VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.9% → 95.6%

Time: 16.8s

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: 76.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.02 \cdot 10^{+214}:\\ \;\;\;\;F \cdot \frac{-1}{F \cdot \sin B} - t\_0\\ \mathbf{elif}\;F \leq 2.4 \cdot 10^{-22}:\\ \;\;\;\;\frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.02e+214)
     (- (* F (/ -1.0 (* F (sin B)))) t_0)
     (if (<= F 2.4e-22)
       (- (/ F (/ (sin B) (pow (fma F F 2.0) -0.5))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.02e+214) {
		tmp = (F * (-1.0 / (F * sin(B)))) - t_0;
	} else if (F <= 2.4e-22) {
		tmp = (F / (sin(B) / pow(fma(F, F, 2.0), -0.5))) - 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 <= -1.02e+214)
		tmp = Float64(Float64(F * Float64(-1.0 / Float64(F * sin(B)))) - t_0);
	elseif (F <= 2.4e-22)
		tmp = Float64(Float64(F / Float64(sin(B) / (fma(F, F, 2.0) ^ -0.5))) - 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, -1.02e+214], N[(N[(F * N[(-1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 2.4e-22], N[(N[(F / N[(N[Sin[B], $MachinePrecision] / N[Power[N[(F * F + 2.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.02e214

   1. Initial program 32.6%

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

   3. Simplified 42.3%

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

   5. Taylor expanded in F around -inf 99.9%

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

   if -1.02e214 < F < 2.40000000000000002e-22

   1. Initial program 96.3%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in x around 0 99.0%

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

      1. associate-*l/ 99.0%

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

      2. *-lft-identity 99.0%

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

      3. +-commutative 99.0%

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

      4. unpow2 99.0%

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

      5. fma-undefine 99.0%

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

   7. Simplified 99.0%

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

      1. clear-num 99.0%

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

      2. un-div-inv 99.0%

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

      3. inv-pow 99.0%

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

      4. sqrt-pow1 99.0%

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

      5. metadata-eval 99.0%

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

   9. Applied egg-rr 99.0%

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

   if 2.40000000000000002e-22 < F

   1. Initial program 60.5%

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

   3. Simplified 68.4%

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

   5. Taylor expanded in x around 0 68.5%

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

      1. associate-*l/ 68.5%

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

      2. *-lft-identity 68.5%

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

      3. +-commutative 68.5%

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

      4. unpow2 68.5%

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

      5. fma-undefine 68.5%

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

   7. Simplified 68.5%

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

   8. Taylor expanded in F around inf 97.6%

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

Alternative 2: 95.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.02 \cdot 10^{+214}:\\ \;\;\;\;F \cdot \frac{-1}{F \cdot \sin B} - t\_0\\ \mathbf{elif}\;F \leq 2.4 \cdot 10^{-22}:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(F, F, 2\right)}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.02e+214)
     (- (* F (/ -1.0 (* F (sin B)))) t_0)
     (if (<= F 2.4e-22)
       (- (/ F (* (sin B) (sqrt (fma F F 2.0)))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.02e+214) {
		tmp = (F * (-1.0 / (F * sin(B)))) - t_0;
	} else if (F <= 2.4e-22) {
		tmp = (F / (sin(B) * sqrt(fma(F, F, 2.0)))) - 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 <= -1.02e+214)
		tmp = Float64(Float64(F * Float64(-1.0 / Float64(F * sin(B)))) - t_0);
	elseif (F <= 2.4e-22)
		tmp = Float64(Float64(F / Float64(sin(B) * sqrt(fma(F, F, 2.0)))) - 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, -1.02e+214], N[(N[(F * N[(-1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 2.4e-22], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(F * F + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.02e214

   1. Initial program 32.6%

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

   3. Simplified 42.3%

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

   5. Taylor expanded in F around -inf 99.9%

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

   if -1.02e214 < F < 2.40000000000000002e-22

   1. Initial program 96.3%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in x around 0 99.0%

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

      1. associate-*l/ 99.0%

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

      2. *-lft-identity 99.0%

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

      3. +-commutative 99.0%

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

      4. unpow2 99.0%

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

      5. fma-undefine 99.0%

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

   7. Simplified 99.0%

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

      1. associate-*r/ 99.0%

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

      2. clear-num 99.0%

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

      3. sqrt-div 99.0%

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

      4. metadata-eval 99.0%

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

      5. un-div-inv 99.0%

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

   9. Applied egg-rr 99.0%

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

       1. associate-/r/ 99.0%

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

       2. times-frac 99.0%

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

       3. *-lft-identity 99.0%

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

   11. Simplified 99.0%

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

   if 2.40000000000000002e-22 < F

   1. Initial program 60.5%

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

   3. Simplified 68.4%

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

   5. Taylor expanded in x around 0 68.5%

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

      1. associate-*l/ 68.5%

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

      2. *-lft-identity 68.5%

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

      3. +-commutative 68.5%

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

      4. unpow2 68.5%

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

      5. fma-undefine 68.5%

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

   7. Simplified 68.5%

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

   8. Taylor expanded in F around inf 97.6%

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

Alternative 3: 96.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.7 \cdot 10^{+142}:\\ \;\;\;\;F \cdot \frac{-1}{F \cdot \sin B} - t\_0\\ \mathbf{elif}\;F \leq 1.33 \cdot 10^{-46}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.7e+142)
     (- (* F (/ -1.0 (* F (sin B)))) t_0)
     (if (<= F 1.33e-46)
       (+
        (/ -1.0 (/ (tan B) x))
        (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)))
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.7e+142) {
		tmp = (F * (-1.0 / (F * sin(B)))) - t_0;
	} else if (F <= 1.33e-46) {
		tmp = (-1.0 / (tan(B) / x)) + ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5));
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.7d+142)) then
        tmp = (f * ((-1.0d0) / (f * sin(b)))) - t_0
    else if (f <= 1.33d-46) then
        tmp = ((-1.0d0) / (tan(b) / x)) + ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)))
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.7e+142) {
		tmp = (F * (-1.0 / (F * Math.sin(B)))) - t_0;
	} else if (F <= 1.33e-46) {
		tmp = (-1.0 / (Math.tan(B) / x)) + ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5));
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.7e+142:
		tmp = (F * (-1.0 / (F * math.sin(B)))) - t_0
	elif F <= 1.33e-46:
		tmp = (-1.0 / (math.tan(B) / x)) + ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5))
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.7e+142)
		tmp = Float64(Float64(F * Float64(-1.0 / Float64(F * sin(B)))) - t_0);
	elseif (F <= 1.33e-46)
		tmp = Float64(Float64(-1.0 / Float64(tan(B) / x)) + Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.7e+142)
		tmp = (F * (-1.0 / (F * sin(B)))) - t_0;
	elseif (F <= 1.33e-46)
		tmp = (-1.0 / (tan(B) / x)) + ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5));
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.7e+142], N[(N[(F * N[(-1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.33e-46], N[(N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.6999999999999999e142

   1. Initial program 48.9%

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

   3. Simplified 60.4%

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

   5. Taylor expanded in F around -inf 99.9%

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

   if -1.6999999999999999e142 < F < 1.33000000000000008e-46

   1. Initial program 98.1%

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

      1. div-inv 98.3%

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

      2. clear-num 98.2%

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

   4. Applied egg-rr 98.2%

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

   if 1.33000000000000008e-46 < F

   1. Initial program 61.0%

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

   3. Simplified 68.8%

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

   5. Taylor expanded in x around 0 68.9%

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

      1. associate-*l/ 68.9%

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

      2. *-lft-identity 68.9%

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

      3. +-commutative 68.9%

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

      4. unpow2 68.9%

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

      5. fma-undefine 68.9%

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

   7. Simplified 68.9%

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

   8. Taylor expanded in F around inf 96.5%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.7 \cdot 10^{+142}:\\ \;\;\;\;F \cdot \frac{-1}{F \cdot \sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.33 \cdot 10^{-46}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.52 \cdot 10^{+142}:\\ \;\;\;\;F \cdot \frac{-1}{F \cdot \sin B} - t\_0\\ \mathbf{elif}\;F \leq 1.33 \cdot 10^{-46}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} + x \cdot \frac{-1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.52e+142)
     (- (* F (/ -1.0 (* F (sin B)))) t_0)
     (if (<= F 1.33e-46)
       (+
        (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5))
        (* x (/ -1.0 (tan B))))
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.52e+142) {
		tmp = (F * (-1.0 / (F * sin(B)))) - t_0;
	} else if (F <= 1.33e-46) {
		tmp = ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) + (x * (-1.0 / tan(B)));
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.52d+142)) then
        tmp = (f * ((-1.0d0) / (f * sin(b)))) - t_0
    else if (f <= 1.33d-46) then
        tmp = ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0))) + (x * ((-1.0d0) / tan(b)))
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.52e+142) {
		tmp = (F * (-1.0 / (F * Math.sin(B)))) - t_0;
	} else if (F <= 1.33e-46) {
		tmp = ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) + (x * (-1.0 / Math.tan(B)));
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.52e+142:
		tmp = (F * (-1.0 / (F * math.sin(B)))) - t_0
	elif F <= 1.33e-46:
		tmp = ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) + (x * (-1.0 / math.tan(B)))
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.52e+142)
		tmp = Float64(Float64(F * Float64(-1.0 / Float64(F * sin(B)))) - t_0);
	elseif (F <= 1.33e-46)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)) + Float64(x * Float64(-1.0 / tan(B))));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.52e+142)
		tmp = (F * (-1.0 / (F * sin(B)))) - t_0;
	elseif (F <= 1.33e-46)
		tmp = ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5)) + (x * (-1.0 / tan(B)));
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.52e+142], N[(N[(F * N[(-1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.33e-46], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] + N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.5199999999999999e142

   1. Initial program 48.9%

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

   3. Simplified 60.4%

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

   5. Taylor expanded in F around -inf 99.9%

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

   if -1.5199999999999999e142 < F < 1.33000000000000008e-46

   1. Initial program 98.1%

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

   if 1.33000000000000008e-46 < F

   1. Initial program 61.0%

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

   3. Simplified 68.8%

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

   5. Taylor expanded in x around 0 68.9%

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

      1. associate-*l/ 68.9%

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

      2. *-lft-identity 68.9%

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

      3. +-commutative 68.9%

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

      4. unpow2 68.9%

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

      5. fma-undefine 68.9%

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

   7. Simplified 68.9%

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

   8. Taylor expanded in F around inf 96.5%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.52 \cdot 10^{+142}:\\ \;\;\;\;F \cdot \frac{-1}{F \cdot \sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.33 \cdot 10^{-46}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} + x \cdot \frac{-1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -260000000000:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 7.8 \cdot 10^{-53}:\\ \;\;\;\;F \cdot \left(t\_0 \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\right) - t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))) (t_1 (/ x (tan B))))
   (if (<= F -260000000000.0)
     (+ (/ -1.0 (/ (tan B) x)) (/ -1.0 (sin B)))
     (if (<= F 7.8e-53)
       (- (* F (* t_0 (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))))) t_1)
       (- t_0 t_1)))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / sin(B);
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -260000000000.0) {
		tmp = (-1.0 / (tan(B) / x)) + (-1.0 / sin(B));
	} else if (F <= 7.8e-53) {
		tmp = (F * (t_0 * sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_1;
	} else {
		tmp = t_0 - t_1;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 / sin(b)
    t_1 = x / tan(b)
    if (f <= (-260000000000.0d0)) then
        tmp = ((-1.0d0) / (tan(b) / x)) + ((-1.0d0) / sin(b))
    else if (f <= 7.8d-53) then
        tmp = (f * (t_0 * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))))) - t_1
    else
        tmp = t_0 - t_1
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = 1.0 / Math.sin(B);
	double t_1 = x / Math.tan(B);
	double tmp;
	if (F <= -260000000000.0) {
		tmp = (-1.0 / (Math.tan(B) / x)) + (-1.0 / Math.sin(B));
	} else if (F <= 7.8e-53) {
		tmp = (F * (t_0 * Math.sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_1;
	} else {
		tmp = t_0 - t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = 1.0 / math.sin(B)
	t_1 = x / math.tan(B)
	tmp = 0
	if F <= -260000000000.0:
		tmp = (-1.0 / (math.tan(B) / x)) + (-1.0 / math.sin(B))
	elif F <= 7.8e-53:
		tmp = (F * (t_0 * math.sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_1
	else:
		tmp = t_0 - t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / sin(B))
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -260000000000.0)
		tmp = Float64(Float64(-1.0 / Float64(tan(B) / x)) + Float64(-1.0 / sin(B)));
	elseif (F <= 7.8e-53)
		tmp = Float64(Float64(F * Float64(t_0 * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0)))))) - t_1);
	else
		tmp = Float64(t_0 - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = 1.0 / sin(B);
	t_1 = x / tan(B);
	tmp = 0.0;
	if (F <= -260000000000.0)
		tmp = (-1.0 / (tan(B) / x)) + (-1.0 / sin(B));
	elseif (F <= 7.8e-53)
		tmp = (F * (t_0 * sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_1;
	else
		tmp = t_0 - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -260000000000.0], N[(N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7.8e-53], N[(N[(F * N[(t$95$0 * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(t$95$0 - t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.6e11

   1. Initial program 68.7%

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

   3. Taylor expanded in F around -inf 99.7%

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

      1. div-inv 68.9%

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

      2. clear-num 68.8%

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

   5. Applied egg-rr 99.8%

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

   if -2.6e11 < F < 7.8000000000000004e-53

   1. Initial program 99.4%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in F around 0 98.8%

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

   if 7.8000000000000004e-53 < F

   1. Initial program 61.9%

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

   3. Simplified 69.5%

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

   5. Taylor expanded in x around 0 69.6%

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

      1. associate-*l/ 69.6%

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

      2. *-lft-identity 69.6%

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

      3. +-commutative 69.6%

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

      4. unpow2 69.6%

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

      5. fma-undefine 69.6%

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

   7. Simplified 69.6%

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

   8. Taylor expanded in F around inf 94.3%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -260000000000:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 7.8 \cdot 10^{-53}:\\ \;\;\;\;F \cdot \left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\right) - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -260000000000:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 7.8 \cdot 10^{-53}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -260000000000.0)
     (+ (/ -1.0 (/ (tan B) x)) (/ -1.0 (sin B)))
     (if (<= F 7.8e-53)
       (- (* F (/ (sqrt 0.5) (sin B))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -260000000000.0) {
		tmp = (-1.0 / (tan(B) / x)) + (-1.0 / sin(B));
	} else if (F <= 7.8e-53) {
		tmp = (F * (sqrt(0.5) / sin(B))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-260000000000.0d0)) then
        tmp = ((-1.0d0) / (tan(b) / x)) + ((-1.0d0) / sin(b))
    else if (f <= 7.8d-53) then
        tmp = (f * (sqrt(0.5d0) / sin(b))) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -260000000000.0) {
		tmp = (-1.0 / (Math.tan(B) / x)) + (-1.0 / Math.sin(B));
	} else if (F <= 7.8e-53) {
		tmp = (F * (Math.sqrt(0.5) / Math.sin(B))) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -260000000000.0:
		tmp = (-1.0 / (math.tan(B) / x)) + (-1.0 / math.sin(B))
	elif F <= 7.8e-53:
		tmp = (F * (math.sqrt(0.5) / math.sin(B))) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -260000000000.0)
		tmp = Float64(Float64(-1.0 / Float64(tan(B) / x)) + Float64(-1.0 / sin(B)));
	elseif (F <= 7.8e-53)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / sin(B))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -260000000000.0)
		tmp = (-1.0 / (tan(B) / x)) + (-1.0 / sin(B));
	elseif (F <= 7.8e-53)
		tmp = (F * (sqrt(0.5) / sin(B))) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -260000000000.0], N[(N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7.8e-53], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.6e11

   1. Initial program 68.7%

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

   3. Taylor expanded in F around -inf 99.7%

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

      1. div-inv 68.9%

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

      2. clear-num 68.8%

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

   5. Applied egg-rr 99.8%

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

   if -2.6e11 < F < 7.8000000000000004e-53

   1. Initial program 99.4%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 99.6%

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

      1. associate-*l/ 99.6%

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

      2. *-lft-identity 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. fma-undefine 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in F around 0 98.8%

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

   if 7.8000000000000004e-53 < F

   1. Initial program 61.9%

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

   3. Simplified 69.5%

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

   5. Taylor expanded in x around 0 69.6%

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

      1. associate-*l/ 69.6%

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

      2. *-lft-identity 69.6%

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

      3. +-commutative 69.6%

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

      4. unpow2 69.6%

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

      5. fma-undefine 69.6%

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

   7. Simplified 69.6%

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

   8. Taylor expanded in F around inf 94.3%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -260000000000:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 7.8 \cdot 10^{-53}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.37:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 2.1 \cdot 10^{-92}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - t\_0\\ \mathbf{elif}\;F \leq 10^{+16}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -0.37)
     (+ (/ -1.0 (/ (tan B) x)) (/ -1.0 (sin B)))
     (if (<= F 2.1e-92)
       (- (* F (/ (sqrt 0.5) B)) t_0)
       (if (<= F 1e+16)
         (- (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)) (/ x B))
         (- (/ 1.0 (sin B)) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -0.37) {
		tmp = (-1.0 / (tan(B) / x)) + (-1.0 / sin(B));
	} else if (F <= 2.1e-92) {
		tmp = (F * (sqrt(0.5) / B)) - t_0;
	} else if (F <= 1e+16) {
		tmp = ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B);
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-0.37d0)) then
        tmp = ((-1.0d0) / (tan(b) / x)) + ((-1.0d0) / sin(b))
    else if (f <= 2.1d-92) then
        tmp = (f * (sqrt(0.5d0) / b)) - t_0
    else if (f <= 1d+16) then
        tmp = ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0))) - (x / b)
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -0.37) {
		tmp = (-1.0 / (Math.tan(B) / x)) + (-1.0 / Math.sin(B));
	} else if (F <= 2.1e-92) {
		tmp = (F * (Math.sqrt(0.5) / B)) - t_0;
	} else if (F <= 1e+16) {
		tmp = ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B);
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -0.37:
		tmp = (-1.0 / (math.tan(B) / x)) + (-1.0 / math.sin(B))
	elif F <= 2.1e-92:
		tmp = (F * (math.sqrt(0.5) / B)) - t_0
	elif F <= 1e+16:
		tmp = ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B)
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -0.37)
		tmp = Float64(Float64(-1.0 / Float64(tan(B) / x)) + Float64(-1.0 / sin(B)));
	elseif (F <= 2.1e-92)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / B)) - t_0);
	elseif (F <= 1e+16)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -0.37)
		tmp = (-1.0 / (tan(B) / x)) + (-1.0 / sin(B));
	elseif (F <= 2.1e-92)
		tmp = (F * (sqrt(0.5) / B)) - t_0;
	elseif (F <= 1e+16)
		tmp = ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5)) - (x / B);
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.37], N[(N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.1e-92], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1e+16], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.37

   1. Initial program 69.2%

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

   3. Taylor expanded in F around -inf 99.7%

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

      1. div-inv 69.3%

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

      2. clear-num 69.2%

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

   5. Applied egg-rr 99.8%

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

   if -0.37 < F < 2.1e-92

   1. Initial program 99.4%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in F around 0 99.7%

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

   6. Taylor expanded in B around 0 89.2%

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

   7. Taylor expanded in x around 0 89.2%

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

      1. associate-/l* 89.2%

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

   9. Simplified 89.2%

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

   if 2.1e-92 < F < 1e16

   1. Initial program 99.3%

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

   3. Taylor expanded in B around 0 80.2%

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

   if 1e16 < F

   1. Initial program 58.9%

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

   3. Simplified 67.1%

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

   5. Taylor expanded in x around 0 67.2%

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

      1. associate-*l/ 67.2%

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

      2. *-lft-identity 67.2%

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

      3. +-commutative 67.2%

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

      4. unpow2 67.2%

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

      5. fma-undefine 67.2%

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

   7. Simplified 67.2%

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

   8. Taylor expanded in F around inf 99.9%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.37:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 2.1 \cdot 10^{-92}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 10^{+16}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -7.4 \cdot 10^{-149}:\\ \;\;\;\;\frac{F}{B \cdot \left(-F\right)} - t\_1\\ \mathbf{elif}\;F \leq -2.8 \cdot 10^{-195}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 9.2 \cdot 10^{-200}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 5.6 \cdot 10^{-142}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 7.3 \cdot 10^{-96}:\\ \;\;\;\;\frac{1}{B} - t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- (* F (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))) x) B))
        (t_1 (/ x (tan B))))
   (if (<= F -7.4e-149)
     (- (/ F (* B (- F))) t_1)
     (if (<= F -2.8e-195)
       t_0
       (if (<= F 9.2e-200)
         (/ x (- (tan B)))
         (if (<= F 5.6e-142)
           t_0
           (if (<= F 7.3e-96) (- (/ 1.0 B) t_1) (- (/ 1.0 (sin B)) t_1))))))))

C

double code(double F, double B, double x) {
	double t_0 = ((F * sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B;
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -7.4e-149) {
		tmp = (F / (B * -F)) - t_1;
	} else if (F <= -2.8e-195) {
		tmp = t_0;
	} else if (F <= 9.2e-200) {
		tmp = x / -tan(B);
	} else if (F <= 5.6e-142) {
		tmp = t_0;
	} else if (F <= 7.3e-96) {
		tmp = (1.0 / B) - t_1;
	} else {
		tmp = (1.0 / 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) :: tmp
    t_0 = ((f * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))) - x) / b
    t_1 = x / tan(b)
    if (f <= (-7.4d-149)) then
        tmp = (f / (b * -f)) - t_1
    else if (f <= (-2.8d-195)) then
        tmp = t_0
    else if (f <= 9.2d-200) then
        tmp = x / -tan(b)
    else if (f <= 5.6d-142) then
        tmp = t_0
    else if (f <= 7.3d-96) then
        tmp = (1.0d0 / b) - t_1
    else
        tmp = (1.0d0 / 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.sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B;
	double t_1 = x / Math.tan(B);
	double tmp;
	if (F <= -7.4e-149) {
		tmp = (F / (B * -F)) - t_1;
	} else if (F <= -2.8e-195) {
		tmp = t_0;
	} else if (F <= 9.2e-200) {
		tmp = x / -Math.tan(B);
	} else if (F <= 5.6e-142) {
		tmp = t_0;
	} else if (F <= 7.3e-96) {
		tmp = (1.0 / B) - t_1;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = ((F * math.sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B
	t_1 = x / math.tan(B)
	tmp = 0
	if F <= -7.4e-149:
		tmp = (F / (B * -F)) - t_1
	elif F <= -2.8e-195:
		tmp = t_0
	elif F <= 9.2e-200:
		tmp = x / -math.tan(B)
	elif F <= 5.6e-142:
		tmp = t_0
	elif F <= 7.3e-96:
		tmp = (1.0 / B) - t_1
	else:
		tmp = (1.0 / math.sin(B)) - t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(Float64(F * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))) - x) / B)
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -7.4e-149)
		tmp = Float64(Float64(F / Float64(B * Float64(-F))) - t_1);
	elseif (F <= -2.8e-195)
		tmp = t_0;
	elseif (F <= 9.2e-200)
		tmp = Float64(x / Float64(-tan(B)));
	elseif (F <= 5.6e-142)
		tmp = t_0;
	elseif (F <= 7.3e-96)
		tmp = Float64(Float64(1.0 / B) - t_1);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = ((F * sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B;
	t_1 = x / tan(B);
	tmp = 0.0;
	if (F <= -7.4e-149)
		tmp = (F / (B * -F)) - t_1;
	elseif (F <= -2.8e-195)
		tmp = t_0;
	elseif (F <= 9.2e-200)
		tmp = x / -tan(B);
	elseif (F <= 5.6e-142)
		tmp = t_0;
	elseif (F <= 7.3e-96)
		tmp = (1.0 / B) - t_1;
	else
		tmp = (1.0 / sin(B)) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(N[(F * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -7.4e-149], N[(N[(F / N[(B * (-F)), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, -2.8e-195], t$95$0, If[LessEqual[F, 9.2e-200], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 5.6e-142], t$95$0, If[LessEqual[F, 7.3e-96], N[(N[(1.0 / B), $MachinePrecision] - t$95$1), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -7.3999999999999998e-149

   1. Initial program 78.1%

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

   3. Simplified 84.5%

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

   5. Taylor expanded in x around 0 84.6%

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

      1. associate-*l/ 84.5%

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

      2. *-lft-identity 84.5%

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

      3. +-commutative 84.5%

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

      4. unpow2 84.5%

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

      5. fma-undefine 84.5%

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

   7. Simplified 84.5%

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

      1. clear-num 84.5%

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

      2. un-div-inv 84.5%

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

      3. inv-pow 84.5%

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

      4. sqrt-pow1 84.5%

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

      5. metadata-eval 84.5%

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

   9. Applied egg-rr 84.5%

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

   10. Taylor expanded in B around 0 72.5%

       \[\leadsto \frac{F}{\color{blue}{B \cdot \sqrt{2 + {F}^{2}}}} - \frac{x}{\tan B} \]
   11. Step-by-step derivation

       1. +-commutative 72.5%

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

       2. unpow2 72.5%

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

       3. fma-undefine 72.5%

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

   12. Simplified 72.5%

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

   13. Taylor expanded in F around -inf 77.1%

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

       1. associate-*r* 77.1%

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

       2. neg-mul-1 77.1%

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

   15. Simplified 77.1%

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

   if -7.3999999999999998e-149 < F < -2.80000000000000003e-195 or 9.2000000000000003e-200 < F < 5.60000000000000009e-142

   1. Initial program 99.2%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in 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 69.3%

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

   if -2.80000000000000003e-195 < F < 9.2000000000000003e-200

   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 38.2%

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

   4. Taylor expanded in x around inf 94.1%

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

      1. mul-1-neg 94.1%

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

      2. associate-/l* 94.0%

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

      3. distribute-lft-neg-in 94.0%

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

   6. Simplified 94.0%

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

      1. add-sqr-sqrt 26.6%

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

      2. sqrt-unprod 20.8%

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

      3. sqr-neg 20.8%

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

      4. sqrt-unprod 0.8%

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

      5. add-sqr-sqrt 1.7%

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

      6. clear-num 1.7%

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

      7. tan-quot 1.7%

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

      8. div-inv 1.7%

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

      9. frac-2neg 1.7%

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

      10. neg-sub0 1.7%

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

      11. div-sub 1.7%

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

      12. add-sqr-sqrt 0.8%

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

      13. sqrt-unprod 20.8%

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

      14. sqr-neg 20.8%

          \[\leadsto \frac{0}{-\tan B} - \frac{\sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}}}{-\tan B} \]

      15. sqrt-unprod 26.8%

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

      16. add-sqr-sqrt 94.2%

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

      17. frac-2neg 94.2%

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

   8. Applied egg-rr 94.2%

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

      1. div0 94.2%

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

      2. neg-sub0 94.2%

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

      3. distribute-frac-neg2 94.2%

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

   10. Simplified 94.2%

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

   if 5.60000000000000009e-142 < F < 7.29999999999999994e-96

   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 99.5%

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

   5. Taylor expanded in x around 0 99.3%

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

      1. associate-*l/ 99.5%

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

      2. *-lft-identity 99.5%

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

      3. +-commutative 99.5%

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

      4. unpow2 99.5%

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

      5. fma-undefine 99.5%

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

   7. Simplified 99.5%

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

      1. clear-num 99.5%

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

      2. un-div-inv 99.5%

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

      3. inv-pow 99.5%

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

      4. sqrt-pow1 99.5%

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

      5. metadata-eval 99.5%

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

   9. Applied egg-rr 99.5%

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

   10. Taylor expanded in B around 0 87.8%

       \[\leadsto \frac{F}{\color{blue}{B \cdot \sqrt{2 + {F}^{2}}}} - \frac{x}{\tan B} \]
   11. Step-by-step derivation

       1. +-commutative 87.8%

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

       2. unpow2 87.8%

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

       3. fma-undefine 87.8%

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

   12. Simplified 87.8%

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

   13. Taylor expanded in F around inf 77.0%

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

   if 7.29999999999999994e-96 < F

   1. Initial program 65.7%

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

   3. Simplified 72.5%

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

   5. Taylor expanded in x around 0 72.6%

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

      1. associate-*l/ 72.6%

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

      2. *-lft-identity 72.6%

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

      3. +-commutative 72.6%

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

      4. unpow2 72.6%

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

      5. fma-undefine 72.6%

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

   7. Simplified 72.6%

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

   8. Taylor expanded in F around inf 88.2%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7.4 \cdot 10^{-149}:\\ \;\;\;\;\frac{F}{B \cdot \left(-F\right)} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -2.8 \cdot 10^{-195}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{elif}\;F \leq 9.2 \cdot 10^{-200}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 5.6 \cdot 10^{-142}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{elif}\;F \leq 7.3 \cdot 10^{-96}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2.5 \cdot 10^{-152}:\\ \;\;\;\;\frac{F}{B \cdot \left(-F\right)} - t\_1\\ \mathbf{elif}\;F \leq -2.9 \cdot 10^{-196}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 4.3 \cdot 10^{-200}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 8.5 \cdot 10^{-145}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{-61}:\\ \;\;\;\;\cos B \cdot \frac{-x}{\sin B}\\ \mathbf{elif}\;F \leq 2.35 \cdot 10^{+212}:\\ \;\;\;\;\frac{1}{B} - t\_1\\ \mathbf{elif}\;F \leq 3.6 \cdot 10^{+280}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{F \cdot B} - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- (* F (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))) x) B))
        (t_1 (/ x (tan B))))
   (if (<= F -2.5e-152)
     (- (/ F (* B (- F))) t_1)
     (if (<= F -2.9e-196)
       t_0
       (if (<= F 4.3e-200)
         (/ x (- (tan B)))
         (if (<= F 8.5e-145)
           t_0
           (if (<= F 2.3e-61)
             (* (cos B) (/ (- x) (sin B)))
             (if (<= F 2.35e+212)
               (- (/ 1.0 B) t_1)
               (if (<= F 3.6e+280)
                 (/ 1.0 (sin B))
                 (- (/ F (* F B)) t_1))))))))))

C

double code(double F, double B, double x) {
	double t_0 = ((F * sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B;
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -2.5e-152) {
		tmp = (F / (B * -F)) - t_1;
	} else if (F <= -2.9e-196) {
		tmp = t_0;
	} else if (F <= 4.3e-200) {
		tmp = x / -tan(B);
	} else if (F <= 8.5e-145) {
		tmp = t_0;
	} else if (F <= 2.3e-61) {
		tmp = cos(B) * (-x / sin(B));
	} else if (F <= 2.35e+212) {
		tmp = (1.0 / B) - t_1;
	} else if (F <= 3.6e+280) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = (F / (F * 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) :: tmp
    t_0 = ((f * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))) - x) / b
    t_1 = x / tan(b)
    if (f <= (-2.5d-152)) then
        tmp = (f / (b * -f)) - t_1
    else if (f <= (-2.9d-196)) then
        tmp = t_0
    else if (f <= 4.3d-200) then
        tmp = x / -tan(b)
    else if (f <= 8.5d-145) then
        tmp = t_0
    else if (f <= 2.3d-61) then
        tmp = cos(b) * (-x / sin(b))
    else if (f <= 2.35d+212) then
        tmp = (1.0d0 / b) - t_1
    else if (f <= 3.6d+280) then
        tmp = 1.0d0 / sin(b)
    else
        tmp = (f / (f * 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.sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B;
	double t_1 = x / Math.tan(B);
	double tmp;
	if (F <= -2.5e-152) {
		tmp = (F / (B * -F)) - t_1;
	} else if (F <= -2.9e-196) {
		tmp = t_0;
	} else if (F <= 4.3e-200) {
		tmp = x / -Math.tan(B);
	} else if (F <= 8.5e-145) {
		tmp = t_0;
	} else if (F <= 2.3e-61) {
		tmp = Math.cos(B) * (-x / Math.sin(B));
	} else if (F <= 2.35e+212) {
		tmp = (1.0 / B) - t_1;
	} else if (F <= 3.6e+280) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = (F / (F * B)) - t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = ((F * math.sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B
	t_1 = x / math.tan(B)
	tmp = 0
	if F <= -2.5e-152:
		tmp = (F / (B * -F)) - t_1
	elif F <= -2.9e-196:
		tmp = t_0
	elif F <= 4.3e-200:
		tmp = x / -math.tan(B)
	elif F <= 8.5e-145:
		tmp = t_0
	elif F <= 2.3e-61:
		tmp = math.cos(B) * (-x / math.sin(B))
	elif F <= 2.35e+212:
		tmp = (1.0 / B) - t_1
	elif F <= 3.6e+280:
		tmp = 1.0 / math.sin(B)
	else:
		tmp = (F / (F * B)) - t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(Float64(F * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))) - x) / B)
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -2.5e-152)
		tmp = Float64(Float64(F / Float64(B * Float64(-F))) - t_1);
	elseif (F <= -2.9e-196)
		tmp = t_0;
	elseif (F <= 4.3e-200)
		tmp = Float64(x / Float64(-tan(B)));
	elseif (F <= 8.5e-145)
		tmp = t_0;
	elseif (F <= 2.3e-61)
		tmp = Float64(cos(B) * Float64(Float64(-x) / sin(B)));
	elseif (F <= 2.35e+212)
		tmp = Float64(Float64(1.0 / B) - t_1);
	elseif (F <= 3.6e+280)
		tmp = Float64(1.0 / sin(B));
	else
		tmp = Float64(Float64(F / Float64(F * B)) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = ((F * sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B;
	t_1 = x / tan(B);
	tmp = 0.0;
	if (F <= -2.5e-152)
		tmp = (F / (B * -F)) - t_1;
	elseif (F <= -2.9e-196)
		tmp = t_0;
	elseif (F <= 4.3e-200)
		tmp = x / -tan(B);
	elseif (F <= 8.5e-145)
		tmp = t_0;
	elseif (F <= 2.3e-61)
		tmp = cos(B) * (-x / sin(B));
	elseif (F <= 2.35e+212)
		tmp = (1.0 / B) - t_1;
	elseif (F <= 3.6e+280)
		tmp = 1.0 / sin(B);
	else
		tmp = (F / (F * B)) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(N[(F * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.5e-152], N[(N[(F / N[(B * (-F)), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, -2.9e-196], t$95$0, If[LessEqual[F, 4.3e-200], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 8.5e-145], t$95$0, If[LessEqual[F, 2.3e-61], N[(N[Cos[B], $MachinePrecision] * N[((-x) / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.35e+212], N[(N[(1.0 / B), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, 3.6e+280], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(F / N[(F * B), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if F < -2.4999999999999998e-152

   1. Initial program 78.1%

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

   3. Simplified 84.5%

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

   5. Taylor expanded in x around 0 84.6%

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

      1. associate-*l/ 84.5%

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

      2. *-lft-identity 84.5%

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

      3. +-commutative 84.5%

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

      4. unpow2 84.5%

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

      5. fma-undefine 84.5%

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

   7. Simplified 84.5%

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

      1. clear-num 84.5%

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

      2. un-div-inv 84.5%

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

      3. inv-pow 84.5%

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

      4. sqrt-pow1 84.5%

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

      5. metadata-eval 84.5%

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

   9. Applied egg-rr 84.5%

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

   10. Taylor expanded in B around 0 72.5%

       \[\leadsto \frac{F}{\color{blue}{B \cdot \sqrt{2 + {F}^{2}}}} - \frac{x}{\tan B} \]
   11. Step-by-step derivation

       1. +-commutative 72.5%

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

       2. unpow2 72.5%

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

       3. fma-undefine 72.5%

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

   12. Simplified 72.5%

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

   13. Taylor expanded in F around -inf 77.1%

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

       1. associate-*r* 77.1%

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

       2. neg-mul-1 77.1%

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

   15. Simplified 77.1%

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

   if -2.4999999999999998e-152 < F < -2.89999999999999987e-196 or 4.29999999999999975e-200 < F < 8.50000000000000043e-145

   1. Initial program 99.2%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in 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 69.3%

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

   if -2.89999999999999987e-196 < F < 4.29999999999999975e-200

   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 38.2%

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

   4. Taylor expanded in x around inf 94.1%

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

      1. mul-1-neg 94.1%

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

      2. associate-/l* 94.0%

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

      3. distribute-lft-neg-in 94.0%

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

   6. Simplified 94.0%

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

      1. add-sqr-sqrt 26.6%

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

      2. sqrt-unprod 20.8%

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

      3. sqr-neg 20.8%

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

      4. sqrt-unprod 0.8%

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

      5. add-sqr-sqrt 1.7%

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

      6. clear-num 1.7%

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

      7. tan-quot 1.7%

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

      8. div-inv 1.7%

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

      9. frac-2neg 1.7%

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

      10. neg-sub0 1.7%

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

      11. div-sub 1.7%

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

      12. add-sqr-sqrt 0.8%

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

      13. sqrt-unprod 20.8%

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

      14. sqr-neg 20.8%

          \[\leadsto \frac{0}{-\tan B} - \frac{\sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}}}{-\tan B} \]

      15. sqrt-unprod 26.8%

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

      16. add-sqr-sqrt 94.2%

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

      17. frac-2neg 94.2%

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

   8. Applied egg-rr 94.2%

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

      1. div0 94.2%

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

      2. neg-sub0 94.2%

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

      3. distribute-frac-neg2 94.2%

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

   10. Simplified 94.2%

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

   if 8.50000000000000043e-145 < F < 2.29999999999999992e-61

   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 29.5%

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

   4. Taylor expanded in x around inf 65.8%

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

      1. mul-1-neg 65.8%

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

      2. associate-/l* 65.9%

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

      3. distribute-lft-neg-in 65.9%

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

   6. Simplified 65.9%

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

      1. clear-num 65.9%

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

      2. tan-quot 65.8%

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

      3. *-un-lft-identity 65.8%

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

   8. Applied egg-rr 65.8%

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

      1. *-lft-identity 65.8%

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

   10. Simplified 65.8%

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

   11. Taylor expanded in x around 0 65.8%

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

       1. mul-1-neg 65.8%

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

       2. distribute-frac-neg2 65.8%

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

       3. *-commutative 65.8%

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

       4. associate-/l* 65.9%

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

   13. Simplified 65.9%

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

   if 2.29999999999999992e-61 < F < 2.34999999999999996e212

   1. Initial program 84.4%

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

   3. Simplified 88.2%

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

   5. Taylor expanded in x around 0 88.3%

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

      1. associate-*l/ 88.3%

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

      2. *-lft-identity 88.3%

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

      3. +-commutative 88.3%

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

      4. unpow2 88.3%

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

      5. fma-undefine 88.3%

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

   7. Simplified 88.3%

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

      1. clear-num 88.2%

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

      2. un-div-inv 88.3%

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

      3. inv-pow 88.3%

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

      4. sqrt-pow1 88.2%

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

      5. metadata-eval 88.2%

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

   9. Applied egg-rr 88.2%

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

   10. Taylor expanded in B around 0 63.8%

       \[\leadsto \frac{F}{\color{blue}{B \cdot \sqrt{2 + {F}^{2}}}} - \frac{x}{\tan B} \]
   11. Step-by-step derivation

       1. +-commutative 63.8%

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

       2. unpow2 63.8%

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

       3. fma-undefine 63.8%

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

   12. Simplified 63.8%

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

   13. Taylor expanded in F around inf 68.4%

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

   if 2.34999999999999996e212 < F < 3.6e280

   1. Initial program 18.0%

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

   3. Taylor expanded in F around -inf 27.1%

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

      1. add-cube-cbrt 27.0%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}} \cdot \sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}\right) \cdot \sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}} \]

      2. pow3 27.0%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}\right)}^{3}} \]

   5. Applied egg-rr 72.6%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{1}{\sin B} + \frac{x}{\tan B}}\right)}^{3}} \]

   6. Taylor expanded in x around 0 73.9%

      \[\leadsto {\color{blue}{\left(\sqrt[3]{\frac{1}{\sin B}}\right)}}^{3} \]

   7. Taylor expanded in B around inf 74.9%

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

   if 3.6e280 < F

   1. Initial program 36.9%

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

   3. Simplified 56.4%

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

   5. Taylor expanded in x around 0 56.4%

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

      1. associate-*l/ 56.4%

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

      2. *-lft-identity 56.4%

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

      3. +-commutative 56.4%

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

      4. unpow2 56.4%

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

      5. fma-undefine 56.4%

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

   7. Simplified 56.4%

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

      1. clear-num 56.4%

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

      2. un-div-inv 56.4%

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

      3. inv-pow 56.4%

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

      4. sqrt-pow1 56.4%

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

      5. metadata-eval 56.4%

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

   9. Applied egg-rr 56.4%

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

   10. Taylor expanded in B around 0 56.4%

       \[\leadsto \frac{F}{\color{blue}{B \cdot \sqrt{2 + {F}^{2}}}} - \frac{x}{\tan B} \]
   11. Step-by-step derivation

       1. +-commutative 56.4%

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

       2. unpow2 56.4%

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

       3. fma-undefine 56.4%

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

   12. Simplified 56.4%

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

   13. Taylor expanded in F around inf 82.5%

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

       1. *-commutative 82.5%

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

   15. Simplified 82.5%

       \[\leadsto \frac{F}{\color{blue}{F \cdot B}} - \frac{x}{\tan B} \]
3. Recombined 7 regimes into one program.

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.5 \cdot 10^{-152}:\\ \;\;\;\;\frac{F}{B \cdot \left(-F\right)} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -2.9 \cdot 10^{-196}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{elif}\;F \leq 4.3 \cdot 10^{-200}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 8.5 \cdot 10^{-145}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{-61}:\\ \;\;\;\;\cos B \cdot \frac{-x}{\sin B}\\ \mathbf{elif}\;F \leq 2.35 \cdot 10^{+212}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 3.6 \cdot 10^{+280}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{F \cdot B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 89.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.37:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{-82}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -0.37)
     (+ (/ -1.0 (/ (tan B) x)) (/ -1.0 (sin B)))
     (if (<= F 4.5e-82)
       (- (* 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 <= -0.37) {
		tmp = (-1.0 / (tan(B) / x)) + (-1.0 / sin(B));
	} else if (F <= 4.5e-82) {
		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 <= (-0.37d0)) then
        tmp = ((-1.0d0) / (tan(b) / x)) + ((-1.0d0) / sin(b))
    else if (f <= 4.5d-82) 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 <= -0.37) {
		tmp = (-1.0 / (Math.tan(B) / x)) + (-1.0 / Math.sin(B));
	} else if (F <= 4.5e-82) {
		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 <= -0.37:
		tmp = (-1.0 / (math.tan(B) / x)) + (-1.0 / math.sin(B))
	elif F <= 4.5e-82:
		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 <= -0.37)
		tmp = Float64(Float64(-1.0 / Float64(tan(B) / x)) + Float64(-1.0 / sin(B)));
	elseif (F <= 4.5e-82)
		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 <= -0.37)
		tmp = (-1.0 / (tan(B) / x)) + (-1.0 / sin(B));
	elseif (F <= 4.5e-82)
		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, -0.37], N[(N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.5e-82], 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 < -0.37

   1. Initial program 69.2%

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

   3. Taylor expanded in F around -inf 99.7%

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

      1. div-inv 69.3%

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

      2. clear-num 69.2%

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

   5. Applied egg-rr 99.8%

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

   if -0.37 < F < 4.4999999999999998e-82

   1. Initial program 99.4%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in 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 88.4%

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

      1. associate-/l* 88.5%

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

   9. Simplified 88.5%

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

   if 4.4999999999999998e-82 < F

   1. Initial program 64.6%

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

   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 71.7%

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

      1. associate-*l/ 71.7%

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

      2. *-lft-identity 71.7%

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

      3. +-commutative 71.7%

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

      4. unpow2 71.7%

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

      5. fma-undefine 71.7%

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

   7. Simplified 71.7%

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

   8. Taylor expanded in F around inf 89.9%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.37:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{-82}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 90.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.5:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 2.15 \cdot 10^{-61}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -0.5)
     (+ (* x (/ -1.0 (tan B))) (/ -1.0 (sin B)))
     (if (<= F 2.15e-61)
       (- (* 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 <= -0.5) {
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / sin(B));
	} else if (F <= 2.15e-61) {
		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 <= (-0.5d0)) then
        tmp = (x * ((-1.0d0) / tan(b))) + ((-1.0d0) / sin(b))
    else if (f <= 2.15d-61) 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 <= -0.5) {
		tmp = (x * (-1.0 / Math.tan(B))) + (-1.0 / Math.sin(B));
	} else if (F <= 2.15e-61) {
		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 <= -0.5:
		tmp = (x * (-1.0 / math.tan(B))) + (-1.0 / math.sin(B))
	elif F <= 2.15e-61:
		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 <= -0.5)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(-1.0 / sin(B)));
	elseif (F <= 2.15e-61)
		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 <= -0.5)
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / sin(B));
	elseif (F <= 2.15e-61)
		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, -0.5], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.15e-61], 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 < -0.5

   1. Initial program 69.2%

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

   3. Taylor expanded in F around -inf 99.7%

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

   if -0.5 < F < 2.1500000000000002e-61

   1. Initial program 99.4%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in 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 88.0%

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

   7. Taylor expanded in x around 0 88.0%

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

      1. associate-/l* 88.0%

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

   9. Simplified 88.0%

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

   if 2.1500000000000002e-61 < F

   1. Initial program 62.8%

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

   3. Simplified 70.3%

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

   5. Taylor expanded in x around 0 70.4%

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

      1. associate-*l/ 70.4%

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

      2. *-lft-identity 70.4%

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

      3. +-commutative 70.4%

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

      4. unpow2 70.4%

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

      5. fma-undefine 70.4%

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

   7. Simplified 70.4%

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

   8. Taylor expanded in F around inf 92.1%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.5:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 2.15 \cdot 10^{-61}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 81.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3.4 \cdot 10^{+66}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq 7.8 \cdot 10^{-53}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -3.4e+66)
     (+ (* x (/ -1.0 (tan B))) (/ -1.0 B))
     (if (<= F 7.8e-53)
       (- (* F (/ (sqrt 0.5) B)) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -3.4e+66) {
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / B);
	} else if (F <= 7.8e-53) {
		tmp = (F * (sqrt(0.5) / B)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-3.4d+66)) then
        tmp = (x * ((-1.0d0) / tan(b))) + ((-1.0d0) / b)
    else if (f <= 7.8d-53) then
        tmp = (f * (sqrt(0.5d0) / b)) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -3.4e+66) {
		tmp = (x * (-1.0 / Math.tan(B))) + (-1.0 / B);
	} else if (F <= 7.8e-53) {
		tmp = (F * (Math.sqrt(0.5) / B)) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -3.4e+66:
		tmp = (x * (-1.0 / math.tan(B))) + (-1.0 / B)
	elif F <= 7.8e-53:
		tmp = (F * (math.sqrt(0.5) / B)) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -3.4e+66)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(-1.0 / B));
	elseif (F <= 7.8e-53)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / B)) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -3.4e+66)
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / B);
	elseif (F <= 7.8e-53)
		tmp = (F * (sqrt(0.5) / B)) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -3.4e+66], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7.8e-53], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.4000000000000003e66

   1. Initial program 63.6%

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

   3. Taylor expanded in F around -inf 99.7%

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

   4. Taylor expanded in B around 0 85.5%

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

   if -3.4000000000000003e66 < F < 7.8000000000000004e-53

   1. Initial program 99.4%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in F around 0 94.1%

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

   6. Taylor expanded in B around 0 83.0%

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

   7. Taylor expanded in x around 0 83.8%

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

      1. associate-/l* 83.9%

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

   9. Simplified 83.9%

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

   if 7.8000000000000004e-53 < F

   1. Initial program 61.9%

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

   3. Simplified 69.5%

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

   5. Taylor expanded in x around 0 69.6%

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

      1. associate-*l/ 69.6%

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

      2. *-lft-identity 69.6%

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

      3. +-commutative 69.6%

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

      4. unpow2 69.6%

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

      5. fma-undefine 69.6%

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

   7. Simplified 69.6%

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

   8. Taylor expanded in F around inf 94.3%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.4 \cdot 10^{+66}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq 7.8 \cdot 10^{-53}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 64.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ t_1 := \frac{x}{-\tan B}\\ t_2 := \frac{1}{\sin B}\\ t_3 := \frac{-1}{\sin B} - \frac{x}{B}\\ t_4 := \frac{1}{B} - t\_0\\ \mathbf{if}\;F \leq -8 \cdot 10^{+209}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;F \leq -5.5 \cdot 10^{+149}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;F \leq -4.2 \cdot 10^{+105}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;F \leq 2.65 \cdot 10^{-61}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;F \leq 9.2 \cdot 10^{+80}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;F \leq 4.2 \cdot 10^{+100}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{+212}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;F \leq 3.5 \cdot 10^{+280}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{F \cdot B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B)))
        (t_1 (/ x (- (tan B))))
        (t_2 (/ 1.0 (sin B)))
        (t_3 (- (/ -1.0 (sin B)) (/ x B)))
        (t_4 (- (/ 1.0 B) t_0)))
   (if (<= F -8e+209)
     t_3
     (if (<= F -5.5e+149)
       t_1
       (if (<= F -4.2e+105)
         t_3
         (if (<= F 2.65e-61)
           t_1
           (if (<= F 9.2e+80)
             t_4
             (if (<= F 4.2e+100)
               t_2
               (if (<= F 2.3e+212)
                 t_4
                 (if (<= F 3.5e+280) t_2 (- (/ F (* F B)) t_0)))))))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double t_1 = x / -tan(B);
	double t_2 = 1.0 / sin(B);
	double t_3 = (-1.0 / sin(B)) - (x / B);
	double t_4 = (1.0 / B) - t_0;
	double tmp;
	if (F <= -8e+209) {
		tmp = t_3;
	} else if (F <= -5.5e+149) {
		tmp = t_1;
	} else if (F <= -4.2e+105) {
		tmp = t_3;
	} else if (F <= 2.65e-61) {
		tmp = t_1;
	} else if (F <= 9.2e+80) {
		tmp = t_4;
	} else if (F <= 4.2e+100) {
		tmp = t_2;
	} else if (F <= 2.3e+212) {
		tmp = t_4;
	} else if (F <= 3.5e+280) {
		tmp = t_2;
	} else {
		tmp = (F / (F * B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = x / tan(b)
    t_1 = x / -tan(b)
    t_2 = 1.0d0 / sin(b)
    t_3 = ((-1.0d0) / sin(b)) - (x / b)
    t_4 = (1.0d0 / b) - t_0
    if (f <= (-8d+209)) then
        tmp = t_3
    else if (f <= (-5.5d+149)) then
        tmp = t_1
    else if (f <= (-4.2d+105)) then
        tmp = t_3
    else if (f <= 2.65d-61) then
        tmp = t_1
    else if (f <= 9.2d+80) then
        tmp = t_4
    else if (f <= 4.2d+100) then
        tmp = t_2
    else if (f <= 2.3d+212) then
        tmp = t_4
    else if (f <= 3.5d+280) then
        tmp = t_2
    else
        tmp = (f / (f * 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 t_1 = x / -Math.tan(B);
	double t_2 = 1.0 / Math.sin(B);
	double t_3 = (-1.0 / Math.sin(B)) - (x / B);
	double t_4 = (1.0 / B) - t_0;
	double tmp;
	if (F <= -8e+209) {
		tmp = t_3;
	} else if (F <= -5.5e+149) {
		tmp = t_1;
	} else if (F <= -4.2e+105) {
		tmp = t_3;
	} else if (F <= 2.65e-61) {
		tmp = t_1;
	} else if (F <= 9.2e+80) {
		tmp = t_4;
	} else if (F <= 4.2e+100) {
		tmp = t_2;
	} else if (F <= 2.3e+212) {
		tmp = t_4;
	} else if (F <= 3.5e+280) {
		tmp = t_2;
	} else {
		tmp = (F / (F * B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	t_1 = x / -math.tan(B)
	t_2 = 1.0 / math.sin(B)
	t_3 = (-1.0 / math.sin(B)) - (x / B)
	t_4 = (1.0 / B) - t_0
	tmp = 0
	if F <= -8e+209:
		tmp = t_3
	elif F <= -5.5e+149:
		tmp = t_1
	elif F <= -4.2e+105:
		tmp = t_3
	elif F <= 2.65e-61:
		tmp = t_1
	elif F <= 9.2e+80:
		tmp = t_4
	elif F <= 4.2e+100:
		tmp = t_2
	elif F <= 2.3e+212:
		tmp = t_4
	elif F <= 3.5e+280:
		tmp = t_2
	else:
		tmp = (F / (F * B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	t_1 = Float64(x / Float64(-tan(B)))
	t_2 = Float64(1.0 / sin(B))
	t_3 = Float64(Float64(-1.0 / sin(B)) - Float64(x / B))
	t_4 = Float64(Float64(1.0 / B) - t_0)
	tmp = 0.0
	if (F <= -8e+209)
		tmp = t_3;
	elseif (F <= -5.5e+149)
		tmp = t_1;
	elseif (F <= -4.2e+105)
		tmp = t_3;
	elseif (F <= 2.65e-61)
		tmp = t_1;
	elseif (F <= 9.2e+80)
		tmp = t_4;
	elseif (F <= 4.2e+100)
		tmp = t_2;
	elseif (F <= 2.3e+212)
		tmp = t_4;
	elseif (F <= 3.5e+280)
		tmp = t_2;
	else
		tmp = Float64(Float64(F / Float64(F * B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	t_1 = x / -tan(B);
	t_2 = 1.0 / sin(B);
	t_3 = (-1.0 / sin(B)) - (x / B);
	t_4 = (1.0 / B) - t_0;
	tmp = 0.0;
	if (F <= -8e+209)
		tmp = t_3;
	elseif (F <= -5.5e+149)
		tmp = t_1;
	elseif (F <= -4.2e+105)
		tmp = t_3;
	elseif (F <= 2.65e-61)
		tmp = t_1;
	elseif (F <= 9.2e+80)
		tmp = t_4;
	elseif (F <= 4.2e+100)
		tmp = t_2;
	elseif (F <= 2.3e+212)
		tmp = t_4;
	elseif (F <= 3.5e+280)
		tmp = t_2;
	else
		tmp = (F / (F * B)) - 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[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[F, -8e+209], t$95$3, If[LessEqual[F, -5.5e+149], t$95$1, If[LessEqual[F, -4.2e+105], t$95$3, If[LessEqual[F, 2.65e-61], t$95$1, If[LessEqual[F, 9.2e+80], t$95$4, If[LessEqual[F, 4.2e+100], t$95$2, If[LessEqual[F, 2.3e+212], t$95$4, If[LessEqual[F, 3.5e+280], t$95$2, N[(N[(F / N[(F * B), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -8.0000000000000006e209 or -5.49999999999999999e149 < F < -4.2000000000000002e105

   1. Initial program 46.9%

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

   3. Taylor expanded in F around -inf 99.8%

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

   4. Taylor expanded in B around 0 75.5%

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

   if -8.0000000000000006e209 < F < -5.49999999999999999e149 or -4.2000000000000002e105 < F < 2.65e-61

   1. Initial program 98.7%

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

   3. Taylor expanded in F around -inf 51.4%

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

   4. Taylor expanded in x around inf 71.3%

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

      1. mul-1-neg 71.3%

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

      2. associate-/l* 71.2%

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

      3. distribute-lft-neg-in 71.2%

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

   6. Simplified 71.2%

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

      1. add-sqr-sqrt 19.2%

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

      2. sqrt-unprod 16.2%

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

      3. sqr-neg 16.2%

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

      4. sqrt-unprod 1.1%

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

      5. add-sqr-sqrt 1.9%

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

      6. clear-num 1.9%

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

      7. tan-quot 1.9%

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

      8. div-inv 1.9%

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

      9. frac-2neg 1.9%

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

      10. neg-sub0 1.9%

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

      11. div-sub 1.9%

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

      12. add-sqr-sqrt 1.1%

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

      13. sqrt-unprod 16.2%

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

      14. sqr-neg 16.2%

          \[\leadsto \frac{0}{-\tan B} - \frac{\sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}}}{-\tan B} \]

      15. sqrt-unprod 19.3%

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

      16. add-sqr-sqrt 71.4%

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

      17. frac-2neg 71.4%

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

   8. Applied egg-rr 71.4%

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

      1. div0 71.4%

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

      2. neg-sub0 71.4%

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

      3. distribute-frac-neg2 71.4%

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

   10. Simplified 71.4%

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

   if 2.65e-61 < F < 9.20000000000000016e80 or 4.1999999999999997e100 < F < 2.2999999999999998e212

   1. Initial program 82.0%

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

   3. Simplified 86.4%

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

   5. Taylor expanded in x around 0 86.5%

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

      1. associate-*l/ 86.6%

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

      2. *-lft-identity 86.6%

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

      3. +-commutative 86.6%

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

      4. unpow2 86.6%

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

      5. fma-undefine 86.6%

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

   7. Simplified 86.6%

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

      1. clear-num 86.4%

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

      2. un-div-inv 86.5%

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

      3. inv-pow 86.5%

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

      4. sqrt-pow1 86.4%

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

      5. metadata-eval 86.4%

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

   9. Applied egg-rr 86.4%

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

   10. Taylor expanded in B around 0 69.0%

       \[\leadsto \frac{F}{\color{blue}{B \cdot \sqrt{2 + {F}^{2}}}} - \frac{x}{\tan B} \]
   11. Step-by-step derivation

       1. +-commutative 69.0%

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

       2. unpow2 69.0%

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

       3. fma-undefine 69.0%

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

   12. Simplified 69.0%

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

   13. Taylor expanded in F around inf 74.3%

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

   if 9.20000000000000016e80 < F < 4.1999999999999997e100 or 2.2999999999999998e212 < F < 3.5000000000000001e280

   1. Initial program 40.0%

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

   3. Taylor expanded in F around -inf 24.0%

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

      1. add-cube-cbrt 23.9%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}} \cdot \sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}\right) \cdot \sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}} \]

      2. pow3 24.0%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}\right)}^{3}} \]

   5. Applied egg-rr 75.7%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{1}{\sin B} + \frac{x}{\tan B}}\right)}^{3}} \]

   6. Taylor expanded in x around 0 76.8%

      \[\leadsto {\color{blue}{\left(\sqrt[3]{\frac{1}{\sin B}}\right)}}^{3} \]

   7. Taylor expanded in B around inf 77.9%

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

   if 3.5000000000000001e280 < F

   1. Initial program 36.9%

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

   3. Simplified 56.4%

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

   5. Taylor expanded in x around 0 56.4%

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

      1. associate-*l/ 56.4%

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

      2. *-lft-identity 56.4%

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

      3. +-commutative 56.4%

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

      4. unpow2 56.4%

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

      5. fma-undefine 56.4%

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

   7. Simplified 56.4%

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

      1. clear-num 56.4%

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

      2. un-div-inv 56.4%

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

      3. inv-pow 56.4%

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

      4. sqrt-pow1 56.4%

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

      5. metadata-eval 56.4%

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

   9. Applied egg-rr 56.4%

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

   10. Taylor expanded in B around 0 56.4%

       \[\leadsto \frac{F}{\color{blue}{B \cdot \sqrt{2 + {F}^{2}}}} - \frac{x}{\tan B} \]
   11. Step-by-step derivation

       1. +-commutative 56.4%

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

       2. unpow2 56.4%

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

       3. fma-undefine 56.4%

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

   12. Simplified 56.4%

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

   13. Taylor expanded in F around inf 82.5%

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

       1. *-commutative 82.5%

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

   15. Simplified 82.5%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -8 \cdot 10^{+209}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -5.5 \cdot 10^{+149}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq -4.2 \cdot 10^{+105}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.65 \cdot 10^{-61}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 9.2 \cdot 10^{+80}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 4.2 \cdot 10^{+100}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{+212}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 3.5 \cdot 10^{+280}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{F \cdot B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 63.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-\tan B}\\ t_1 := \frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{if}\;F \leq -1 \cdot 10^{+212}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;F \leq -5.5 \cdot 10^{+149}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq -4.2 \cdot 10^{+105}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;F \leq 3.2 \cdot 10^{-61}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 8.5 \cdot 10^{+78} \lor \neg \left(F \leq 1.5 \cdot 10^{+101}\right) \land \left(F \leq 2.15 \cdot 10^{+212} \lor \neg \left(F \leq 3.9 \cdot 10^{+280}\right)\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (- (tan B)))) (t_1 (- (/ -1.0 (sin B)) (/ x B))))
   (if (<= F -1e+212)
     t_1
     (if (<= F -5.5e+149)
       t_0
       (if (<= F -4.2e+105)
         t_1
         (if (<= F 3.2e-61)
           t_0
           (if (or (<= F 8.5e+78)
                   (and (not (<= F 1.5e+101))
                        (or (<= F 2.15e+212) (not (<= F 3.9e+280)))))
             (- (/ 1.0 B) (/ x (tan B)))
             (/ 1.0 (sin B)))))))))

C

double code(double F, double B, double x) {
	double t_0 = x / -tan(B);
	double t_1 = (-1.0 / sin(B)) - (x / B);
	double tmp;
	if (F <= -1e+212) {
		tmp = t_1;
	} else if (F <= -5.5e+149) {
		tmp = t_0;
	} else if (F <= -4.2e+105) {
		tmp = t_1;
	} else if (F <= 3.2e-61) {
		tmp = t_0;
	} else if ((F <= 8.5e+78) || (!(F <= 1.5e+101) && ((F <= 2.15e+212) || !(F <= 3.9e+280)))) {
		tmp = (1.0 / B) - (x / tan(B));
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x / -tan(b)
    t_1 = ((-1.0d0) / sin(b)) - (x / b)
    if (f <= (-1d+212)) then
        tmp = t_1
    else if (f <= (-5.5d+149)) then
        tmp = t_0
    else if (f <= (-4.2d+105)) then
        tmp = t_1
    else if (f <= 3.2d-61) then
        tmp = t_0
    else if ((f <= 8.5d+78) .or. (.not. (f <= 1.5d+101)) .and. (f <= 2.15d+212) .or. (.not. (f <= 3.9d+280))) then
        tmp = (1.0d0 / b) - (x / tan(b))
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / -Math.tan(B);
	double t_1 = (-1.0 / Math.sin(B)) - (x / B);
	double tmp;
	if (F <= -1e+212) {
		tmp = t_1;
	} else if (F <= -5.5e+149) {
		tmp = t_0;
	} else if (F <= -4.2e+105) {
		tmp = t_1;
	} else if (F <= 3.2e-61) {
		tmp = t_0;
	} else if ((F <= 8.5e+78) || (!(F <= 1.5e+101) && ((F <= 2.15e+212) || !(F <= 3.9e+280)))) {
		tmp = (1.0 / B) - (x / Math.tan(B));
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / -math.tan(B)
	t_1 = (-1.0 / math.sin(B)) - (x / B)
	tmp = 0
	if F <= -1e+212:
		tmp = t_1
	elif F <= -5.5e+149:
		tmp = t_0
	elif F <= -4.2e+105:
		tmp = t_1
	elif F <= 3.2e-61:
		tmp = t_0
	elif (F <= 8.5e+78) or (not (F <= 1.5e+101) and ((F <= 2.15e+212) or not (F <= 3.9e+280))):
		tmp = (1.0 / B) - (x / math.tan(B))
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / Float64(-tan(B)))
	t_1 = Float64(Float64(-1.0 / sin(B)) - Float64(x / B))
	tmp = 0.0
	if (F <= -1e+212)
		tmp = t_1;
	elseif (F <= -5.5e+149)
		tmp = t_0;
	elseif (F <= -4.2e+105)
		tmp = t_1;
	elseif (F <= 3.2e-61)
		tmp = t_0;
	elseif ((F <= 8.5e+78) || (!(F <= 1.5e+101) && ((F <= 2.15e+212) || !(F <= 3.9e+280))))
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / -tan(B);
	t_1 = (-1.0 / sin(B)) - (x / B);
	tmp = 0.0;
	if (F <= -1e+212)
		tmp = t_1;
	elseif (F <= -5.5e+149)
		tmp = t_0;
	elseif (F <= -4.2e+105)
		tmp = t_1;
	elseif (F <= 3.2e-61)
		tmp = t_0;
	elseif ((F <= 8.5e+78) || (~((F <= 1.5e+101)) && ((F <= 2.15e+212) || ~((F <= 3.9e+280)))))
		tmp = (1.0 / B) - (x / tan(B));
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1e+212], t$95$1, If[LessEqual[F, -5.5e+149], t$95$0, If[LessEqual[F, -4.2e+105], t$95$1, If[LessEqual[F, 3.2e-61], t$95$0, If[Or[LessEqual[F, 8.5e+78], And[N[Not[LessEqual[F, 1.5e+101]], $MachinePrecision], Or[LessEqual[F, 2.15e+212], N[Not[LessEqual[F, 3.9e+280]], $MachinePrecision]]]], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -9.9999999999999991e211 or -5.49999999999999999e149 < F < -4.2000000000000002e105

   1. Initial program 46.9%

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

   3. Taylor expanded in F around -inf 99.8%

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

   4. Taylor expanded in B around 0 75.5%

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

   if -9.9999999999999991e211 < F < -5.49999999999999999e149 or -4.2000000000000002e105 < F < 3.2000000000000001e-61

   1. Initial program 98.7%

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

   3. Taylor expanded in F around -inf 51.4%

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

   4. Taylor expanded in x around inf 71.3%

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

      1. mul-1-neg 71.3%

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

      2. associate-/l* 71.2%

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

      3. distribute-lft-neg-in 71.2%

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

   6. Simplified 71.2%

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

      1. add-sqr-sqrt 19.2%

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

      2. sqrt-unprod 16.2%

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

      3. sqr-neg 16.2%

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

      4. sqrt-unprod 1.1%

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

      5. add-sqr-sqrt 1.9%

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

      6. clear-num 1.9%

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

      7. tan-quot 1.9%

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

      8. div-inv 1.9%

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

      9. frac-2neg 1.9%

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

      10. neg-sub0 1.9%

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

      11. div-sub 1.9%

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

      12. add-sqr-sqrt 1.1%

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

      13. sqrt-unprod 16.2%

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

      14. sqr-neg 16.2%

          \[\leadsto \frac{0}{-\tan B} - \frac{\sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}}}{-\tan B} \]

      15. sqrt-unprod 19.3%

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

      16. add-sqr-sqrt 71.4%

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

      17. frac-2neg 71.4%

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

   8. Applied egg-rr 71.4%

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

      1. div0 71.4%

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

      2. neg-sub0 71.4%

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

      3. distribute-frac-neg2 71.4%

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

   10. Simplified 71.4%

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

   if 3.2000000000000001e-61 < F < 8.50000000000000079e78 or 1.49999999999999997e101 < F < 2.1499999999999998e212 or 3.90000000000000029e280 < F

   1. Initial program 73.0%

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

   3. Simplified 80.4%

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

   5. Taylor expanded in x around 0 80.5%

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

      1. associate-*l/ 80.5%

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

      2. *-lft-identity 80.5%

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

      3. +-commutative 80.5%

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

      4. unpow2 80.5%

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

      5. fma-undefine 80.5%

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

   7. Simplified 80.5%

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

      1. clear-num 80.4%

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

      2. un-div-inv 80.5%

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

      3. inv-pow 80.5%

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

      4. sqrt-pow1 80.4%

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

      5. metadata-eval 80.4%

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

   9. Applied egg-rr 80.4%

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

   10. Taylor expanded in B around 0 66.5%

       \[\leadsto \frac{F}{\color{blue}{B \cdot \sqrt{2 + {F}^{2}}}} - \frac{x}{\tan B} \]
   11. Step-by-step derivation

       1. +-commutative 66.5%

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

       2. unpow2 66.5%

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

       3. fma-undefine 66.5%

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

   12. Simplified 66.5%

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

   13. Taylor expanded in F around inf 75.9%

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

   if 8.50000000000000079e78 < F < 1.49999999999999997e101 or 2.1499999999999998e212 < F < 3.90000000000000029e280

   1. Initial program 40.0%

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

   3. Taylor expanded in F around -inf 24.0%

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

      1. add-cube-cbrt 23.9%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}} \cdot \sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}\right) \cdot \sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}} \]

      2. pow3 24.0%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}\right)}^{3}} \]

   5. Applied egg-rr 75.7%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{1}{\sin B} + \frac{x}{\tan B}}\right)}^{3}} \]

   6. Taylor expanded in x around 0 76.8%

      \[\leadsto {\color{blue}{\left(\sqrt[3]{\frac{1}{\sin B}}\right)}}^{3} \]

   7. Taylor expanded in B around inf 77.9%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{+212}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -5.5 \cdot 10^{+149}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq -4.2 \cdot 10^{+105}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.2 \cdot 10^{-61}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 8.5 \cdot 10^{+78} \lor \neg \left(F \leq 1.5 \cdot 10^{+101}\right) \land \left(F \leq 2.15 \cdot 10^{+212} \lor \neg \left(F \leq 3.9 \cdot 10^{+280}\right)\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 66.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -4.5 \cdot 10^{-152}:\\ \;\;\;\;\frac{F}{B \cdot \left(-F\right)} - t\_1\\ \mathbf{elif}\;F \leq -8.5 \cdot 10^{-196}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 9 \cdot 10^{-200}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 9.2 \cdot 10^{-145}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 3.2 \cdot 10^{-61}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B}\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{+212}:\\ \;\;\;\;\frac{1}{B} - t\_1\\ \mathbf{elif}\;F \leq 3.9 \cdot 10^{+280}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{F \cdot B} - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- (* F (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))) x) B))
        (t_1 (/ x (tan B))))
   (if (<= F -4.5e-152)
     (- (/ F (* B (- F))) t_1)
     (if (<= F -8.5e-196)
       t_0
       (if (<= F 9e-200)
         (/ x (- (tan B)))
         (if (<= F 9.2e-145)
           t_0
           (if (<= F 3.2e-61)
             (* x (/ -1.0 (tan B)))
             (if (<= F 2.3e+212)
               (- (/ 1.0 B) t_1)
               (if (<= F 3.9e+280)
                 (/ 1.0 (sin B))
                 (- (/ F (* F B)) t_1))))))))))

C

double code(double F, double B, double x) {
	double t_0 = ((F * sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B;
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -4.5e-152) {
		tmp = (F / (B * -F)) - t_1;
	} else if (F <= -8.5e-196) {
		tmp = t_0;
	} else if (F <= 9e-200) {
		tmp = x / -tan(B);
	} else if (F <= 9.2e-145) {
		tmp = t_0;
	} else if (F <= 3.2e-61) {
		tmp = x * (-1.0 / tan(B));
	} else if (F <= 2.3e+212) {
		tmp = (1.0 / B) - t_1;
	} else if (F <= 3.9e+280) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = (F / (F * 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) :: tmp
    t_0 = ((f * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))) - x) / b
    t_1 = x / tan(b)
    if (f <= (-4.5d-152)) then
        tmp = (f / (b * -f)) - t_1
    else if (f <= (-8.5d-196)) then
        tmp = t_0
    else if (f <= 9d-200) then
        tmp = x / -tan(b)
    else if (f <= 9.2d-145) then
        tmp = t_0
    else if (f <= 3.2d-61) then
        tmp = x * ((-1.0d0) / tan(b))
    else if (f <= 2.3d+212) then
        tmp = (1.0d0 / b) - t_1
    else if (f <= 3.9d+280) then
        tmp = 1.0d0 / sin(b)
    else
        tmp = (f / (f * 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.sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B;
	double t_1 = x / Math.tan(B);
	double tmp;
	if (F <= -4.5e-152) {
		tmp = (F / (B * -F)) - t_1;
	} else if (F <= -8.5e-196) {
		tmp = t_0;
	} else if (F <= 9e-200) {
		tmp = x / -Math.tan(B);
	} else if (F <= 9.2e-145) {
		tmp = t_0;
	} else if (F <= 3.2e-61) {
		tmp = x * (-1.0 / Math.tan(B));
	} else if (F <= 2.3e+212) {
		tmp = (1.0 / B) - t_1;
	} else if (F <= 3.9e+280) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = (F / (F * B)) - t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = ((F * math.sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B
	t_1 = x / math.tan(B)
	tmp = 0
	if F <= -4.5e-152:
		tmp = (F / (B * -F)) - t_1
	elif F <= -8.5e-196:
		tmp = t_0
	elif F <= 9e-200:
		tmp = x / -math.tan(B)
	elif F <= 9.2e-145:
		tmp = t_0
	elif F <= 3.2e-61:
		tmp = x * (-1.0 / math.tan(B))
	elif F <= 2.3e+212:
		tmp = (1.0 / B) - t_1
	elif F <= 3.9e+280:
		tmp = 1.0 / math.sin(B)
	else:
		tmp = (F / (F * B)) - t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(Float64(F * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))) - x) / B)
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -4.5e-152)
		tmp = Float64(Float64(F / Float64(B * Float64(-F))) - t_1);
	elseif (F <= -8.5e-196)
		tmp = t_0;
	elseif (F <= 9e-200)
		tmp = Float64(x / Float64(-tan(B)));
	elseif (F <= 9.2e-145)
		tmp = t_0;
	elseif (F <= 3.2e-61)
		tmp = Float64(x * Float64(-1.0 / tan(B)));
	elseif (F <= 2.3e+212)
		tmp = Float64(Float64(1.0 / B) - t_1);
	elseif (F <= 3.9e+280)
		tmp = Float64(1.0 / sin(B));
	else
		tmp = Float64(Float64(F / Float64(F * B)) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = ((F * sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B;
	t_1 = x / tan(B);
	tmp = 0.0;
	if (F <= -4.5e-152)
		tmp = (F / (B * -F)) - t_1;
	elseif (F <= -8.5e-196)
		tmp = t_0;
	elseif (F <= 9e-200)
		tmp = x / -tan(B);
	elseif (F <= 9.2e-145)
		tmp = t_0;
	elseif (F <= 3.2e-61)
		tmp = x * (-1.0 / tan(B));
	elseif (F <= 2.3e+212)
		tmp = (1.0 / B) - t_1;
	elseif (F <= 3.9e+280)
		tmp = 1.0 / sin(B);
	else
		tmp = (F / (F * B)) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(N[(F * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -4.5e-152], N[(N[(F / N[(B * (-F)), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, -8.5e-196], t$95$0, If[LessEqual[F, 9e-200], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 9.2e-145], t$95$0, If[LessEqual[F, 3.2e-61], N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.3e+212], N[(N[(1.0 / B), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, 3.9e+280], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(F / N[(F * B), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if F < -4.5000000000000004e-152

   1. Initial program 78.1%

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

   3. Simplified 84.5%

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

   5. Taylor expanded in x around 0 84.6%

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

      1. associate-*l/ 84.5%

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

      2. *-lft-identity 84.5%

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

      3. +-commutative 84.5%

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

      4. unpow2 84.5%

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

      5. fma-undefine 84.5%

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

   7. Simplified 84.5%

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

      1. clear-num 84.5%

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

      2. un-div-inv 84.5%

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

      3. inv-pow 84.5%

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

      4. sqrt-pow1 84.5%

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

      5. metadata-eval 84.5%

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

   9. Applied egg-rr 84.5%

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

   10. Taylor expanded in B around 0 72.5%

       \[\leadsto \frac{F}{\color{blue}{B \cdot \sqrt{2 + {F}^{2}}}} - \frac{x}{\tan B} \]
   11. Step-by-step derivation

       1. +-commutative 72.5%

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

       2. unpow2 72.5%

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

       3. fma-undefine 72.5%

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

   12. Simplified 72.5%

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

   13. Taylor expanded in F around -inf 77.1%

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

       1. associate-*r* 77.1%

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

       2. neg-mul-1 77.1%

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

   15. Simplified 77.1%

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

   if -4.5000000000000004e-152 < F < -8.50000000000000004e-196 or 9.0000000000000004e-200 < F < 9.20000000000000028e-145

   1. Initial program 99.2%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in 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 69.3%

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

   if -8.50000000000000004e-196 < F < 9.0000000000000004e-200

   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 38.2%

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

   4. Taylor expanded in x around inf 94.1%

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

      1. mul-1-neg 94.1%

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

      2. associate-/l* 94.0%

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

      3. distribute-lft-neg-in 94.0%

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

   6. Simplified 94.0%

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

      1. add-sqr-sqrt 26.6%

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

      2. sqrt-unprod 20.8%

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

      3. sqr-neg 20.8%

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

      4. sqrt-unprod 0.8%

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

      5. add-sqr-sqrt 1.7%

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

      6. clear-num 1.7%

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

      7. tan-quot 1.7%

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

      8. div-inv 1.7%

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

      9. frac-2neg 1.7%

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

      10. neg-sub0 1.7%

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

      11. div-sub 1.7%

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

      12. add-sqr-sqrt 0.8%

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

      13. sqrt-unprod 20.8%

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

      14. sqr-neg 20.8%

          \[\leadsto \frac{0}{-\tan B} - \frac{\sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}}}{-\tan B} \]

      15. sqrt-unprod 26.8%

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

      16. add-sqr-sqrt 94.2%

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

      17. frac-2neg 94.2%

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

   8. Applied egg-rr 94.2%

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

      1. div0 94.2%

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

      2. neg-sub0 94.2%

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

      3. distribute-frac-neg2 94.2%

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

   10. Simplified 94.2%

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

   if 9.20000000000000028e-145 < F < 3.2000000000000001e-61

   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 29.5%

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

   4. Taylor expanded in x around inf 65.8%

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

      1. mul-1-neg 65.8%

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

      2. associate-/l* 65.9%

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

      3. distribute-lft-neg-in 65.9%

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

   6. Simplified 65.9%

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

      1. clear-num 65.9%

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

      2. tan-quot 65.8%

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

      3. *-un-lft-identity 65.8%

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

   8. Applied egg-rr 65.8%

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

      1. *-lft-identity 65.8%

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

   10. Simplified 65.8%

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

   if 3.2000000000000001e-61 < F < 2.2999999999999998e212

   1. Initial program 84.4%

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

   3. Simplified 88.2%

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

   5. Taylor expanded in x around 0 88.3%

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

      1. associate-*l/ 88.3%

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

      2. *-lft-identity 88.3%

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

      3. +-commutative 88.3%

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

      4. unpow2 88.3%

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

      5. fma-undefine 88.3%

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

   7. Simplified 88.3%

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

      1. clear-num 88.2%

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

      2. un-div-inv 88.3%

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

      3. inv-pow 88.3%

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

      4. sqrt-pow1 88.2%

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

      5. metadata-eval 88.2%

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

   9. Applied egg-rr 88.2%

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

   10. Taylor expanded in B around 0 63.8%

       \[\leadsto \frac{F}{\color{blue}{B \cdot \sqrt{2 + {F}^{2}}}} - \frac{x}{\tan B} \]
   11. Step-by-step derivation

       1. +-commutative 63.8%

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

       2. unpow2 63.8%

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

       3. fma-undefine 63.8%

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

   12. Simplified 63.8%

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

   13. Taylor expanded in F around inf 68.4%

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

   if 2.2999999999999998e212 < F < 3.90000000000000029e280

   1. Initial program 18.0%

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

   3. Taylor expanded in F around -inf 27.1%

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

      1. add-cube-cbrt 27.0%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}} \cdot \sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}\right) \cdot \sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}} \]

      2. pow3 27.0%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}\right)}^{3}} \]

   5. Applied egg-rr 72.6%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{1}{\sin B} + \frac{x}{\tan B}}\right)}^{3}} \]

   6. Taylor expanded in x around 0 73.9%

      \[\leadsto {\color{blue}{\left(\sqrt[3]{\frac{1}{\sin B}}\right)}}^{3} \]

   7. Taylor expanded in B around inf 74.9%

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

   if 3.90000000000000029e280 < F

   1. Initial program 36.9%

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

   3. Simplified 56.4%

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

   5. Taylor expanded in x around 0 56.4%

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

      1. associate-*l/ 56.4%

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

      2. *-lft-identity 56.4%

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

      3. +-commutative 56.4%

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

      4. unpow2 56.4%

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

      5. fma-undefine 56.4%

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

   7. Simplified 56.4%

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

      1. clear-num 56.4%

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

      2. un-div-inv 56.4%

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

      3. inv-pow 56.4%

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

      4. sqrt-pow1 56.4%

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

      5. metadata-eval 56.4%

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

   9. Applied egg-rr 56.4%

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

   10. Taylor expanded in B around 0 56.4%

       \[\leadsto \frac{F}{\color{blue}{B \cdot \sqrt{2 + {F}^{2}}}} - \frac{x}{\tan B} \]
   11. Step-by-step derivation

       1. +-commutative 56.4%

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

       2. unpow2 56.4%

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

       3. fma-undefine 56.4%

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

   12. Simplified 56.4%

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

   13. Taylor expanded in F around inf 82.5%

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

       1. *-commutative 82.5%

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

   15. Simplified 82.5%

       \[\leadsto \frac{F}{\color{blue}{F \cdot B}} - \frac{x}{\tan B} \]
3. Recombined 7 regimes into one program.

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.5 \cdot 10^{-152}:\\ \;\;\;\;\frac{F}{B \cdot \left(-F\right)} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -8.5 \cdot 10^{-196}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{elif}\;F \leq 9 \cdot 10^{-200}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 9.2 \cdot 10^{-145}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{elif}\;F \leq 3.2 \cdot 10^{-61}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B}\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{+212}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 3.9 \cdot 10^{+280}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{F \cdot B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 61.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-\tan B}\\ \mathbf{if}\;F \leq -1.25 \cdot 10^{+149}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq -4.1 \cdot 10^{+83}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.85 \cdot 10^{-61}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 4.4 \cdot 10^{+76} \lor \neg \left(F \leq 6.2 \cdot 10^{+100} \lor \neg \left(F \leq 2.3 \cdot 10^{+212}\right) \land F \leq 3.9 \cdot 10^{+280}\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (- (tan B)))))
   (if (<= F -1.25e+149)
     t_0
     (if (<= F -4.1e+83)
       (/ (- -1.0 x) B)
       (if (<= F 2.85e-61)
         t_0
         (if (or (<= F 4.4e+76)
                 (not
                  (or (<= F 6.2e+100)
                      (and (not (<= F 2.3e+212)) (<= F 3.9e+280)))))
           (- (/ 1.0 B) (/ x (tan B)))
           (/ 1.0 (sin B))))))))

C

double code(double F, double B, double x) {
	double t_0 = x / -tan(B);
	double tmp;
	if (F <= -1.25e+149) {
		tmp = t_0;
	} else if (F <= -4.1e+83) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 2.85e-61) {
		tmp = t_0;
	} else if ((F <= 4.4e+76) || !((F <= 6.2e+100) || (!(F <= 2.3e+212) && (F <= 3.9e+280)))) {
		tmp = (1.0 / B) - (x / tan(B));
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / -tan(b)
    if (f <= (-1.25d+149)) then
        tmp = t_0
    else if (f <= (-4.1d+83)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 2.85d-61) then
        tmp = t_0
    else if ((f <= 4.4d+76) .or. (.not. (f <= 6.2d+100) .or. (.not. (f <= 2.3d+212)) .and. (f <= 3.9d+280))) then
        tmp = (1.0d0 / b) - (x / tan(b))
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / -Math.tan(B);
	double tmp;
	if (F <= -1.25e+149) {
		tmp = t_0;
	} else if (F <= -4.1e+83) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 2.85e-61) {
		tmp = t_0;
	} else if ((F <= 4.4e+76) || !((F <= 6.2e+100) || (!(F <= 2.3e+212) && (F <= 3.9e+280)))) {
		tmp = (1.0 / B) - (x / Math.tan(B));
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / -math.tan(B)
	tmp = 0
	if F <= -1.25e+149:
		tmp = t_0
	elif F <= -4.1e+83:
		tmp = (-1.0 - x) / B
	elif F <= 2.85e-61:
		tmp = t_0
	elif (F <= 4.4e+76) or not ((F <= 6.2e+100) or (not (F <= 2.3e+212) and (F <= 3.9e+280))):
		tmp = (1.0 / B) - (x / math.tan(B))
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / Float64(-tan(B)))
	tmp = 0.0
	if (F <= -1.25e+149)
		tmp = t_0;
	elseif (F <= -4.1e+83)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 2.85e-61)
		tmp = t_0;
	elseif ((F <= 4.4e+76) || !((F <= 6.2e+100) || (!(F <= 2.3e+212) && (F <= 3.9e+280))))
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / -tan(B);
	tmp = 0.0;
	if (F <= -1.25e+149)
		tmp = t_0;
	elseif (F <= -4.1e+83)
		tmp = (-1.0 - x) / B;
	elseif (F <= 2.85e-61)
		tmp = t_0;
	elseif ((F <= 4.4e+76) || ~(((F <= 6.2e+100) || (~((F <= 2.3e+212)) && (F <= 3.9e+280)))))
		tmp = (1.0 / B) - (x / tan(B));
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision]}, If[LessEqual[F, -1.25e+149], t$95$0, If[LessEqual[F, -4.1e+83], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 2.85e-61], t$95$0, If[Or[LessEqual[F, 4.4e+76], N[Not[Or[LessEqual[F, 6.2e+100], And[N[Not[LessEqual[F, 2.3e+212]], $MachinePrecision], LessEqual[F, 3.9e+280]]]], $MachinePrecision]], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.24999999999999998e149 or -4.1000000000000001e83 < F < 2.85000000000000003e-61

   1. Initial program 87.6%

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

   3. Taylor expanded in F around -inf 58.1%

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

   4. Taylor expanded in x around inf 66.7%

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

      1. mul-1-neg 66.7%

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

      2. associate-/l* 66.7%

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

      3. distribute-lft-neg-in 66.7%

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

   6. Simplified 66.7%

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

      1. add-sqr-sqrt 21.1%

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

      2. sqrt-unprod 18.1%

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

      3. sqr-neg 18.1%

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

      4. sqrt-unprod 1.1%

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

      5. add-sqr-sqrt 2.0%

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

      6. clear-num 2.0%

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

      7. tan-quot 2.0%

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

      8. div-inv 2.0%

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

      9. frac-2neg 2.0%

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

      10. neg-sub0 2.0%

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

      11. div-sub 2.0%

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

      12. add-sqr-sqrt 1.1%

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

      13. sqrt-unprod 18.2%

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

      14. sqr-neg 18.2%

          \[\leadsto \frac{0}{-\tan B} - \frac{\sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}}}{-\tan B} \]

      15. sqrt-unprod 21.2%

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

      16. add-sqr-sqrt 66.9%

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

      17. frac-2neg 66.9%

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

   8. Applied egg-rr 66.9%

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

      1. div0 66.9%

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

      2. neg-sub0 66.9%

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

      3. distribute-frac-neg2 66.9%

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

   10. Simplified 66.9%

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

   if -1.24999999999999998e149 < F < -4.1000000000000001e83

   1. Initial program 82.9%

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

   3. Taylor expanded in F around -inf 99.9%

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

   4. Taylor expanded in B around 0 71.5%

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

      1. mul-1-neg 71.5%

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

      2. distribute-neg-frac2 71.5%

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

   6. Simplified 71.5%

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

   if 2.85000000000000003e-61 < F < 4.4000000000000001e76 or 6.20000000000000014e100 < F < 2.2999999999999998e212 or 3.90000000000000029e280 < F

   1. Initial program 73.0%

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

   3. Simplified 80.4%

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

   5. Taylor expanded in x around 0 80.5%

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

      1. associate-*l/ 80.5%

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

      2. *-lft-identity 80.5%

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

      3. +-commutative 80.5%

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

      4. unpow2 80.5%

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

      5. fma-undefine 80.5%

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

   7. Simplified 80.5%

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

      1. clear-num 80.4%

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

      2. un-div-inv 80.5%

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

      3. inv-pow 80.5%

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

      4. sqrt-pow1 80.4%

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

      5. metadata-eval 80.4%

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

   9. Applied egg-rr 80.4%

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

   10. Taylor expanded in B around 0 66.5%

       \[\leadsto \frac{F}{\color{blue}{B \cdot \sqrt{2 + {F}^{2}}}} - \frac{x}{\tan B} \]
   11. Step-by-step derivation

       1. +-commutative 66.5%

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

       2. unpow2 66.5%

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

       3. fma-undefine 66.5%

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

   12. Simplified 66.5%

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

   13. Taylor expanded in F around inf 75.9%

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

   if 4.4000000000000001e76 < F < 6.20000000000000014e100 or 2.2999999999999998e212 < F < 3.90000000000000029e280

   1. Initial program 40.0%

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

   3. Taylor expanded in F around -inf 24.0%

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

      1. add-cube-cbrt 23.9%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}} \cdot \sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}\right) \cdot \sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}} \]

      2. pow3 24.0%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}\right)}^{3}} \]

   5. Applied egg-rr 75.7%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{1}{\sin B} + \frac{x}{\tan B}}\right)}^{3}} \]

   6. Taylor expanded in x around 0 76.8%

      \[\leadsto {\color{blue}{\left(\sqrt[3]{\frac{1}{\sin B}}\right)}}^{3} \]

   7. Taylor expanded in B around inf 77.9%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.25 \cdot 10^{+149}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq -4.1 \cdot 10^{+83}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.85 \cdot 10^{-61}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 4.4 \cdot 10^{+76} \lor \neg \left(F \leq 6.2 \cdot 10^{+100} \lor \neg \left(F \leq 2.3 \cdot 10^{+212}\right) \land F \leq 3.9 \cdot 10^{+280}\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 67.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ t_1 := \frac{1}{B} - t\_0\\ t_2 := \frac{1}{\sin B}\\ \mathbf{if}\;F \leq -2.2 \cdot 10^{-7}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq 2.45 \cdot 10^{-145}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 4.4 \cdot 10^{+57}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;F \leq 8.8 \cdot 10^{+101}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;F \leq 2.35 \cdot 10^{+212}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;F \leq 3.5 \cdot 10^{+280}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{F \cdot B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))) (t_1 (- (/ 1.0 B) t_0)) (t_2 (/ 1.0 (sin B))))
   (if (<= F -2.2e-7)
     (+ (* x (/ -1.0 (tan B))) (/ -1.0 B))
     (if (<= F 2.45e-145)
       (/ x (- (tan B)))
       (if (<= F 4.4e+57)
         t_1
         (if (<= F 8.8e+101)
           t_2
           (if (<= F 2.35e+212)
             t_1
             (if (<= F 3.5e+280) t_2 (- (/ F (* F B)) t_0)))))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double t_1 = (1.0 / B) - t_0;
	double t_2 = 1.0 / sin(B);
	double tmp;
	if (F <= -2.2e-7) {
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / B);
	} else if (F <= 2.45e-145) {
		tmp = x / -tan(B);
	} else if (F <= 4.4e+57) {
		tmp = t_1;
	} else if (F <= 8.8e+101) {
		tmp = t_2;
	} else if (F <= 2.35e+212) {
		tmp = t_1;
	} else if (F <= 3.5e+280) {
		tmp = t_2;
	} else {
		tmp = (F / (F * B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = x / tan(b)
    t_1 = (1.0d0 / b) - t_0
    t_2 = 1.0d0 / sin(b)
    if (f <= (-2.2d-7)) then
        tmp = (x * ((-1.0d0) / tan(b))) + ((-1.0d0) / b)
    else if (f <= 2.45d-145) then
        tmp = x / -tan(b)
    else if (f <= 4.4d+57) then
        tmp = t_1
    else if (f <= 8.8d+101) then
        tmp = t_2
    else if (f <= 2.35d+212) then
        tmp = t_1
    else if (f <= 3.5d+280) then
        tmp = t_2
    else
        tmp = (f / (f * 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 t_1 = (1.0 / B) - t_0;
	double t_2 = 1.0 / Math.sin(B);
	double tmp;
	if (F <= -2.2e-7) {
		tmp = (x * (-1.0 / Math.tan(B))) + (-1.0 / B);
	} else if (F <= 2.45e-145) {
		tmp = x / -Math.tan(B);
	} else if (F <= 4.4e+57) {
		tmp = t_1;
	} else if (F <= 8.8e+101) {
		tmp = t_2;
	} else if (F <= 2.35e+212) {
		tmp = t_1;
	} else if (F <= 3.5e+280) {
		tmp = t_2;
	} else {
		tmp = (F / (F * B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	t_1 = (1.0 / B) - t_0
	t_2 = 1.0 / math.sin(B)
	tmp = 0
	if F <= -2.2e-7:
		tmp = (x * (-1.0 / math.tan(B))) + (-1.0 / B)
	elif F <= 2.45e-145:
		tmp = x / -math.tan(B)
	elif F <= 4.4e+57:
		tmp = t_1
	elif F <= 8.8e+101:
		tmp = t_2
	elif F <= 2.35e+212:
		tmp = t_1
	elif F <= 3.5e+280:
		tmp = t_2
	else:
		tmp = (F / (F * B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	t_1 = Float64(Float64(1.0 / B) - t_0)
	t_2 = Float64(1.0 / sin(B))
	tmp = 0.0
	if (F <= -2.2e-7)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(-1.0 / B));
	elseif (F <= 2.45e-145)
		tmp = Float64(x / Float64(-tan(B)));
	elseif (F <= 4.4e+57)
		tmp = t_1;
	elseif (F <= 8.8e+101)
		tmp = t_2;
	elseif (F <= 2.35e+212)
		tmp = t_1;
	elseif (F <= 3.5e+280)
		tmp = t_2;
	else
		tmp = Float64(Float64(F / Float64(F * B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	t_1 = (1.0 / B) - t_0;
	t_2 = 1.0 / sin(B);
	tmp = 0.0;
	if (F <= -2.2e-7)
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / B);
	elseif (F <= 2.45e-145)
		tmp = x / -tan(B);
	elseif (F <= 4.4e+57)
		tmp = t_1;
	elseif (F <= 8.8e+101)
		tmp = t_2;
	elseif (F <= 2.35e+212)
		tmp = t_1;
	elseif (F <= 3.5e+280)
		tmp = t_2;
	else
		tmp = (F / (F * B)) - 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[(1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.2e-7], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.45e-145], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 4.4e+57], t$95$1, If[LessEqual[F, 8.8e+101], t$95$2, If[LessEqual[F, 2.35e+212], t$95$1, If[LessEqual[F, 3.5e+280], t$95$2, N[(N[(F / N[(F * B), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -2.2000000000000001e-7

   1. Initial program 69.2%

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

   3. Taylor expanded in F around -inf 99.7%

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

   4. Taylor expanded in B around 0 83.7%

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

   if -2.2000000000000001e-7 < F < 2.44999999999999984e-145

   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 38.8%

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

   4. Taylor expanded in x around inf 74.6%

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

      1. mul-1-neg 74.6%

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

      2. associate-/l* 74.4%

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

      3. distribute-lft-neg-in 74.4%

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

   6. Simplified 74.4%

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

      1. add-sqr-sqrt 16.6%

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

      2. sqrt-unprod 14.6%

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

      3. sqr-neg 14.6%

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

      4. sqrt-unprod 1.3%

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

      5. add-sqr-sqrt 2.0%

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

      6. clear-num 2.0%

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

      7. tan-quot 2.0%

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

      8. div-inv 2.0%

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

      9. frac-2neg 2.0%

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

      10. neg-sub0 2.0%

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

      11. div-sub 2.0%

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

      12. add-sqr-sqrt 1.3%

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

      13. sqrt-unprod 14.6%

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

      14. sqr-neg 14.6%

          \[\leadsto \frac{0}{-\tan B} - \frac{\sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}}}{-\tan B} \]

      15. sqrt-unprod 16.7%

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

      16. add-sqr-sqrt 74.7%

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

      17. frac-2neg 74.7%

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

   8. Applied egg-rr 74.7%

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

      1. div0 74.7%

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

      2. neg-sub0 74.7%

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

      3. distribute-frac-neg2 74.7%

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

   10. Simplified 74.7%

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

   if 2.44999999999999984e-145 < F < 4.4000000000000001e57 or 8.8000000000000003e101 < F < 2.34999999999999996e212

   1. Initial program 86.0%

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

   3. Simplified 89.3%

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

   5. Taylor expanded in x around 0 89.4%

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

      1. associate-*l/ 89.4%

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

      2. *-lft-identity 89.4%

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

      3. +-commutative 89.4%

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

      4. unpow2 89.4%

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

      5. fma-undefine 89.4%

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

   7. Simplified 89.4%

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

      1. clear-num 89.3%

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

      2. un-div-inv 89.4%

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

      3. inv-pow 89.4%

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

      4. sqrt-pow1 89.4%

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

      5. metadata-eval 89.4%

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

   9. Applied egg-rr 89.4%

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

   10. Taylor expanded in B around 0 72.3%

       \[\leadsto \frac{F}{\color{blue}{B \cdot \sqrt{2 + {F}^{2}}}} - \frac{x}{\tan B} \]
   11. Step-by-step derivation

       1. +-commutative 72.3%

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

       2. unpow2 72.3%

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

       3. fma-undefine 72.3%

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

   12. Simplified 72.3%

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

   13. Taylor expanded in F around inf 70.6%

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

   if 4.4000000000000001e57 < F < 8.8000000000000003e101 or 2.34999999999999996e212 < F < 3.5000000000000001e280

   1. Initial program 51.1%

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

   3. Taylor expanded in F around -inf 25.9%

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

      1. add-cube-cbrt 25.6%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}} \cdot \sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}\right) \cdot \sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}} \]

      2. pow3 25.7%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}\right)}^{3}} \]

   5. Applied egg-rr 73.8%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{1}{\sin B} + \frac{x}{\tan B}}\right)}^{3}} \]

   6. Taylor expanded in x around 0 74.9%

      \[\leadsto {\color{blue}{\left(\sqrt[3]{\frac{1}{\sin B}}\right)}}^{3} \]

   7. Taylor expanded in B around inf 76.0%

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

   if 3.5000000000000001e280 < F

   1. Initial program 36.9%

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

   3. Simplified 56.4%

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

   5. Taylor expanded in x around 0 56.4%

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

      1. associate-*l/ 56.4%

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

      2. *-lft-identity 56.4%

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

      3. +-commutative 56.4%

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

      4. unpow2 56.4%

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

      5. fma-undefine 56.4%

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

   7. Simplified 56.4%

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

      1. clear-num 56.4%

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

      2. un-div-inv 56.4%

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

      3. inv-pow 56.4%

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

      4. sqrt-pow1 56.4%

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

      5. metadata-eval 56.4%

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

   9. Applied egg-rr 56.4%

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

   10. Taylor expanded in B around 0 56.4%

       \[\leadsto \frac{F}{\color{blue}{B \cdot \sqrt{2 + {F}^{2}}}} - \frac{x}{\tan B} \]
   11. Step-by-step derivation

       1. +-commutative 56.4%

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

       2. unpow2 56.4%

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

       3. fma-undefine 56.4%

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

   12. Simplified 56.4%

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

   13. Taylor expanded in F around inf 82.5%

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

       1. *-commutative 82.5%

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

   15. Simplified 82.5%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.2 \cdot 10^{-7}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq 2.45 \cdot 10^{-145}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 4.4 \cdot 10^{+57}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 8.8 \cdot 10^{+101}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{elif}\;F \leq 2.35 \cdot 10^{+212}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 3.5 \cdot 10^{+280}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{F \cdot B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 39.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1 - x}{B}\\ \mathbf{if}\;F \leq -2.2 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{-222}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 5.3 \cdot 10^{+129} \lor \neg \left(F \leq 6.2 \cdot 10^{+155}\right):\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- -1.0 x) B)))
   (if (<= F -2.2e-7)
     t_0
     (if (<= F 2.3e-222)
       (/ (- x) B)
       (if (or (<= F 5.3e+129) (not (<= F 6.2e+155))) (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 - x) / B;
	double tmp;
	if (F <= -2.2e-7) {
		tmp = t_0;
	} else if (F <= 2.3e-222) {
		tmp = -x / B;
	} else if ((F <= 5.3e+129) || !(F <= 6.2e+155)) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-1.0d0) - x) / b
    if (f <= (-2.2d-7)) then
        tmp = t_0
    else if (f <= 2.3d-222) then
        tmp = -x / b
    else if ((f <= 5.3d+129) .or. (.not. (f <= 6.2d+155))) then
        tmp = 1.0d0 / sin(b)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (-1.0 - x) / B;
	double tmp;
	if (F <= -2.2e-7) {
		tmp = t_0;
	} else if (F <= 2.3e-222) {
		tmp = -x / B;
	} else if ((F <= 5.3e+129) || !(F <= 6.2e+155)) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (-1.0 - x) / B
	tmp = 0
	if F <= -2.2e-7:
		tmp = t_0
	elif F <= 2.3e-222:
		tmp = -x / B
	elif (F <= 5.3e+129) or not (F <= 6.2e+155):
		tmp = 1.0 / math.sin(B)
	else:
		tmp = t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 - x) / B)
	tmp = 0.0
	if (F <= -2.2e-7)
		tmp = t_0;
	elseif (F <= 2.3e-222)
		tmp = Float64(Float64(-x) / B);
	elseif ((F <= 5.3e+129) || !(F <= 6.2e+155))
		tmp = Float64(1.0 / sin(B));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (-1.0 - x) / B;
	tmp = 0.0;
	if (F <= -2.2e-7)
		tmp = t_0;
	elseif (F <= 2.3e-222)
		tmp = -x / B;
	elseif ((F <= 5.3e+129) || ~((F <= 6.2e+155)))
		tmp = 1.0 / sin(B);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision]}, If[LessEqual[F, -2.2e-7], t$95$0, If[LessEqual[F, 2.3e-222], N[((-x) / B), $MachinePrecision], If[Or[LessEqual[F, 5.3e+129], N[Not[LessEqual[F, 6.2e+155]], $MachinePrecision]], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.2000000000000001e-7 or 5.2999999999999999e129 < F < 6.19999999999999978e155

   1. Initial program 72.3%

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

   3. Taylor expanded in F around -inf 99.2%

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

   4. Taylor expanded in B around 0 54.8%

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

      1. mul-1-neg 54.8%

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

      2. distribute-neg-frac2 54.8%

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

   6. Simplified 54.8%

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

   if -2.2000000000000001e-7 < F < 2.3000000000000001e-222

   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 41.1%

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

   4. Taylor expanded in B around 0 23.6%

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

      1. mul-1-neg 23.6%

         \[\leadsto \color{blue}{-\frac{1 + x}{B}} \]

      2. distribute-neg-frac2 23.6%

         \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   6. Simplified 23.6%

      \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   7. Taylor expanded in x around inf 43.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   8. Step-by-step derivation

      1. associate-*r/ 43.6%

         \[\leadsto \color{blue}{\frac{-1 \cdot x}{B}} \]

      2. mul-1-neg 43.6%

         \[\leadsto \frac{\color{blue}{-x}}{B} \]

   9. Simplified 43.6%

      \[\leadsto \color{blue}{\frac{-x}{B}} \]

   if 2.3000000000000001e-222 < F < 5.2999999999999999e129 or 6.19999999999999978e155 < F

   1. Initial program 70.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf 30.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 30.0%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}} \cdot \sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}\right) \cdot \sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}} \]

      2. pow3 30.0%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}\right)}^{3}} \]

   5. Applied egg-rr 43.0%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{1}{\sin B} + \frac{x}{\tan B}}\right)}^{3}} \]

   6. Taylor expanded in x around 0 44.1%

      \[\leadsto {\color{blue}{\left(\sqrt[3]{\frac{1}{\sin B}}\right)}}^{3} \]

   7. Taylor expanded in B around inf 44.8%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{-222}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 5.3 \cdot 10^{+129} \lor \neg \left(F \leq 6.2 \cdot 10^{+155}\right):\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 42.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1 - x}{B}\\ \mathbf{if}\;B \leq 9.5 \cdot 10^{-102}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;B \leq 9.6 \cdot 10^{-52}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;B \leq 3 \cdot 10^{-12}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- -1.0 x) B)))
   (if (<= B 9.5e-102)
     t_0
     (if (<= B 9.6e-52) (/ (- x) B) (if (<= B 3e-12) 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 <= 9.5e-102) {
		tmp = t_0;
	} else if (B <= 9.6e-52) {
		tmp = -x / B;
	} else if (B <= 3e-12) {
		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 <= 9.5d-102) then
        tmp = t_0
    else if (b <= 9.6d-52) then
        tmp = -x / b
    else if (b <= 3d-12) 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 <= 9.5e-102) {
		tmp = t_0;
	} else if (B <= 9.6e-52) {
		tmp = -x / B;
	} else if (B <= 3e-12) {
		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 <= 9.5e-102:
		tmp = t_0
	elif B <= 9.6e-52:
		tmp = -x / B
	elif B <= 3e-12:
		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 <= 9.5e-102)
		tmp = t_0;
	elseif (B <= 9.6e-52)
		tmp = Float64(Float64(-x) / B);
	elseif (B <= 3e-12)
		tmp = t_0;
	else
		tmp = Float64(x / Float64(-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 <= 9.5e-102)
		tmp = t_0;
	elseif (B <= 9.6e-52)
		tmp = -x / B;
	elseif (B <= 3e-12)
		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, 9.5e-102], t$95$0, If[LessEqual[B, 9.6e-52], N[((-x) / B), $MachinePrecision], If[LessEqual[B, 3e-12], t$95$0, N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if B < 9.50000000000000025e-102 or 9.6000000000000007e-52 < B < 3.0000000000000001e-12

   1. Initial program 76.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf 54.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   4. Taylor expanded in B around 0 35.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   5. Step-by-step derivation

      1. mul-1-neg 35.7%

         \[\leadsto \color{blue}{-\frac{1 + x}{B}} \]

      2. distribute-neg-frac2 35.7%

         \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   6. Simplified 35.7%

      \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   if 9.50000000000000025e-102 < B < 9.6000000000000007e-52

   1. Initial program 83.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf 48.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   4. Taylor expanded in B around 0 48.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   5. Step-by-step derivation

      1. mul-1-neg 48.5%

         \[\leadsto \color{blue}{-\frac{1 + x}{B}} \]

      2. distribute-neg-frac2 48.5%

         \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   6. Simplified 48.5%

      \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   7. Taylor expanded in x around inf 65.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   8. Step-by-step derivation

      1. associate-*r/ 65.3%

         \[\leadsto \color{blue}{\frac{-1 \cdot x}{B}} \]

      2. mul-1-neg 65.3%

         \[\leadsto \frac{\color{blue}{-x}}{B} \]

   9. Simplified 65.3%

      \[\leadsto \color{blue}{\frac{-x}{B}} \]

   if 3.0000000000000001e-12 < B

   1. Initial program 87.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf 55.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   4. Taylor expanded in x around inf 60.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   5. Step-by-step derivation

      1. mul-1-neg 60.6%

         \[\leadsto \color{blue}{-\frac{x \cdot \cos B}{\sin B}} \]

      2. associate-/l* 60.7%

         \[\leadsto -\color{blue}{x \cdot \frac{\cos B}{\sin B}} \]

      3. distribute-lft-neg-in 60.7%

         \[\leadsto \color{blue}{\left(-x\right) \cdot \frac{\cos B}{\sin B}} \]

   6. Simplified 60.7%

      \[\leadsto \color{blue}{\left(-x\right) \cdot \frac{\cos B}{\sin B}} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 28.3%

         \[\leadsto \color{blue}{\left(\sqrt{-x} \cdot \sqrt{-x}\right)} \cdot \frac{\cos B}{\sin B} \]

      2. sqrt-unprod 20.7%

         \[\leadsto \color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}} \cdot \frac{\cos B}{\sin B} \]

      3. sqr-neg 20.7%

         \[\leadsto \sqrt{\color{blue}{x \cdot x}} \cdot \frac{\cos B}{\sin B} \]

      4. sqrt-unprod 1.0%

         \[\leadsto \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \frac{\cos B}{\sin B} \]

      5. add-sqr-sqrt 2.2%

         \[\leadsto \color{blue}{x} \cdot \frac{\cos B}{\sin B} \]

      6. clear-num 2.2%

         \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{\sin B}{\cos B}}} \]

      7. tan-quot 2.2%

         \[\leadsto x \cdot \frac{1}{\color{blue}{\tan B}} \]

      8. div-inv 2.2%

         \[\leadsto \color{blue}{\frac{x}{\tan B}} \]

      9. frac-2neg 2.2%

         \[\leadsto \color{blue}{\frac{-x}{-\tan B}} \]

      10. neg-sub0 2.2%

          \[\leadsto \frac{\color{blue}{0 - x}}{-\tan B} \]

      11. div-sub 2.2%

          \[\leadsto \color{blue}{\frac{0}{-\tan B} - \frac{x}{-\tan B}} \]

      12. add-sqr-sqrt 1.0%

          \[\leadsto \frac{0}{-\tan B} - \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{-\tan B} \]

      13. sqrt-unprod 20.7%

          \[\leadsto \frac{0}{-\tan B} - \frac{\color{blue}{\sqrt{x \cdot x}}}{-\tan B} \]

      14. sqr-neg 20.7%

          \[\leadsto \frac{0}{-\tan B} - \frac{\sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}}}{-\tan B} \]

      15. sqrt-unprod 28.4%

          \[\leadsto \frac{0}{-\tan B} - \frac{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}{-\tan B} \]

      16. add-sqr-sqrt 60.8%

          \[\leadsto \frac{0}{-\tan B} - \frac{\color{blue}{-x}}{-\tan B} \]

      17. frac-2neg 60.8%

          \[\leadsto \frac{0}{-\tan B} - \color{blue}{\frac{x}{\tan B}} \]

   8. Applied egg-rr 60.8%

      \[\leadsto \color{blue}{\frac{0}{-\tan B} - \frac{x}{\tan B}} \]
   9. Step-by-step derivation

      1. div0 60.8%

         \[\leadsto \color{blue}{0} - \frac{x}{\tan B} \]

      2. neg-sub0 60.8%

         \[\leadsto \color{blue}{-\frac{x}{\tan B}} \]

      3. distribute-frac-neg2 60.8%

         \[\leadsto \color{blue}{\frac{x}{-\tan B}} \]

   10. Simplified 60.8%

       \[\leadsto \color{blue}{\frac{x}{-\tan B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 9.5 \cdot 10^{-102}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;B \leq 9.6 \cdot 10^{-52}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;B \leq 3 \cdot 10^{-12}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 34.2% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3 \cdot 10^{+56}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 9.5 \cdot 10^{-79} \lor \neg \left(F \leq 4.5 \cdot 10^{+121}\right) \land F \leq 4.6 \cdot 10^{+152}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3e+56)
   (/ (- -1.0 x) B)
   (if (or (<= F 9.5e-79) (and (not (<= F 4.5e+121)) (<= F 4.6e+152)))
     (/ (- x) B)
     (/ (+ x 1.0) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3e+56) {
		tmp = (-1.0 - x) / B;
	} else if ((F <= 9.5e-79) || (!(F <= 4.5e+121) && (F <= 4.6e+152))) {
		tmp = -x / B;
	} else {
		tmp = (x + 1.0) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-3d+56)) then
        tmp = ((-1.0d0) - x) / b
    else if ((f <= 9.5d-79) .or. (.not. (f <= 4.5d+121)) .and. (f <= 4.6d+152)) then
        tmp = -x / b
    else
        tmp = (x + 1.0d0) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -3e+56) {
		tmp = (-1.0 - x) / B;
	} else if ((F <= 9.5e-79) || (!(F <= 4.5e+121) && (F <= 4.6e+152))) {
		tmp = -x / B;
	} else {
		tmp = (x + 1.0) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3e+56:
		tmp = (-1.0 - x) / B
	elif (F <= 9.5e-79) or (not (F <= 4.5e+121) and (F <= 4.6e+152)):
		tmp = -x / B
	else:
		tmp = (x + 1.0) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3e+56)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif ((F <= 9.5e-79) || (!(F <= 4.5e+121) && (F <= 4.6e+152)))
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(x + 1.0) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -3e+56)
		tmp = (-1.0 - x) / B;
	elseif ((F <= 9.5e-79) || (~((F <= 4.5e+121)) && (F <= 4.6e+152)))
		tmp = -x / B;
	else
		tmp = (x + 1.0) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3e+56], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[Or[LessEqual[F, 9.5e-79], And[N[Not[LessEqual[F, 4.5e+121]], $MachinePrecision], LessEqual[F, 4.6e+152]]], N[((-x) / B), $MachinePrecision], N[(N[(x + 1.0), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.00000000000000006e56

   1. Initial program 64.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf 99.7%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   4. Taylor expanded in B around 0 50.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   5. Step-by-step derivation

      1. mul-1-neg 50.2%

         \[\leadsto \color{blue}{-\frac{1 + x}{B}} \]

      2. distribute-neg-frac2 50.2%

         \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   6. Simplified 50.2%

      \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   if -3.00000000000000006e56 < F < 9.4999999999999997e-79 or 4.5000000000000003e121 < F < 4.5999999999999997e152

   1. Initial program 98.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf 44.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   4. Taylor expanded in B around 0 27.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   5. Step-by-step derivation

      1. mul-1-neg 27.8%

         \[\leadsto \color{blue}{-\frac{1 + x}{B}} \]

      2. distribute-neg-frac2 27.8%

         \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   6. Simplified 27.8%

      \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   7. Taylor expanded in x around inf 39.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   8. Step-by-step derivation

      1. associate-*r/ 39.7%

         \[\leadsto \color{blue}{\frac{-1 \cdot x}{B}} \]

      2. mul-1-neg 39.7%

         \[\leadsto \frac{\color{blue}{-x}}{B} \]

   9. Simplified 39.7%

      \[\leadsto \color{blue}{\frac{-x}{B}} \]

   if 9.4999999999999997e-79 < F < 4.5000000000000003e121 or 4.5999999999999997e152 < F

   1. Initial program 61.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf 33.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 33.4%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}} \cdot \sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}\right) \cdot \sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}} \]

      2. pow3 33.4%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}\right)}^{3}} \]

   5. Applied egg-rr 54.7%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{1}{\sin B} + \frac{x}{\tan B}}\right)}^{3}} \]

   6. Taylor expanded in B around 0 28.0%

      \[\leadsto \color{blue}{\frac{1 + x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 38.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3 \cdot 10^{+56}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 9.5 \cdot 10^{-79} \lor \neg \left(F \leq 4.5 \cdot 10^{+121}\right) \land F \leq 4.6 \cdot 10^{+152}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 28.5% accurate, 21.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.3 \cdot 10^{+235}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq 1.05 \cdot 10^{-78}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.3e+235)
   (/ -1.0 B)
   (if (<= F 1.05e-78) (/ (- x) B) (/ (+ x 1.0) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.3e+235) {
		tmp = -1.0 / B;
	} else if (F <= 1.05e-78) {
		tmp = -x / B;
	} else {
		tmp = (x + 1.0) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.3d+235)) then
        tmp = (-1.0d0) / b
    else if (f <= 1.05d-78) then
        tmp = -x / b
    else
        tmp = (x + 1.0d0) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.3e+235) {
		tmp = -1.0 / B;
	} else if (F <= 1.05e-78) {
		tmp = -x / B;
	} else {
		tmp = (x + 1.0) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.3e+235:
		tmp = -1.0 / B
	elif F <= 1.05e-78:
		tmp = -x / B
	else:
		tmp = (x + 1.0) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.3e+235)
		tmp = Float64(-1.0 / B);
	elseif (F <= 1.05e-78)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(x + 1.0) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.3e+235)
		tmp = -1.0 / B;
	elseif (F <= 1.05e-78)
		tmp = -x / B;
	else
		tmp = (x + 1.0) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.3e+235], N[(-1.0 / B), $MachinePrecision], If[LessEqual[F, 1.05e-78], N[((-x) / B), $MachinePrecision], N[(N[(x + 1.0), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.2999999999999999e235

   1. Initial program 32.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf 99.8%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   4. Taylor expanded in B around 0 54.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   5. Step-by-step derivation

      1. mul-1-neg 54.8%

         \[\leadsto \color{blue}{-\frac{1 + x}{B}} \]

      2. distribute-neg-frac2 54.8%

         \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   6. Simplified 54.8%

      \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   7. Taylor expanded in x around 0 39.8%

      \[\leadsto \color{blue}{\frac{-1}{B}} \]

   if -1.2999999999999999e235 < F < 1.05e-78

   1. Initial program 93.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf 58.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   4. Taylor expanded in B around 0 32.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   5. Step-by-step derivation

      1. mul-1-neg 32.3%

         \[\leadsto \color{blue}{-\frac{1 + x}{B}} \]

      2. distribute-neg-frac2 32.3%

         \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   6. Simplified 32.3%

      \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   7. Taylor expanded in x around inf 33.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   8. Step-by-step derivation

      1. associate-*r/ 33.9%

         \[\leadsto \color{blue}{\frac{-1 \cdot x}{B}} \]

      2. mul-1-neg 33.9%

         \[\leadsto \frac{\color{blue}{-x}}{B} \]

   9. Simplified 33.9%

      \[\leadsto \color{blue}{\frac{-x}{B}} \]

   if 1.05e-78 < F

   1. Initial program 64.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf 37.7%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]
   4. Step-by-step derivation

      1. add-cube-cbrt 37.2%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}} \cdot \sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}\right) \cdot \sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}} \]

      2. pow3 37.3%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B}}\right)}^{3}} \]

   5. Applied egg-rr 51.9%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{1}{\sin B} + \frac{x}{\tan B}}\right)}^{3}} \]

   6. Taylor expanded in B around 0 27.8%

      \[\leadsto \color{blue}{\frac{1 + x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 32.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.3 \cdot 10^{+235}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq 1.05 \cdot 10^{-78}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 28.8% accurate, 35.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.16 \cdot 10^{+235}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.16e+235) (/ -1.0 B) (/ (- x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.16e+235) {
		tmp = -1.0 / B;
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.16d+235)) then
        tmp = (-1.0d0) / b
    else
        tmp = -x / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.16e+235) {
		tmp = -1.0 / B;
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.16e+235:
		tmp = -1.0 / B
	else:
		tmp = -x / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.16e+235)
		tmp = Float64(-1.0 / B);
	else
		tmp = Float64(Float64(-x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.16e+235)
		tmp = -1.0 / B;
	else
		tmp = -x / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.16e+235], N[(-1.0 / B), $MachinePrecision], N[((-x) / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -1.15999999999999994e235

   1. Initial program 32.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf 99.8%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   4. Taylor expanded in B around 0 54.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   5. Step-by-step derivation

      1. mul-1-neg 54.8%

         \[\leadsto \color{blue}{-\frac{1 + x}{B}} \]

      2. distribute-neg-frac2 54.8%

         \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   6. Simplified 54.8%

      \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   7. Taylor expanded in x around 0 39.8%

      \[\leadsto \color{blue}{\frac{-1}{B}} \]

   if -1.15999999999999994e235 < F

   1. Initial program 83.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf 51.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   4. Taylor expanded in B around 0 26.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   5. Step-by-step derivation

      1. mul-1-neg 26.6%

         \[\leadsto \color{blue}{-\frac{1 + x}{B}} \]

      2. distribute-neg-frac2 26.6%

         \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   6. Simplified 26.6%

      \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   7. Taylor expanded in x around inf 27.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   8. Step-by-step derivation

      1. associate-*r/ 27.9%

         \[\leadsto \color{blue}{\frac{-1 \cdot x}{B}} \]

      2. mul-1-neg 27.9%

         \[\leadsto \frac{\color{blue}{-x}}{B} \]

   9. Simplified 27.9%

      \[\leadsto \color{blue}{\frac{-x}{B}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 10.5% accurate, 108.0× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (/ -1.0 B))

C

double code(double F, double B, double x) {
	return -1.0 / B;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = (-1.0d0) / b
end function

Java

public static double code(double F, double B, double x) {
	return -1.0 / B;
}

Python

def code(F, B, x):
	return -1.0 / B

Julia

function code(F, B, x)
	return Float64(-1.0 / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = -1.0 / B;
end

Wolfram

code[F_, B_, x_] := N[(-1.0 / B), $MachinePrecision]

Derivation

1. Initial program 79.3%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
2. Add Preprocessing

3. Taylor expanded in F around -inf 54.6%

   \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

4. Taylor expanded in B around 0 28.7%

   \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
5. Step-by-step derivation

   1. mul-1-neg 28.7%

      \[\leadsto \color{blue}{-\frac{1 + x}{B}} \]

   2. distribute-neg-frac2 28.7%

      \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

6. Simplified 28.7%

   \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

7. Taylor expanded in x around 0 11.2%

   \[\leadsto \color{blue}{\frac{-1}{B}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024103 
(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))))))