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

      5. fma-undefine 78.4%

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

   7. Simplified 78.4%

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

      1. associate-*r/ 78.6%

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

      2. sqrt-div 78.6%

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

      3. metadata-eval 78.6%

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

      4. un-div-inv 78.6%

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

   9. Applied egg-rr 78.6%

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

   10. Taylor expanded in F around inf 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{\frac{1}{{F}^{2}} + -1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 3.6 \cdot 10^{-13}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{F}{F + \frac{1}{F}}}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + 2 \cdot x}}\\ t_1 := \frac{x}{\tan B}\\ t_2 := F \cdot \frac{\sqrt{0.5}}{B} - t\_1\\ \mathbf{if}\;F \leq -6 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{F}{\frac{-1}{F} - F}}{\sin B} - t\_1\\ \mathbf{elif}\;F \leq -1.62 \cdot 10^{-52}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq -6.4 \cdot 10^{-112}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;F \leq -5.5 \cdot 10^{-142}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 3.6 \cdot 10^{-13}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{F}{F + \frac{1}{F}}}{\sin B} - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (/ F (sin B)) (sqrt (/ 1.0 (+ 2.0 (* 2.0 x))))))
        (t_1 (/ x (tan B)))
        (t_2 (- (* F (/ (sqrt 0.5) B)) t_1)))
   (if (<= F -6e-21)
     (- (/ (/ F (- (/ -1.0 F) F)) (sin B)) t_1)
     (if (<= F -1.62e-52)
       t_0
       (if (<= F -6.4e-112)
         t_2
         (if (<= F -5.5e-142)
           t_0
           (if (<= F 3.6e-13)
             t_2
             (- (/ (/ F (+ F (/ 1.0 F))) (sin B)) t_1))))))))

C

double code(double F, double B, double x) {
	double t_0 = (F / sin(B)) * sqrt((1.0 / (2.0 + (2.0 * x))));
	double t_1 = x / tan(B);
	double t_2 = (F * (sqrt(0.5) / B)) - t_1;
	double tmp;
	if (F <= -6e-21) {
		tmp = ((F / ((-1.0 / F) - F)) / sin(B)) - t_1;
	} else if (F <= -1.62e-52) {
		tmp = t_0;
	} else if (F <= -6.4e-112) {
		tmp = t_2;
	} else if (F <= -5.5e-142) {
		tmp = t_0;
	} else if (F <= 3.6e-13) {
		tmp = t_2;
	} else {
		tmp = ((F / (F + (1.0 / F))) / sin(B)) - 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) :: t_2
    real(8) :: tmp
    t_0 = (f / sin(b)) * sqrt((1.0d0 / (2.0d0 + (2.0d0 * x))))
    t_1 = x / tan(b)
    t_2 = (f * (sqrt(0.5d0) / b)) - t_1
    if (f <= (-6d-21)) then
        tmp = ((f / (((-1.0d0) / f) - f)) / sin(b)) - t_1
    else if (f <= (-1.62d-52)) then
        tmp = t_0
    else if (f <= (-6.4d-112)) then
        tmp = t_2
    else if (f <= (-5.5d-142)) then
        tmp = t_0
    else if (f <= 3.6d-13) then
        tmp = t_2
    else
        tmp = ((f / (f + (1.0d0 / f))) / sin(b)) - t_1
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (F / Math.sin(B)) * Math.sqrt((1.0 / (2.0 + (2.0 * x))));
	double t_1 = x / Math.tan(B);
	double t_2 = (F * (Math.sqrt(0.5) / B)) - t_1;
	double tmp;
	if (F <= -6e-21) {
		tmp = ((F / ((-1.0 / F) - F)) / Math.sin(B)) - t_1;
	} else if (F <= -1.62e-52) {
		tmp = t_0;
	} else if (F <= -6.4e-112) {
		tmp = t_2;
	} else if (F <= -5.5e-142) {
		tmp = t_0;
	} else if (F <= 3.6e-13) {
		tmp = t_2;
	} else {
		tmp = ((F / (F + (1.0 / F))) / Math.sin(B)) - t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (F / math.sin(B)) * math.sqrt((1.0 / (2.0 + (2.0 * x))))
	t_1 = x / math.tan(B)
	t_2 = (F * (math.sqrt(0.5) / B)) - t_1
	tmp = 0
	if F <= -6e-21:
		tmp = ((F / ((-1.0 / F) - F)) / math.sin(B)) - t_1
	elif F <= -1.62e-52:
		tmp = t_0
	elif F <= -6.4e-112:
		tmp = t_2
	elif F <= -5.5e-142:
		tmp = t_0
	elif F <= 3.6e-13:
		tmp = t_2
	else:
		tmp = ((F / (F + (1.0 / F))) / math.sin(B)) - t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(F / sin(B)) * sqrt(Float64(1.0 / Float64(2.0 + Float64(2.0 * x)))))
	t_1 = Float64(x / tan(B))
	t_2 = Float64(Float64(F * Float64(sqrt(0.5) / B)) - t_1)
	tmp = 0.0
	if (F <= -6e-21)
		tmp = Float64(Float64(Float64(F / Float64(Float64(-1.0 / F) - F)) / sin(B)) - t_1);
	elseif (F <= -1.62e-52)
		tmp = t_0;
	elseif (F <= -6.4e-112)
		tmp = t_2;
	elseif (F <= -5.5e-142)
		tmp = t_0;
	elseif (F <= 3.6e-13)
		tmp = t_2;
	else
		tmp = Float64(Float64(Float64(F / Float64(F + Float64(1.0 / F))) / sin(B)) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (F / sin(B)) * sqrt((1.0 / (2.0 + (2.0 * x))));
	t_1 = x / tan(B);
	t_2 = (F * (sqrt(0.5) / B)) - t_1;
	tmp = 0.0;
	if (F <= -6e-21)
		tmp = ((F / ((-1.0 / F) - F)) / sin(B)) - t_1;
	elseif (F <= -1.62e-52)
		tmp = t_0;
	elseif (F <= -6.4e-112)
		tmp = t_2;
	elseif (F <= -5.5e-142)
		tmp = t_0;
	elseif (F <= 3.6e-13)
		tmp = t_2;
	else
		tmp = ((F / (F + (1.0 / F))) / sin(B)) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 + N[(2.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]}, If[LessEqual[F, -6e-21], N[(N[(N[(F / N[(N[(-1.0 / F), $MachinePrecision] - F), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, -1.62e-52], t$95$0, If[LessEqual[F, -6.4e-112], t$95$2, If[LessEqual[F, -5.5e-142], t$95$0, If[LessEqual[F, 3.6e-13], t$95$2, N[(N[(N[(F / N[(F + N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -5.99999999999999982e-21

   1. Initial program 59.7%

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

   3. Simplified 71.6%

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

   5. Taylor expanded in x around 0 71.6%

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

      1. associate-*l/ 71.6%

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

      2. *-lft-identity 71.6%

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

      3. +-commutative 71.6%

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

      4. unpow2 71.6%

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

      5. fma-undefine 71.6%

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

   7. Simplified 71.6%

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

      1. associate-*r/ 71.6%

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

      2. sqrt-div 71.7%

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

      3. metadata-eval 71.7%

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

      4. un-div-inv 71.7%

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

   9. Applied egg-rr 71.7%

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

   10. Taylor expanded in F around -inf 98.6%

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

       1. neg-mul-1 98.6%

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

   12. Simplified 98.6%

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

   if -5.99999999999999982e-21 < F < -1.61999999999999995e-52 or -6.39999999999999986e-112 < F < -5.50000000000000023e-142

   1. Initial program 99.5%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in F around 0 99.3%

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

   6. Taylor expanded in F around inf 91.7%

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

   if -1.61999999999999995e-52 < F < -6.39999999999999986e-112 or -5.50000000000000023e-142 < F < 3.5999999999999998e-13

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in F around 0 99.6%

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

   6. Taylor expanded in B around 0 88.5%

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

   7. Taylor expanded in x around 0 88.5%

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

   if 3.5999999999999998e-13 < F

   1. Initial program 64.7%

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

   3. Simplified 78.4%

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

   5. Taylor expanded in x around 0 78.5%

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

      1. associate-*l/ 78.4%

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

      2. *-lft-identity 78.4%

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

      3. +-commutative 78.4%

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

      4. unpow2 78.4%

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

      5. fma-undefine 78.4%

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

   7. Simplified 78.4%

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

      1. associate-*r/ 78.6%

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

      2. sqrt-div 78.6%

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

      3. metadata-eval 78.6%

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

      4. un-div-inv 78.6%

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

   9. Applied egg-rr 78.6%

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

   10. Taylor expanded in F around inf 99.7%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{F}{\frac{-1}{F} - F}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.62 \cdot 10^{-52}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + 2 \cdot x}}\\ \mathbf{elif}\;F \leq -6.4 \cdot 10^{-112}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -5.5 \cdot 10^{-142}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + 2 \cdot x}}\\ \mathbf{elif}\;F \leq 3.6 \cdot 10^{-13}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{F}{F + \frac{1}{F}}}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{2 + 2 \cdot x}}\\ t_1 := \frac{F}{\sin B} \cdot t\_0\\ t_2 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2.2 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{F}{\frac{-1}{F} - F}}{\sin B} - t\_2\\ \mathbf{elif}\;F \leq -1.46 \cdot 10^{-52}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;F \leq -4.2 \cdot 10^{-112}:\\ \;\;\;\;t\_0 \cdot \frac{F}{B} - t\_2\\ \mathbf{elif}\;F \leq -5.5 \cdot 10^{-142}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;F \leq 3.6 \cdot 10^{-13}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{F}{F + \frac{1}{F}}}{\sin B} - t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (+ 2.0 (* 2.0 x)))))
        (t_1 (* (/ F (sin B)) t_0))
        (t_2 (/ x (tan B))))
   (if (<= F -2.2e-20)
     (- (/ (/ F (- (/ -1.0 F) F)) (sin B)) t_2)
     (if (<= F -1.46e-52)
       t_1
       (if (<= F -4.2e-112)
         (- (* t_0 (/ F B)) t_2)
         (if (<= F -5.5e-142)
           t_1
           (if (<= F 3.6e-13)
             (- (* F (/ (sqrt 0.5) B)) t_2)
             (- (/ (/ F (+ F (/ 1.0 F))) (sin B)) t_2))))))))

C

double code(double F, double B, double x) {
	double t_0 = sqrt((1.0 / (2.0 + (2.0 * x))));
	double t_1 = (F / sin(B)) * t_0;
	double t_2 = x / tan(B);
	double tmp;
	if (F <= -2.2e-20) {
		tmp = ((F / ((-1.0 / F) - F)) / sin(B)) - t_2;
	} else if (F <= -1.46e-52) {
		tmp = t_1;
	} else if (F <= -4.2e-112) {
		tmp = (t_0 * (F / B)) - t_2;
	} else if (F <= -5.5e-142) {
		tmp = t_1;
	} else if (F <= 3.6e-13) {
		tmp = (F * (sqrt(0.5) / B)) - t_2;
	} else {
		tmp = ((F / (F + (1.0 / F))) / sin(B)) - t_2;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sqrt((1.0d0 / (2.0d0 + (2.0d0 * x))))
    t_1 = (f / sin(b)) * t_0
    t_2 = x / tan(b)
    if (f <= (-2.2d-20)) then
        tmp = ((f / (((-1.0d0) / f) - f)) / sin(b)) - t_2
    else if (f <= (-1.46d-52)) then
        tmp = t_1
    else if (f <= (-4.2d-112)) then
        tmp = (t_0 * (f / b)) - t_2
    else if (f <= (-5.5d-142)) then
        tmp = t_1
    else if (f <= 3.6d-13) then
        tmp = (f * (sqrt(0.5d0) / b)) - t_2
    else
        tmp = ((f / (f + (1.0d0 / f))) / sin(b)) - t_2
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = Math.sqrt((1.0 / (2.0 + (2.0 * x))));
	double t_1 = (F / Math.sin(B)) * t_0;
	double t_2 = x / Math.tan(B);
	double tmp;
	if (F <= -2.2e-20) {
		tmp = ((F / ((-1.0 / F) - F)) / Math.sin(B)) - t_2;
	} else if (F <= -1.46e-52) {
		tmp = t_1;
	} else if (F <= -4.2e-112) {
		tmp = (t_0 * (F / B)) - t_2;
	} else if (F <= -5.5e-142) {
		tmp = t_1;
	} else if (F <= 3.6e-13) {
		tmp = (F * (Math.sqrt(0.5) / B)) - t_2;
	} else {
		tmp = ((F / (F + (1.0 / F))) / Math.sin(B)) - t_2;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = math.sqrt((1.0 / (2.0 + (2.0 * x))))
	t_1 = (F / math.sin(B)) * t_0
	t_2 = x / math.tan(B)
	tmp = 0
	if F <= -2.2e-20:
		tmp = ((F / ((-1.0 / F) - F)) / math.sin(B)) - t_2
	elif F <= -1.46e-52:
		tmp = t_1
	elif F <= -4.2e-112:
		tmp = (t_0 * (F / B)) - t_2
	elif F <= -5.5e-142:
		tmp = t_1
	elif F <= 3.6e-13:
		tmp = (F * (math.sqrt(0.5) / B)) - t_2
	else:
		tmp = ((F / (F + (1.0 / F))) / math.sin(B)) - t_2
	return tmp

Julia

function code(F, B, x)
	t_0 = sqrt(Float64(1.0 / Float64(2.0 + Float64(2.0 * x))))
	t_1 = Float64(Float64(F / sin(B)) * t_0)
	t_2 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -2.2e-20)
		tmp = Float64(Float64(Float64(F / Float64(Float64(-1.0 / F) - F)) / sin(B)) - t_2);
	elseif (F <= -1.46e-52)
		tmp = t_1;
	elseif (F <= -4.2e-112)
		tmp = Float64(Float64(t_0 * Float64(F / B)) - t_2);
	elseif (F <= -5.5e-142)
		tmp = t_1;
	elseif (F <= 3.6e-13)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / B)) - t_2);
	else
		tmp = Float64(Float64(Float64(F / Float64(F + Float64(1.0 / F))) / sin(B)) - t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = sqrt((1.0 / (2.0 + (2.0 * x))));
	t_1 = (F / sin(B)) * t_0;
	t_2 = x / tan(B);
	tmp = 0.0;
	if (F <= -2.2e-20)
		tmp = ((F / ((-1.0 / F) - F)) / sin(B)) - t_2;
	elseif (F <= -1.46e-52)
		tmp = t_1;
	elseif (F <= -4.2e-112)
		tmp = (t_0 * (F / B)) - t_2;
	elseif (F <= -5.5e-142)
		tmp = t_1;
	elseif (F <= 3.6e-13)
		tmp = (F * (sqrt(0.5) / B)) - t_2;
	else
		tmp = ((F / (F + (1.0 / F))) / sin(B)) - t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(2.0 + N[(2.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.2e-20], N[(N[(N[(F / N[(N[(-1.0 / F), $MachinePrecision] - F), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision], If[LessEqual[F, -1.46e-52], t$95$1, If[LessEqual[F, -4.2e-112], N[(N[(t$95$0 * N[(F / B), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision], If[LessEqual[F, -5.5e-142], t$95$1, If[LessEqual[F, 3.6e-13], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision], N[(N[(N[(F / N[(F + N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -2.19999999999999991e-20

   1. Initial program 59.7%

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

   3. Simplified 71.6%

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

   5. Taylor expanded in x around 0 71.6%

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

      1. associate-*l/ 71.6%

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

      2. *-lft-identity 71.6%

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

      3. +-commutative 71.6%

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

      4. unpow2 71.6%

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

      5. fma-undefine 71.6%

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

   7. Simplified 71.6%

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

      1. associate-*r/ 71.6%

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

      2. sqrt-div 71.7%

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

      3. metadata-eval 71.7%

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

      4. un-div-inv 71.7%

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

   9. Applied egg-rr 71.7%

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

   10. Taylor expanded in F around -inf 98.6%

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

       1. neg-mul-1 98.6%

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

   12. Simplified 98.6%

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

   if -2.19999999999999991e-20 < F < -1.46000000000000003e-52 or -4.2000000000000001e-112 < F < -5.50000000000000023e-142

   1. Initial program 99.5%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in F around 0 99.3%

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

   6. Taylor expanded in F around inf 91.7%

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

   if -1.46000000000000003e-52 < F < -4.2000000000000001e-112

   1. Initial program 99.6%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in F around 0 99.6%

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

   6. Taylor expanded in B around 0 75.6%

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

   if -5.50000000000000023e-142 < F < 3.5999999999999998e-13

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in F around 0 99.6%

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

   6. Taylor expanded in B around 0 90.6%

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

   7. Taylor expanded in x around 0 90.6%

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

   if 3.5999999999999998e-13 < F

   1. Initial program 64.7%

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

   3. Simplified 78.4%

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

   5. Taylor expanded in x around 0 78.5%

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

      1. associate-*l/ 78.4%

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

      2. *-lft-identity 78.4%

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

      3. +-commutative 78.4%

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

      4. unpow2 78.4%

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

      5. fma-undefine 78.4%

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

   7. Simplified 78.4%

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

      1. associate-*r/ 78.6%

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

      2. sqrt-div 78.6%

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

      3. metadata-eval 78.6%

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

      4. un-div-inv 78.6%

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

   9. Applied egg-rr 78.6%

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

   10. Taylor expanded in F around inf 99.7%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.2 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{F}{\frac{-1}{F} - F}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.46 \cdot 10^{-52}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + 2 \cdot x}}\\ \mathbf{elif}\;F \leq -4.2 \cdot 10^{-112}:\\ \;\;\;\;\sqrt{\frac{1}{2 + 2 \cdot x}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -5.5 \cdot 10^{-142}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + 2 \cdot x}}\\ \mathbf{elif}\;F \leq 3.6 \cdot 10^{-13}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{F}{F + \frac{1}{F}}}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{1}{F}}{\frac{\sin B}{F}} - \frac{x}{B}\\ \mathbf{if}\;F \leq -25500000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 7.6 \cdot 10^{-23}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{+102}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 6 \cdot 10^{+215}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 8.5 \cdot 10^{+296}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\sin B} \cdot \left(-\cos B\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ (/ 1.0 F) (/ (sin B) F)) (/ x B))))
   (if (<= F -25500000.0)
     (- (/ -1.0 (sin B)) (/ x B))
     (if (<= F 7.6e-23)
       (/ (- x) (tan B))
       (if (<= F 4.5e+102)
         t_0
         (if (<= F 6e+215)
           (- (/ 1.0 B) (/ x (tan B)))
           (if (<= F 8.5e+296) t_0 (* (/ x (sin B)) (- (cos B))))))))))

C

double code(double F, double B, double x) {
	double t_0 = ((1.0 / F) / (sin(B) / F)) - (x / B);
	double tmp;
	if (F <= -25500000.0) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 7.6e-23) {
		tmp = -x / tan(B);
	} else if (F <= 4.5e+102) {
		tmp = t_0;
	} else if (F <= 6e+215) {
		tmp = (1.0 / B) - (x / tan(B));
	} else if (F <= 8.5e+296) {
		tmp = t_0;
	} else {
		tmp = (x / sin(B)) * -cos(B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((1.0d0 / f) / (sin(b) / f)) - (x / b)
    if (f <= (-25500000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 7.6d-23) then
        tmp = -x / tan(b)
    else if (f <= 4.5d+102) then
        tmp = t_0
    else if (f <= 6d+215) then
        tmp = (1.0d0 / b) - (x / tan(b))
    else if (f <= 8.5d+296) then
        tmp = t_0
    else
        tmp = (x / sin(b)) * -cos(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = ((1.0 / F) / (Math.sin(B) / F)) - (x / B);
	double tmp;
	if (F <= -25500000.0) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 7.6e-23) {
		tmp = -x / Math.tan(B);
	} else if (F <= 4.5e+102) {
		tmp = t_0;
	} else if (F <= 6e+215) {
		tmp = (1.0 / B) - (x / Math.tan(B));
	} else if (F <= 8.5e+296) {
		tmp = t_0;
	} else {
		tmp = (x / Math.sin(B)) * -Math.cos(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = ((1.0 / F) / (math.sin(B) / F)) - (x / B)
	tmp = 0
	if F <= -25500000.0:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 7.6e-23:
		tmp = -x / math.tan(B)
	elif F <= 4.5e+102:
		tmp = t_0
	elif F <= 6e+215:
		tmp = (1.0 / B) - (x / math.tan(B))
	elif F <= 8.5e+296:
		tmp = t_0
	else:
		tmp = (x / math.sin(B)) * -math.cos(B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(Float64(1.0 / F) / Float64(sin(B) / F)) - Float64(x / B))
	tmp = 0.0
	if (F <= -25500000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 7.6e-23)
		tmp = Float64(Float64(-x) / tan(B));
	elseif (F <= 4.5e+102)
		tmp = t_0;
	elseif (F <= 6e+215)
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	elseif (F <= 8.5e+296)
		tmp = t_0;
	else
		tmp = Float64(Float64(x / sin(B)) * Float64(-cos(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = ((1.0 / F) / (sin(B) / F)) - (x / B);
	tmp = 0.0;
	if (F <= -25500000.0)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 7.6e-23)
		tmp = -x / tan(B);
	elseif (F <= 4.5e+102)
		tmp = t_0;
	elseif (F <= 6e+215)
		tmp = (1.0 / B) - (x / tan(B));
	elseif (F <= 8.5e+296)
		tmp = t_0;
	else
		tmp = (x / sin(B)) * -cos(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(N[(1.0 / F), $MachinePrecision] / N[(N[Sin[B], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -25500000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7.6e-23], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.5e+102], t$95$0, If[LessEqual[F, 6e+215], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 8.5e+296], t$95$0, N[(N[(x / N[Sin[B], $MachinePrecision]), $MachinePrecision] * (-N[Cos[B], $MachinePrecision])), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -2.55e7

   1. Initial program 55.9%

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

   3. Taylor expanded in F around -inf 99.6%

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

   4. Taylor expanded in B around 0 74.7%

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

   if -2.55e7 < F < 7.60000000000000023e-23

   1. Initial program 99.5%

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

   3. Taylor expanded in F around -inf 36.8%

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

   4. Taylor expanded in x around inf 69.2%

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

      1. associate-/l* 69.2%

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

      2. tan-quot 69.2%

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

      3. div-inv 69.2%

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

   6. Applied egg-rr 69.2%

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

      1. associate-*r/ 69.2%

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

      2. *-rgt-identity 69.2%

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

   8. Simplified 69.2%

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

   if 7.60000000000000023e-23 < F < 4.50000000000000021e102 or 5.9999999999999998e215 < F < 8.5000000000000003e296

   1. Initial program 66.4%

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

   3. Taylor expanded in F around inf 86.5%

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

      1. *-commutative 86.5%

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

      2. clear-num 86.6%

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

      3. un-div-inv 89.1%

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

   5. Applied egg-rr 89.1%

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

   6. Taylor expanded in B around 0 82.6%

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

   if 4.50000000000000021e102 < F < 5.9999999999999998e215

   1. Initial program 62.6%

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

   3. Taylor expanded in F around inf 74.4%

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

   4. Taylor expanded in B around 0 82.1%

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

      1. +-commutative 82.1%

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

      2. div-inv 82.2%

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

      3. unsub-neg 82.2%

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

   6. Applied egg-rr 82.2%

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

   if 8.5000000000000003e296 < F

   1. Initial program 98.4%

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

   3. Taylor expanded in F around -inf 98.4%

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

   4. Taylor expanded in x around inf 98.4%

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

      1. mul-1-neg 98.4%

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

      2. associate-*l/ 99.2%

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

      3. distribute-rgt-neg-in 99.2%

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

   6. Simplified 99.2%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -25500000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 7.6 \cdot 10^{-23}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{1}{F}}{\frac{\sin B}{F}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 6 \cdot 10^{+215}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 8.5 \cdot 10^{+296}:\\ \;\;\;\;\frac{\frac{1}{F}}{\frac{\sin B}{F}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\sin B} \cdot \left(-\cos B\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{1}{F}}{\frac{\sin B}{F}} - \frac{x}{B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.000155:\\ \;\;\;\;\frac{-1}{\sin B} - t\_1\\ \mathbf{elif}\;F \leq 1.42 \cdot 10^{-24}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 3.8 \cdot 10^{+102}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 1.36 \cdot 10^{+216}:\\ \;\;\;\;\frac{1}{B} - t\_1\\ \mathbf{elif}\;F \leq 1.7 \cdot 10^{+296}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\sin B} \cdot \left(-\cos B\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ (/ 1.0 F) (/ (sin B) F)) (/ x B))) (t_1 (/ x (tan B))))
   (if (<= F -0.000155)
     (- (/ -1.0 (sin B)) t_1)
     (if (<= F 1.42e-24)
       (/ (- x) (tan B))
       (if (<= F 3.8e+102)
         t_0
         (if (<= F 1.36e+216)
           (- (/ 1.0 B) t_1)
           (if (<= F 1.7e+296) t_0 (* (/ x (sin B)) (- (cos B))))))))))

C

double code(double F, double B, double x) {
	double t_0 = ((1.0 / F) / (sin(B) / F)) - (x / B);
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -0.000155) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= 1.42e-24) {
		tmp = -x / tan(B);
	} else if (F <= 3.8e+102) {
		tmp = t_0;
	} else if (F <= 1.36e+216) {
		tmp = (1.0 / B) - t_1;
	} else if (F <= 1.7e+296) {
		tmp = t_0;
	} else {
		tmp = (x / sin(B)) * -cos(B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((1.0d0 / f) / (sin(b) / f)) - (x / b)
    t_1 = x / tan(b)
    if (f <= (-0.000155d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_1
    else if (f <= 1.42d-24) then
        tmp = -x / tan(b)
    else if (f <= 3.8d+102) then
        tmp = t_0
    else if (f <= 1.36d+216) then
        tmp = (1.0d0 / b) - t_1
    else if (f <= 1.7d+296) then
        tmp = t_0
    else
        tmp = (x / sin(b)) * -cos(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = ((1.0 / F) / (Math.sin(B) / F)) - (x / B);
	double t_1 = x / Math.tan(B);
	double tmp;
	if (F <= -0.000155) {
		tmp = (-1.0 / Math.sin(B)) - t_1;
	} else if (F <= 1.42e-24) {
		tmp = -x / Math.tan(B);
	} else if (F <= 3.8e+102) {
		tmp = t_0;
	} else if (F <= 1.36e+216) {
		tmp = (1.0 / B) - t_1;
	} else if (F <= 1.7e+296) {
		tmp = t_0;
	} else {
		tmp = (x / Math.sin(B)) * -Math.cos(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = ((1.0 / F) / (math.sin(B) / F)) - (x / B)
	t_1 = x / math.tan(B)
	tmp = 0
	if F <= -0.000155:
		tmp = (-1.0 / math.sin(B)) - t_1
	elif F <= 1.42e-24:
		tmp = -x / math.tan(B)
	elif F <= 3.8e+102:
		tmp = t_0
	elif F <= 1.36e+216:
		tmp = (1.0 / B) - t_1
	elif F <= 1.7e+296:
		tmp = t_0
	else:
		tmp = (x / math.sin(B)) * -math.cos(B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(Float64(1.0 / F) / Float64(sin(B) / F)) - Float64(x / B))
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -0.000155)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= 1.42e-24)
		tmp = Float64(Float64(-x) / tan(B));
	elseif (F <= 3.8e+102)
		tmp = t_0;
	elseif (F <= 1.36e+216)
		tmp = Float64(Float64(1.0 / B) - t_1);
	elseif (F <= 1.7e+296)
		tmp = t_0;
	else
		tmp = Float64(Float64(x / sin(B)) * Float64(-cos(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = ((1.0 / F) / (sin(B) / F)) - (x / B);
	t_1 = x / tan(B);
	tmp = 0.0;
	if (F <= -0.000155)
		tmp = (-1.0 / sin(B)) - t_1;
	elseif (F <= 1.42e-24)
		tmp = -x / tan(B);
	elseif (F <= 3.8e+102)
		tmp = t_0;
	elseif (F <= 1.36e+216)
		tmp = (1.0 / B) - t_1;
	elseif (F <= 1.7e+296)
		tmp = t_0;
	else
		tmp = (x / sin(B)) * -cos(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(N[(1.0 / F), $MachinePrecision] / N[(N[Sin[B], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.000155], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, 1.42e-24], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.8e+102], t$95$0, If[LessEqual[F, 1.36e+216], N[(N[(1.0 / B), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, 1.7e+296], t$95$0, N[(N[(x / N[Sin[B], $MachinePrecision]), $MachinePrecision] * (-N[Cos[B], $MachinePrecision])), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -1.55e-4

   1. Initial program 57.6%

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

   3. Simplified 70.1%

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

   5. Taylor expanded in x around 0 70.1%

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

      1. associate-*l/ 70.1%

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

      2. *-lft-identity 70.1%

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

      3. +-commutative 70.1%

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

      4. unpow2 70.1%

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

      5. fma-undefine 70.1%

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

   7. Simplified 70.1%

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

   8. Taylor expanded in F around -inf 99.5%

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

   if -1.55e-4 < F < 1.42e-24

   1. Initial program 99.5%

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

   3. Taylor expanded in F around -inf 35.1%

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

   4. Taylor expanded in x around inf 69.2%

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

      1. associate-/l* 69.2%

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

      2. tan-quot 69.3%

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

      3. div-inv 69.3%

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

   6. Applied egg-rr 69.3%

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

      1. associate-*r/ 69.3%

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

      2. *-rgt-identity 69.3%

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

   8. Simplified 69.3%

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

   if 1.42e-24 < F < 3.79999999999999979e102 or 1.36000000000000007e216 < F < 1.7000000000000001e296

   1. Initial program 66.4%

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

   3. Taylor expanded in F around inf 86.5%

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

      1. *-commutative 86.5%

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

      2. clear-num 86.6%

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

      3. un-div-inv 89.1%

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

   5. Applied egg-rr 89.1%

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

   6. Taylor expanded in B around 0 82.6%

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

   if 3.79999999999999979e102 < F < 1.36000000000000007e216

   1. Initial program 62.6%

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

   3. Taylor expanded in F around inf 74.4%

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

   4. Taylor expanded in B around 0 82.1%

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

      1. +-commutative 82.1%

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

      2. div-inv 82.2%

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

      3. unsub-neg 82.2%

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

   6. Applied egg-rr 82.2%

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

   if 1.7000000000000001e296 < F

   1. Initial program 98.4%

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

   3. Taylor expanded in F around -inf 98.4%

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

   4. Taylor expanded in x around inf 98.4%

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

      1. mul-1-neg 98.4%

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

      2. associate-*l/ 99.2%

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

      3. distribute-rgt-neg-in 99.2%

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

   6. Simplified 99.2%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.000155:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.42 \cdot 10^{-24}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 3.8 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{1}{F}}{\frac{\sin B}{F}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.36 \cdot 10^{+216}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.7 \cdot 10^{+296}:\\ \;\;\;\;\frac{\frac{1}{F}}{\frac{\sin B}{F}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\sin B} \cdot \left(-\cos B\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2.7 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{F}{\frac{-1}{F} - F}}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq -4.4 \cdot 10^{-196}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + 2 \cdot x\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.6 \cdot 10^{-13}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{F}{F + \frac{1}{F}}}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -2.7e-20)
     (- (/ (/ F (- (/ -1.0 F) F)) (sin B)) t_0)
     (if (<= F -4.4e-196)
       (- (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* 2.0 x)) -0.5)) (/ x B))
       (if (<= F 3.6e-13)
         (- (* F (/ (sqrt 0.5) 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 <= -2.7e-20) {
		tmp = ((F / ((-1.0 / F) - F)) / sin(B)) - t_0;
	} else if (F <= -4.4e-196) {
		tmp = ((F / sin(B)) * pow(((2.0 + (F * F)) + (2.0 * x)), -0.5)) - (x / B);
	} else if (F <= 3.6e-13) {
		tmp = (F * (sqrt(0.5) / 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 <= (-2.7d-20)) then
        tmp = ((f / (((-1.0d0) / f) - f)) / sin(b)) - t_0
    else if (f <= (-4.4d-196)) then
        tmp = ((f / sin(b)) * (((2.0d0 + (f * f)) + (2.0d0 * x)) ** (-0.5d0))) - (x / b)
    else if (f <= 3.6d-13) then
        tmp = (f * (sqrt(0.5d0) / 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 <= -2.7e-20) {
		tmp = ((F / ((-1.0 / F) - F)) / Math.sin(B)) - t_0;
	} else if (F <= -4.4e-196) {
		tmp = ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (2.0 * x)), -0.5)) - (x / B);
	} else if (F <= 3.6e-13) {
		tmp = (F * (Math.sqrt(0.5) / 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 <= -2.7e-20:
		tmp = ((F / ((-1.0 / F) - F)) / math.sin(B)) - t_0
	elif F <= -4.4e-196:
		tmp = ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (2.0 * x)), -0.5)) - (x / B)
	elif F <= 3.6e-13:
		tmp = (F * (math.sqrt(0.5) / 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 <= -2.7e-20)
		tmp = Float64(Float64(Float64(F / Float64(Float64(-1.0 / F) - F)) / sin(B)) - t_0);
	elseif (F <= -4.4e-196)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(2.0 * x)) ^ -0.5)) - Float64(x / B));
	elseif (F <= 3.6e-13)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / 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 <= -2.7e-20)
		tmp = ((F / ((-1.0 / F) - F)) / sin(B)) - t_0;
	elseif (F <= -4.4e-196)
		tmp = ((F / sin(B)) * (((2.0 + (F * F)) + (2.0 * x)) ^ -0.5)) - (x / B);
	elseif (F <= 3.6e-13)
		tmp = (F * (sqrt(0.5) / 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, -2.7e-20], N[(N[(N[(F / N[(N[(-1.0 / F), $MachinePrecision] - F), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -4.4e-196], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.6e-13], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / 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 4 regimes

2. if F < -2.7e-20

   1. Initial program 59.7%

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

   3. Simplified 71.6%

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

   5. Taylor expanded in x around 0 71.6%

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

      1. associate-*l/ 71.6%

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

      2. *-lft-identity 71.6%

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

      3. +-commutative 71.6%

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

      4. unpow2 71.6%

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

      5. fma-undefine 71.6%

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

   7. Simplified 71.6%

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

      1. associate-*r/ 71.6%

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

      2. sqrt-div 71.7%

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

      3. metadata-eval 71.7%

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

      4. un-div-inv 71.7%

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

   9. Applied egg-rr 71.7%

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

   10. Taylor expanded in F around -inf 98.6%

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

       1. neg-mul-1 98.6%

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

   12. Simplified 98.6%

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

   if -2.7e-20 < F < -4.4000000000000003e-196

   1. Initial program 99.5%

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

   3. Taylor expanded in B around 0 80.5%

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

   if -4.4000000000000003e-196 < F < 3.5999999999999998e-13

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in F around 0 99.5%

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

   6. Taylor expanded in B around 0 94.8%

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

   7. Taylor expanded in x around 0 94.8%

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

   if 3.5999999999999998e-13 < F

   1. Initial program 64.7%

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

   3. Simplified 78.4%

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

   5. Taylor expanded in x around 0 78.5%

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

      1. associate-*l/ 78.4%

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

      2. *-lft-identity 78.4%

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

      3. +-commutative 78.4%

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

      4. unpow2 78.4%

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

      5. fma-undefine 78.4%

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

   7. Simplified 78.4%

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

      1. associate-*r/ 78.6%

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

      2. sqrt-div 78.6%

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

      3. metadata-eval 78.6%

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

      4. un-div-inv 78.6%

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

   9. Applied egg-rr 78.6%

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

   10. Taylor expanded in F around inf 99.7%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.7 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{F}{\frac{-1}{F} - F}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -4.4 \cdot 10^{-196}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + 2 \cdot x\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.6 \cdot 10^{-13}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{F}{F + \frac{1}{F}}}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 92.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.115:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 3.6 \cdot 10^{-13}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{F}{F + \frac{1}{F}}}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -0.115)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 3.6e-13)
       (- (* F (/ (sqrt 0.5) 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 <= -0.115) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 3.6e-13) {
		tmp = (F * (sqrt(0.5) / 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 <= (-0.115d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 3.6d-13) then
        tmp = (f * (sqrt(0.5d0) / 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 <= -0.115) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 3.6e-13) {
		tmp = (F * (Math.sqrt(0.5) / 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 <= -0.115:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 3.6e-13:
		tmp = (F * (math.sqrt(0.5) / 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 <= -0.115)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 3.6e-13)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / 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 <= -0.115)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 3.6e-13)
		tmp = (F * (sqrt(0.5) / 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, -0.115], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 3.6e-13], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / 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 < -0.115000000000000005

   1. Initial program 57.6%

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

   3. Simplified 70.1%

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

   5. Taylor expanded in x around 0 70.1%

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

      1. associate-*l/ 70.1%

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

      2. *-lft-identity 70.1%

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

      3. +-commutative 70.1%

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

      4. unpow2 70.1%

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

      5. fma-undefine 70.1%

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

   7. Simplified 70.1%

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

   8. Taylor expanded in F around -inf 99.5%

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

   if -0.115000000000000005 < F < 3.5999999999999998e-13

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in F around 0 99.6%

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

   6. Taylor expanded in B around 0 82.8%

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

   7. Taylor expanded in x around 0 82.9%

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

   if 3.5999999999999998e-13 < F

   1. Initial program 64.7%

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

   3. Simplified 78.4%

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

   5. Taylor expanded in x around 0 78.5%

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

      1. associate-*l/ 78.4%

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

      2. *-lft-identity 78.4%

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

      3. +-commutative 78.4%

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

      4. unpow2 78.4%

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

      5. fma-undefine 78.4%

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

   7. Simplified 78.4%

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

      1. associate-*r/ 78.6%

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

      2. sqrt-div 78.6%

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

      3. metadata-eval 78.6%

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

      4. un-div-inv 78.6%

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

   9. Applied egg-rr 78.6%

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

   10. Taylor expanded in F around inf 99.7%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.115:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 3.6 \cdot 10^{-13}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{F}{F + \frac{1}{F}}}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 92.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.17:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 3.6 \cdot 10^{-13}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -0.17)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 3.6e-13)
       (- (* F (/ (sqrt 0.5) B)) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -0.17) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 3.6e-13) {
		tmp = (F * (sqrt(0.5) / B)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-0.17d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 3.6d-13) then
        tmp = (f * (sqrt(0.5d0) / b)) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -0.17) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 3.6e-13) {
		tmp = (F * (Math.sqrt(0.5) / B)) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -0.17:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 3.6e-13:
		tmp = (F * (math.sqrt(0.5) / B)) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -0.17)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 3.6e-13)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / B)) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -0.17)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 3.6e-13)
		tmp = (F * (sqrt(0.5) / B)) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.17], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 3.6e-13], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.170000000000000012

   1. Initial program 57.6%

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

   3. Simplified 70.1%

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

   5. Taylor expanded in x around 0 70.1%

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

      1. associate-*l/ 70.1%

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

      2. *-lft-identity 70.1%

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

      3. +-commutative 70.1%

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

      4. unpow2 70.1%

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

      5. fma-undefine 70.1%

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

   7. Simplified 70.1%

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

   8. Taylor expanded in F around -inf 99.5%

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

   if -0.170000000000000012 < F < 3.5999999999999998e-13

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in F around 0 99.6%

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

   6. Taylor expanded in B around 0 82.8%

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

   7. Taylor expanded in x around 0 82.9%

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

   if 3.5999999999999998e-13 < F

   1. Initial program 64.7%

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

   3. Simplified 78.4%

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

   5. Taylor expanded in x around 0 78.5%

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

      1. associate-*l/ 78.4%

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

      2. *-lft-identity 78.4%

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

      3. +-commutative 78.4%

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

      4. unpow2 78.4%

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

      5. fma-undefine 78.4%

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

   7. Simplified 78.4%

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

   8. Taylor expanded in F around inf 99.5%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.17:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 3.6 \cdot 10^{-13}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 85.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.000155:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1.58 \cdot 10^{-86}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -0.000155)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.58e-86) (/ (- x) (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 <= -0.000155) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.58e-86) {
		tmp = -x / 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 <= (-0.000155d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.58d-86) then
        tmp = -x / 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 <= -0.000155) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.58e-86) {
		tmp = -x / 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 <= -0.000155:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.58e-86:
		tmp = -x / 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 <= -0.000155)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.58e-86)
		tmp = Float64(Float64(-x) / 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 <= -0.000155)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.58e-86)
		tmp = -x / 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, -0.000155], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.58e-86], N[((-x) / N[Tan[B], $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 < -1.55e-4

   1. Initial program 57.6%

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

   3. Simplified 70.1%

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

   5. Taylor expanded in x around 0 70.1%

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

      1. associate-*l/ 70.1%

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

      2. *-lft-identity 70.1%

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

      3. +-commutative 70.1%

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

      4. unpow2 70.1%

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

      5. fma-undefine 70.1%

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

   7. Simplified 70.1%

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

   8. Taylor expanded in F around -inf 99.5%

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

   if -1.55e-4 < F < 1.58000000000000007e-86

   1. Initial program 99.4%

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

   3. Taylor expanded in F around -inf 32.4%

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

   4. Taylor expanded in x around inf 69.0%

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

      1. associate-/l* 69.0%

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

      2. tan-quot 69.1%

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

      3. div-inv 69.0%

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

   6. Applied egg-rr 69.0%

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

      1. associate-*r/ 69.1%

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

      2. *-rgt-identity 69.1%

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

   8. Simplified 69.1%

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

   if 1.58000000000000007e-86 < F

   1. Initial program 68.4%

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

   3. Simplified 80.6%

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

   5. Taylor expanded in x around 0 80.7%

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

      1. associate-*l/ 80.6%

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

      2. *-lft-identity 80.6%

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

      3. +-commutative 80.6%

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

      4. unpow2 80.6%

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

      5. fma-undefine 80.6%

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

   7. Simplified 80.6%

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

   8. Taylor expanded in F around inf 95.3%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.000155:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.58 \cdot 10^{-86}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 69.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{\tan B}\\ t_1 := \frac{\frac{1}{F}}{\frac{\sin B}{F}} - \frac{x}{B}\\ \mathbf{if}\;F \leq -25500000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 7.6 \cdot 10^{-23}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{+102}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;F \leq 4.05 \cdot 10^{+217}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 8.5 \cdot 10^{+296}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))) (t_1 (- (/ (/ 1.0 F) (/ (sin B) F)) (/ x B))))
   (if (<= F -25500000.0)
     (- (/ -1.0 (sin B)) (/ x B))
     (if (<= F 7.6e-23)
       t_0
       (if (<= F 7.5e+102)
         t_1
         (if (<= F 4.05e+217)
           (- (/ 1.0 B) (/ x (tan B)))
           (if (<= F 8.5e+296) t_1 t_0)))))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double t_1 = ((1.0 / F) / (sin(B) / F)) - (x / B);
	double tmp;
	if (F <= -25500000.0) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 7.6e-23) {
		tmp = t_0;
	} else if (F <= 7.5e+102) {
		tmp = t_1;
	} else if (F <= 4.05e+217) {
		tmp = (1.0 / B) - (x / tan(B));
	} else if (F <= 8.5e+296) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = -x / tan(b)
    t_1 = ((1.0d0 / f) / (sin(b) / f)) - (x / b)
    if (f <= (-25500000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 7.6d-23) then
        tmp = t_0
    else if (f <= 7.5d+102) then
        tmp = t_1
    else if (f <= 4.05d+217) then
        tmp = (1.0d0 / b) - (x / tan(b))
    else if (f <= 8.5d+296) then
        tmp = t_1
    else
        tmp = 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 t_1 = ((1.0 / F) / (Math.sin(B) / F)) - (x / B);
	double tmp;
	if (F <= -25500000.0) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 7.6e-23) {
		tmp = t_0;
	} else if (F <= 7.5e+102) {
		tmp = t_1;
	} else if (F <= 4.05e+217) {
		tmp = (1.0 / B) - (x / Math.tan(B));
	} else if (F <= 8.5e+296) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -x / math.tan(B)
	t_1 = ((1.0 / F) / (math.sin(B) / F)) - (x / B)
	tmp = 0
	if F <= -25500000.0:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 7.6e-23:
		tmp = t_0
	elif F <= 7.5e+102:
		tmp = t_1
	elif F <= 4.05e+217:
		tmp = (1.0 / B) - (x / math.tan(B))
	elif F <= 8.5e+296:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	t_1 = Float64(Float64(Float64(1.0 / F) / Float64(sin(B) / F)) - Float64(x / B))
	tmp = 0.0
	if (F <= -25500000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 7.6e-23)
		tmp = t_0;
	elseif (F <= 7.5e+102)
		tmp = t_1;
	elseif (F <= 4.05e+217)
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	elseif (F <= 8.5e+296)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = -x / tan(B);
	t_1 = ((1.0 / F) / (sin(B) / F)) - (x / B);
	tmp = 0.0;
	if (F <= -25500000.0)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 7.6e-23)
		tmp = t_0;
	elseif (F <= 7.5e+102)
		tmp = t_1;
	elseif (F <= 4.05e+217)
		tmp = (1.0 / B) - (x / tan(B));
	elseif (F <= 8.5e+296)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(1.0 / F), $MachinePrecision] / N[(N[Sin[B], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -25500000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7.6e-23], t$95$0, If[LessEqual[F, 7.5e+102], t$95$1, If[LessEqual[F, 4.05e+217], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 8.5e+296], t$95$1, t$95$0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.55e7

   1. Initial program 55.9%

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

   3. Taylor expanded in F around -inf 99.6%

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

   4. Taylor expanded in B around 0 74.7%

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

   if -2.55e7 < F < 7.60000000000000023e-23 or 8.5000000000000003e296 < F

   1. Initial program 99.4%

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

   3. Taylor expanded in F around -inf 38.0%

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

   4. Taylor expanded in x around inf 69.7%

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

      1. associate-/l* 69.7%

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

      2. tan-quot 69.8%

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

      3. div-inv 69.7%

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

   6. Applied egg-rr 69.7%

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

      1. associate-*r/ 69.8%

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

      2. *-rgt-identity 69.8%

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

   8. Simplified 69.8%

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

   if 7.60000000000000023e-23 < F < 7.5e102 or 4.0499999999999998e217 < F < 8.5000000000000003e296

   1. Initial program 66.4%

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

   3. Taylor expanded in F around inf 86.5%

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

      1. *-commutative 86.5%

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

      2. clear-num 86.6%

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

      3. un-div-inv 89.1%

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

   5. Applied egg-rr 89.1%

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

   6. Taylor expanded in B around 0 82.6%

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

   if 7.5e102 < F < 4.0499999999999998e217

   1. Initial program 62.6%

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

   3. Taylor expanded in F around inf 74.4%

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

   4. Taylor expanded in B around 0 82.1%

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

      1. +-commutative 82.1%

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

      2. div-inv 82.2%

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

      3. unsub-neg 82.2%

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

   6. Applied egg-rr 82.2%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -25500000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 7.6 \cdot 10^{-23}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{1}{F}}{\frac{\sin B}{F}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4.05 \cdot 10^{+217}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 8.5 \cdot 10^{+296}:\\ \;\;\;\;\frac{\frac{1}{F}}{\frac{\sin B}{F}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 71.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -25500000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 7.6 \cdot 10^{-23}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 7.4 \cdot 10^{+102}:\\ \;\;\;\;\frac{1}{F} \cdot \frac{F}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -25500000.0)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 7.6e-23)
     (/ (- x) (tan B))
     (if (<= F 7.4e+102)
       (- (* (/ 1.0 F) (/ F (sin B))) (/ x B))
       (- (/ 1.0 B) (/ x (tan B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -25500000.0) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 7.6e-23) {
		tmp = -x / tan(B);
	} else if (F <= 7.4e+102) {
		tmp = ((1.0 / F) * (F / sin(B))) - (x / B);
	} else {
		tmp = (1.0 / B) - (x / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-25500000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 7.6d-23) then
        tmp = -x / tan(b)
    else if (f <= 7.4d+102) then
        tmp = ((1.0d0 / f) * (f / sin(b))) - (x / b)
    else
        tmp = (1.0d0 / b) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -25500000.0) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 7.6e-23) {
		tmp = -x / Math.tan(B);
	} else if (F <= 7.4e+102) {
		tmp = ((1.0 / F) * (F / Math.sin(B))) - (x / B);
	} else {
		tmp = (1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -25500000.0:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 7.6e-23:
		tmp = -x / math.tan(B)
	elif F <= 7.4e+102:
		tmp = ((1.0 / F) * (F / math.sin(B))) - (x / B)
	else:
		tmp = (1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -25500000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 7.6e-23)
		tmp = Float64(Float64(-x) / tan(B));
	elseif (F <= 7.4e+102)
		tmp = Float64(Float64(Float64(1.0 / F) * Float64(F / sin(B))) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -25500000.0)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 7.6e-23)
		tmp = -x / tan(B);
	elseif (F <= 7.4e+102)
		tmp = ((1.0 / F) * (F / sin(B))) - (x / B);
	else
		tmp = (1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -25500000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7.6e-23], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7.4e+102], N[(N[(N[(1.0 / F), $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.55e7

   1. Initial program 55.9%

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

   3. Taylor expanded in F around -inf 99.6%

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

   4. Taylor expanded in B around 0 74.7%

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

   if -2.55e7 < F < 7.60000000000000023e-23

   1. Initial program 99.5%

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

   3. Taylor expanded in F around -inf 36.8%

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

   4. Taylor expanded in x around inf 69.2%

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

      1. associate-/l* 69.2%

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

      2. tan-quot 69.2%

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

      3. div-inv 69.2%

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

   6. Applied egg-rr 69.2%

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

      1. associate-*r/ 69.2%

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

      2. *-rgt-identity 69.2%

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

   8. Simplified 69.2%

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

   if 7.60000000000000023e-23 < F < 7.40000000000000045e102

   1. Initial program 96.2%

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

   3. Taylor expanded in F around inf 88.8%

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

   4. Taylor expanded in B around 0 81.9%

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

   if 7.40000000000000045e102 < F

   1. Initial program 50.0%

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

   3. Taylor expanded in F around inf 78.0%

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

   4. Taylor expanded in B around 0 73.8%

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

      1. +-commutative 73.8%

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

      2. div-inv 73.9%

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

      3. unsub-neg 73.9%

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

   6. Applied egg-rr 73.9%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -25500000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 7.6 \cdot 10^{-23}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 7.4 \cdot 10^{+102}:\\ \;\;\;\;\frac{1}{F} \cdot \frac{F}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 43.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 - x}{B}\\ \mathbf{if}\;B \leq 1.1 \cdot 10^{-174}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;B \leq 1.606 \cdot 10^{-74}:\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{elif}\;B \leq 8.8 \cdot 10^{-21}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- 1.0 x) B)))
   (if (<= B 1.1e-174)
     t_0
     (if (<= B 1.606e-74)
       (- (/ x B))
       (if (<= B 8.8e-21) t_0 (* x (/ -1.0 (tan B))))))))

C

double code(double F, double B, double x) {
	double t_0 = (1.0 - x) / B;
	double tmp;
	if (B <= 1.1e-174) {
		tmp = t_0;
	} else if (B <= 1.606e-74) {
		tmp = -(x / B);
	} else if (B <= 8.8e-21) {
		tmp = t_0;
	} else {
		tmp = x * (-1.0 / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 - x) / b
    if (b <= 1.1d-174) then
        tmp = t_0
    else if (b <= 1.606d-74) then
        tmp = -(x / b)
    else if (b <= 8.8d-21) then
        tmp = t_0
    else
        tmp = x * ((-1.0d0) / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (1.0 - x) / B;
	double tmp;
	if (B <= 1.1e-174) {
		tmp = t_0;
	} else if (B <= 1.606e-74) {
		tmp = -(x / B);
	} else if (B <= 8.8e-21) {
		tmp = t_0;
	} else {
		tmp = x * (-1.0 / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (1.0 - x) / B
	tmp = 0
	if B <= 1.1e-174:
		tmp = t_0
	elif B <= 1.606e-74:
		tmp = -(x / B)
	elif B <= 8.8e-21:
		tmp = t_0
	else:
		tmp = x * (-1.0 / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(1.0 - x) / B)
	tmp = 0.0
	if (B <= 1.1e-174)
		tmp = t_0;
	elseif (B <= 1.606e-74)
		tmp = Float64(-Float64(x / B));
	elseif (B <= 8.8e-21)
		tmp = t_0;
	else
		tmp = Float64(x * Float64(-1.0 / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (1.0 - x) / B;
	tmp = 0.0;
	if (B <= 1.1e-174)
		tmp = t_0;
	elseif (B <= 1.606e-74)
		tmp = -(x / B);
	elseif (B <= 8.8e-21)
		tmp = t_0;
	else
		tmp = x * (-1.0 / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]}, If[LessEqual[B, 1.1e-174], t$95$0, If[LessEqual[B, 1.606e-74], (-N[(x / B), $MachinePrecision]), If[LessEqual[B, 8.8e-21], t$95$0, N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.10000000000000011e-174 or 1.6060000000000001e-74 < B < 8.8000000000000002e-21

   1. Initial program 75.3%

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

   3. Taylor expanded in F around inf 48.4%

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

   4. Taylor expanded in B around 0 58.0%

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

   5. Taylor expanded in B around 0 39.9%

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

   if 1.10000000000000011e-174 < B < 1.6060000000000001e-74

   1. Initial program 86.4%

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

   3. Taylor expanded in F around -inf 67.7%

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

   4. Taylor expanded in x around inf 74.7%

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

   5. Taylor expanded in B around 0 74.7%

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

   if 8.8000000000000002e-21 < B

   1. Initial program 77.9%

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

   3. Taylor expanded in F around -inf 57.5%

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

   4. Taylor expanded in x around inf 47.5%

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

      1. associate-/l* 47.4%

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

      2. tan-quot 47.6%

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

      3. clear-num 47.6%

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

      4. associate-/r/ 47.7%

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

   6. Applied egg-rr 47.7%

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

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.1 \cdot 10^{-174}:\\ \;\;\;\;\frac{1 - x}{B}\\ \mathbf{elif}\;B \leq 1.606 \cdot 10^{-74}:\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{elif}\;B \leq 8.8 \cdot 10^{-21}:\\ \;\;\;\;\frac{1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 43.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 - x}{B}\\ \mathbf{if}\;B \leq 6.2 \cdot 10^{-174}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;B \leq 1.606 \cdot 10^{-74}:\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{elif}\;B \leq 1.4 \cdot 10^{-20}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- 1.0 x) B)))
   (if (<= B 6.2e-174)
     t_0
     (if (<= B 1.606e-74)
       (- (/ x B))
       (if (<= B 1.4e-20) t_0 (/ (- x) (tan B)))))))

C

double code(double F, double B, double x) {
	double t_0 = (1.0 - x) / B;
	double tmp;
	if (B <= 6.2e-174) {
		tmp = t_0;
	} else if (B <= 1.606e-74) {
		tmp = -(x / B);
	} else if (B <= 1.4e-20) {
		tmp = t_0;
	} else {
		tmp = -x / tan(B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 - x) / b
    if (b <= 6.2d-174) then
        tmp = t_0
    else if (b <= 1.606d-74) then
        tmp = -(x / b)
    else if (b <= 1.4d-20) then
        tmp = t_0
    else
        tmp = -x / tan(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (1.0 - x) / B;
	double tmp;
	if (B <= 6.2e-174) {
		tmp = t_0;
	} else if (B <= 1.606e-74) {
		tmp = -(x / B);
	} else if (B <= 1.4e-20) {
		tmp = t_0;
	} else {
		tmp = -x / Math.tan(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (1.0 - x) / B
	tmp = 0
	if B <= 6.2e-174:
		tmp = t_0
	elif B <= 1.606e-74:
		tmp = -(x / B)
	elif B <= 1.4e-20:
		tmp = t_0
	else:
		tmp = -x / math.tan(B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(1.0 - x) / B)
	tmp = 0.0
	if (B <= 6.2e-174)
		tmp = t_0;
	elseif (B <= 1.606e-74)
		tmp = Float64(-Float64(x / B));
	elseif (B <= 1.4e-20)
		tmp = t_0;
	else
		tmp = Float64(Float64(-x) / tan(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (1.0 - x) / B;
	tmp = 0.0;
	if (B <= 6.2e-174)
		tmp = t_0;
	elseif (B <= 1.606e-74)
		tmp = -(x / B);
	elseif (B <= 1.4e-20)
		tmp = t_0;
	else
		tmp = -x / tan(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]}, If[LessEqual[B, 6.2e-174], t$95$0, If[LessEqual[B, 1.606e-74], (-N[(x / B), $MachinePrecision]), If[LessEqual[B, 1.4e-20], t$95$0, N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if B < 6.1999999999999998e-174 or 1.6060000000000001e-74 < B < 1.4000000000000001e-20

   1. Initial program 75.3%

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

   3. Taylor expanded in F around inf 48.4%

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

   4. Taylor expanded in B around 0 58.0%

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

   5. Taylor expanded in B around 0 39.9%

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

   if 6.1999999999999998e-174 < B < 1.6060000000000001e-74

   1. Initial program 86.4%

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

   3. Taylor expanded in F around -inf 67.7%

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

   4. Taylor expanded in x around inf 74.7%

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

   5. Taylor expanded in B around 0 74.7%

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

   if 1.4000000000000001e-20 < B

   1. Initial program 77.9%

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

   3. Taylor expanded in F around -inf 57.5%

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

   4. Taylor expanded in x around inf 47.5%

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

      1. associate-/l* 47.4%

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

      2. tan-quot 47.6%

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

      3. div-inv 47.7%

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

   6. Applied egg-rr 47.7%

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

      1. associate-*r/ 47.6%

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

      2. *-rgt-identity 47.6%

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

   8. Simplified 47.6%

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

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 6.2 \cdot 10^{-174}:\\ \;\;\;\;\frac{1 - x}{B}\\ \mathbf{elif}\;B \leq 1.606 \cdot 10^{-74}:\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{elif}\;B \leq 1.4 \cdot 10^{-20}:\\ \;\;\;\;\frac{1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 71.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -25500000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4.3 \cdot 10^{-83}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -25500000.0)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 4.3e-83) (/ (- x) (tan B)) (- (/ 1.0 B) (/ x (tan B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -25500000.0) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 4.3e-83) {
		tmp = -x / tan(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 <= (-25500000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 4.3d-83) then
        tmp = -x / tan(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 <= -25500000.0) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 4.3e-83) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -25500000.0:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 4.3e-83:
		tmp = -x / math.tan(B)
	else:
		tmp = (1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -25500000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 4.3e-83)
		tmp = Float64(Float64(-x) / tan(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 <= -25500000.0)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 4.3e-83)
		tmp = -x / tan(B);
	else
		tmp = (1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -25500000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.3e-83], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.55e7

   1. Initial program 55.9%

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

   3. Taylor expanded in F around -inf 99.6%

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

   4. Taylor expanded in B around 0 74.7%

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

   if -2.55e7 < F < 4.30000000000000033e-83

   1. Initial program 99.4%

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

   3. Taylor expanded in F around -inf 34.3%

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

   4. Taylor expanded in x around inf 68.9%

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

      1. associate-/l* 68.9%

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

      2. tan-quot 69.0%

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

      3. div-inv 68.9%

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

   6. Applied egg-rr 68.9%

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

      1. associate-*r/ 69.0%

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

      2. *-rgt-identity 69.0%

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

   8. Simplified 69.0%

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

   if 4.30000000000000033e-83 < F

   1. Initial program 68.4%

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

   3. Taylor expanded in F around inf 81.0%

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

   4. Taylor expanded in B around 0 70.7%

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

      1. +-commutative 70.7%

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

      2. div-inv 70.7%

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

      3. unsub-neg 70.7%

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

   6. Applied egg-rr 70.7%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -25500000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4.3 \cdot 10^{-83}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 44.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.008:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 6.5 \cdot 10^{-77}:\\ \;\;\;\;\frac{-x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.008)
   (/ (- -1.0 x) B)
   (if (<= F 6.5e-77) (/ (- x) (sin B)) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.008) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 6.5e-77) {
		tmp = -x / sin(B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-0.008d0)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 6.5d-77) then
        tmp = -x / sin(b)
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.008) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 6.5e-77) {
		tmp = -x / Math.sin(B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -0.008:
		tmp = (-1.0 - x) / B
	elif F <= 6.5e-77:
		tmp = -x / math.sin(B)
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.008)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 6.5e-77)
		tmp = Float64(Float64(-x) / sin(B));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -0.008)
		tmp = (-1.0 - x) / B;
	elseif (F <= 6.5e-77)
		tmp = -x / sin(B);
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.008], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 6.5e-77], N[((-x) / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.0080000000000000002

   1. Initial program 57.6%

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

   3. Taylor expanded in F around -inf 99.4%

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

   4. Taylor expanded in B around 0 41.7%

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

      1. associate-*r/ 41.7%

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

      2. distribute-lft-in 41.7%

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

      3. metadata-eval 41.7%

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

      4. neg-mul-1 41.7%

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

   6. Simplified 41.7%

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

   if -0.0080000000000000002 < F < 6.4999999999999999e-77

   1. Initial program 99.4%

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

   3. Taylor expanded in F around -inf 32.4%

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

   4. Taylor expanded in x around inf 69.0%

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

   5. Taylor expanded in B around 0 37.9%

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

   if 6.4999999999999999e-77 < F

   1. Initial program 68.4%

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

   3. Taylor expanded in F around inf 81.0%

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

   4. Taylor expanded in B around 0 70.7%

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

   5. Taylor expanded in B around 0 51.3%

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

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.008:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 6.5 \cdot 10^{-77}:\\ \;\;\;\;\frac{-x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 62.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 1.32 \cdot 10^{-83}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F 1.32e-83) (/ (- x) (tan B)) (- (/ 1.0 B) (/ x (tan B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= 1.32e-83) {
		tmp = -x / tan(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 <= 1.32d-83) then
        tmp = -x / tan(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 <= 1.32e-83) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= 1.32e-83:
		tmp = -x / math.tan(B)
	else:
		tmp = (1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= 1.32e-83)
		tmp = Float64(Float64(-x) / tan(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 <= 1.32e-83)
		tmp = -x / tan(B);
	else
		tmp = (1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, 1.32e-83], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 1.31999999999999994e-83

   1. Initial program 80.9%

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

   3. Taylor expanded in F around -inf 62.1%

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

   4. Taylor expanded in x around inf 58.9%

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

      1. associate-/l* 58.8%

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

      2. tan-quot 58.9%

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

      3. div-inv 58.8%

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

   6. Applied egg-rr 58.8%

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

      1. associate-*r/ 58.9%

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

      2. *-rgt-identity 58.9%

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

   8. Simplified 58.9%

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

   if 1.31999999999999994e-83 < F

   1. Initial program 68.4%

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

   3. Taylor expanded in F around inf 81.0%

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

   4. Taylor expanded in B around 0 70.7%

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

      1. +-commutative 70.7%

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

      2. div-inv 70.7%

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

      3. unsub-neg 70.7%

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

   6. Applied egg-rr 70.7%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.32 \cdot 10^{-83}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 43.9% accurate, 18.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{-12}:\\ \;\;\;\;\frac{-1 - x}{B} + B \cdot \left(x \cdot 0.3333333333333333 - 0.16666666666666666\right)\\ \mathbf{elif}\;F \leq 6.4 \cdot 10^{-78}:\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2e-12)
   (+ (/ (- -1.0 x) B) (* B (- (* x 0.3333333333333333) 0.16666666666666666)))
   (if (<= F 6.4e-78) (- (/ x B)) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2e-12) {
		tmp = ((-1.0 - x) / B) + (B * ((x * 0.3333333333333333) - 0.16666666666666666));
	} else if (F <= 6.4e-78) {
		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 <= (-2d-12)) then
        tmp = (((-1.0d0) - x) / b) + (b * ((x * 0.3333333333333333d0) - 0.16666666666666666d0))
    else if (f <= 6.4d-78) 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 <= -2e-12) {
		tmp = ((-1.0 - x) / B) + (B * ((x * 0.3333333333333333) - 0.16666666666666666));
	} else if (F <= 6.4e-78) {
		tmp = -(x / B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2e-12:
		tmp = ((-1.0 - x) / B) + (B * ((x * 0.3333333333333333) - 0.16666666666666666))
	elif F <= 6.4e-78:
		tmp = -(x / B)
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2e-12)
		tmp = Float64(Float64(Float64(-1.0 - x) / B) + Float64(B * Float64(Float64(x * 0.3333333333333333) - 0.16666666666666666)));
	elseif (F <= 6.4e-78)
		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 <= -2e-12)
		tmp = ((-1.0 - x) / B) + (B * ((x * 0.3333333333333333) - 0.16666666666666666));
	elseif (F <= 6.4e-78)
		tmp = -(x / B);
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2e-12], N[(N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision] + N[(B * N[(N[(x * 0.3333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 6.4e-78], (-N[(x / B), $MachinePrecision]), N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.99999999999999996e-12

   1. Initial program 58.2%

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

   3. Taylor expanded in F around -inf 99.3%

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

   4. Taylor expanded in B around 0 41.3%

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

   if -1.99999999999999996e-12 < F < 6.4e-78

   1. Initial program 99.5%

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

   3. Taylor expanded in F around -inf 31.7%

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

   4. Taylor expanded in x around inf 68.7%

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

   5. Taylor expanded in B around 0 35.3%

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

   if 6.4e-78 < F

   1. Initial program 68.4%

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

   3. Taylor expanded in F around inf 81.0%

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

   4. Taylor expanded in B around 0 70.7%

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

   5. Taylor expanded in B around 0 51.3%

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

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{-12}:\\ \;\;\;\;\frac{-1 - x}{B} + B \cdot \left(x \cdot 0.3333333333333333 - 0.16666666666666666\right)\\ \mathbf{elif}\;F \leq 6.4 \cdot 10^{-78}:\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 43.8% accurate, 21.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.000155:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 3.65 \cdot 10^{-79}:\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.000155)
   (/ (- -1.0 x) B)
   (if (<= F 3.65e-79) (- (/ x B)) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.000155) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 3.65e-79) {
		tmp = -(x / B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-0.000155d0)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 3.65d-79) then
        tmp = -(x / b)
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.000155) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 3.65e-79) {
		tmp = -(x / B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -0.000155:
		tmp = (-1.0 - x) / B
	elif F <= 3.65e-79:
		tmp = -(x / B)
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.000155)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 3.65e-79)
		tmp = Float64(-Float64(x / B));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -0.000155)
		tmp = (-1.0 - x) / B;
	elseif (F <= 3.65e-79)
		tmp = -(x / B);
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.000155], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 3.65e-79], (-N[(x / B), $MachinePrecision]), N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.55e-4

   1. Initial program 57.6%

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

   3. Taylor expanded in F around -inf 99.4%

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

   4. Taylor expanded in B around 0 41.7%

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

      1. associate-*r/ 41.7%

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

      2. distribute-lft-in 41.7%

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

      3. metadata-eval 41.7%

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

      4. neg-mul-1 41.7%

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

   6. Simplified 41.7%

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

   if -1.55e-4 < F < 3.6499999999999999e-79

   1. Initial program 99.4%

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

   3. Taylor expanded in F around -inf 32.4%

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

   4. Taylor expanded in x around inf 69.0%

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

   5. Taylor expanded in B around 0 34.9%

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

   if 3.6499999999999999e-79 < F

   1. Initial program 68.4%

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

   3. Taylor expanded in F around inf 81.0%

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

   4. Taylor expanded in B around 0 70.7%

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

   5. Taylor expanded in B around 0 51.3%

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

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.000155:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 3.65 \cdot 10^{-79}:\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 31.0% accurate, 23.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{-144} \lor \neg \left(x \leq 8.8 \cdot 10^{-69}\right):\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (or (<= x -6.6e-144) (not (<= x 8.8e-69))) (- (/ x B)) (/ 1.0 B)))

C

double code(double F, double B, double x) {
	double tmp;
	if ((x <= -6.6e-144) || !(x <= 8.8e-69)) {
		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 <= (-6.6d-144)) .or. (.not. (x <= 8.8d-69))) 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 <= -6.6e-144) || !(x <= 8.8e-69)) {
		tmp = -(x / B);
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if (x <= -6.6e-144) or not (x <= 8.8e-69):
		tmp = -(x / B)
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if ((x <= -6.6e-144) || !(x <= 8.8e-69))
		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 <= -6.6e-144) || ~((x <= 8.8e-69)))
		tmp = -(x / B);
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[Or[LessEqual[x, -6.6e-144], N[Not[LessEqual[x, 8.8e-69]], $MachinePrecision]], (-N[(x / B), $MachinePrecision]), N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.5999999999999999e-144 or 8.8000000000000001e-69 < x

   1. Initial program 81.8%

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

   3. Taylor expanded in F around -inf 78.9%

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

   4. Taylor expanded in x around inf 82.1%

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

   5. Taylor expanded in B around 0 44.5%

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

   if -6.5999999999999999e-144 < x < 8.8000000000000001e-69

   1. Initial program 69.9%

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

   3. Taylor expanded in F around inf 27.9%

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

   4. Taylor expanded in B around 0 25.6%

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

   5. Taylor expanded in x around 0 20.3%

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

4. Final simplification 34.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{-144} \lor \neg \left(x \leq 8.8 \cdot 10^{-69}\right):\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 36.3% accurate, 32.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 6.2 \cdot 10^{-89}:\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F 6.2e-89) (- (/ x B)) (/ (- 1.0 x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= 6.2e-89) {
		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 <= 6.2d-89) 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 <= 6.2e-89) {
		tmp = -(x / B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= 6.2e-89:
		tmp = -(x / B)
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= 6.2e-89)
		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 <= 6.2e-89)
		tmp = -(x / B);
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, 6.2e-89], (-N[(x / B), $MachinePrecision]), N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 6.19999999999999993e-89

   1. Initial program 80.9%

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

   3. Taylor expanded in F around -inf 62.1%

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

   4. Taylor expanded in x around inf 58.9%

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

   5. Taylor expanded in B around 0 29.4%

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

   if 6.19999999999999993e-89 < F

   1. Initial program 68.4%

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

   3. Taylor expanded in F around inf 81.0%

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

   4. Taylor expanded in B around 0 70.7%

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

   5. Taylor expanded in B around 0 51.3%

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

4. Final simplification 36.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 6.2 \cdot 10^{-89}:\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 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 76.6%

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

3. Taylor expanded in F around inf 50.5%

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

4. Taylor expanded in B around 0 54.3%

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

5. Taylor expanded in x around 0 11.1%

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

6. Final simplification 11.1%

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

Reproduce

herbie shell --seed 2024034 
(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))))))