VandenBroeck and Keller, Equation (23)

Percentage Accurate: 77.0% → 99.7%

Time: 19.4s

Alternatives: 25

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.9 \cdot 10^{+53}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 50000000:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.9e+53)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 50000000.0)
       (- (* F (/ (sqrt (/ 1.0 (fma F F 2.0))) (sin B))) t_0)
       (+ (* x (/ -1.0 (tan B))) (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.9e+53) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 50000000.0) {
		tmp = (F * (sqrt((1.0 / fma(F, F, 2.0))) / sin(B))) - t_0;
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.9e+53)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 50000000.0)
		tmp = Float64(Float64(F * Float64(sqrt(Float64(1.0 / fma(F, F, 2.0))) / sin(B))) - t_0);
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.9e+53], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 50000000.0], N[(N[(F * N[(N[Sqrt[N[(1.0 / N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.89999999999999999e53

   1. Initial program 50.2%

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

   3. Simplified 64.0%

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

   5. Taylor expanded in x around 0 64.1%

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

      1. associate-*l/ 64.0%

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

      2. *-lft-identity 64.0%

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

      3. +-commutative 64.0%

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

      4. unpow2 64.0%

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

      5. fma-undefine 64.0%

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

   7. Simplified 64.0%

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

   8. Taylor expanded in F around -inf 99.8%

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

   if -1.89999999999999999e53 < F < 5e7

   1. Initial program 98.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 99.7%

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

      1. associate-*l/ 99.7%

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

      2. *-lft-identity 99.7%

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

      3. +-commutative 99.7%

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

      4. unpow2 99.7%

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

      5. fma-undefine 99.7%

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

   7. Simplified 99.7%

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

   if 5e7 < F

   1. Initial program 56.2%

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

   3. Taylor expanded in F around inf 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.9 \cdot 10^{+53}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 50000000:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -14000000000:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 50000000:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(F, F, 2\right)}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -14000000000.0)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 50000000.0)
       (- (/ F (* (sin B) (sqrt (fma F F 2.0)))) t_0)
       (+ (* x (/ -1.0 (tan B))) (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -14000000000.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 50000000.0) {
		tmp = (F / (sin(B) * sqrt(fma(F, F, 2.0)))) - t_0;
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -14000000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 50000000.0)
		tmp = Float64(Float64(F / Float64(sin(B) * sqrt(fma(F, F, 2.0)))) - t_0);
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -14000000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 50000000.0], 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[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.4e10

   1. Initial program 57.3%

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

   3. Simplified 70.2%

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

   5. Taylor expanded in x around 0 70.3%

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

      1. associate-*l/ 70.2%

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

      2. *-lft-identity 70.2%

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

      3. +-commutative 70.2%

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

      4. unpow2 70.2%

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

      5. fma-undefine 70.2%

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

   7. Simplified 70.2%

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

   8. Taylor expanded in F around -inf 99.8%

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

   if -1.4e10 < F < 5e7

   1. Initial program 99.4%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 99.7%

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

      1. associate-*l/ 99.7%

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

      2. *-lft-identity 99.7%

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

      3. +-commutative 99.7%

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

      4. unpow2 99.7%

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

      5. fma-undefine 99.7%

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

   7. Simplified 99.7%

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

      1. associate-*r/ 99.6%

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

      2. sqrt-div 99.6%

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

      3. metadata-eval 99.6%

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

      4. un-div-inv 99.6%

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

   9. Applied egg-rr 99.6%

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

       1. associate-/l/ 99.6%

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

   11. Simplified 99.6%

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

   if 5e7 < F

   1. Initial program 56.2%

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

   3. Taylor expanded in F around inf 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -14000000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 50000000:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(F, F, 2\right)}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2 \cdot 10^{+156}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 9200000000:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -2e+156)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 9200000000.0)
       (- (/ (/ F (sqrt (fma F F 2.0))) (sin B)) t_0)
       (+ (* x (/ -1.0 (tan B))) (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -2e+156) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 9200000000.0) {
		tmp = ((F / sqrt(fma(F, F, 2.0))) / sin(B)) - t_0;
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -2e+156)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 9200000000.0)
		tmp = Float64(Float64(Float64(F / sqrt(fma(F, F, 2.0))) / sin(B)) - t_0);
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2e+156], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 9200000000.0], N[(N[(N[(F / N[Sqrt[N[(F * F + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2e156

   1. Initial program 17.9%

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

   3. Simplified 32.2%

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

   5. Taylor expanded in x around 0 32.2%

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

      1. associate-*l/ 32.2%

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

      2. *-lft-identity 32.2%

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

      3. +-commutative 32.2%

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

      4. unpow2 32.2%

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

      5. fma-undefine 32.2%

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

   7. Simplified 32.2%

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

   8. Taylor expanded in F around -inf 99.8%

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

   if -2e156 < F < 9.2e9

   1. Initial program 96.4%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 99.6%

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

      1. associate-*l/ 99.7%

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

      2. *-lft-identity 99.7%

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

      3. +-commutative 99.7%

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

      4. unpow2 99.7%

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

      5. fma-undefine 99.7%

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

   7. Simplified 99.7%

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

      1. associate-*r/ 99.6%

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

      2. sqrt-div 99.6%

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

      3. metadata-eval 99.6%

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

      4. un-div-inv 99.6%

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

   9. Applied egg-rr 99.6%

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

   if 9.2e9 < F

   1. Initial program 56.2%

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

   3. Taylor expanded in F around inf 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{+156}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 9200000000:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -2.5 \cdot 10^{+37}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 150000000:\\ \;\;\;\;t\_0 + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B)))))
   (if (<= F -2.5e+37)
     (- (/ -1.0 (sin B)) (/ x (tan B)))
     (if (<= F 150000000.0)
       (+ t_0 (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)))
       (+ t_0 (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double tmp;
	if (F <= -2.5e+37) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 150000000.0) {
		tmp = t_0 + ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5));
	} else {
		tmp = t_0 + (1.0 / sin(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * ((-1.0d0) / tan(b))
    if (f <= (-2.5d+37)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 150000000.0d0) then
        tmp = t_0 + ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)))
    else
        tmp = t_0 + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / Math.tan(B));
	double tmp;
	if (F <= -2.5e+37) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 150000000.0) {
		tmp = t_0 + ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5));
	} else {
		tmp = t_0 + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * (-1.0 / math.tan(B))
	tmp = 0
	if F <= -2.5e+37:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 150000000.0:
		tmp = t_0 + ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5))
	else:
		tmp = t_0 + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	tmp = 0.0
	if (F <= -2.5e+37)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 150000000.0)
		tmp = Float64(t_0 + Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)));
	else
		tmp = Float64(t_0 + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x * (-1.0 / tan(B));
	tmp = 0.0;
	if (F <= -2.5e+37)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 150000000.0)
		tmp = t_0 + ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5));
	else
		tmp = t_0 + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.5e+37], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 150000000.0], N[(t$95$0 + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.49999999999999994e37

   1. Initial program 54.0%

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

   3. Simplified 68.0%

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

   5. Taylor expanded in x around 0 68.0%

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

      1. associate-*l/ 68.0%

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

      2. *-lft-identity 68.0%

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

      3. +-commutative 68.0%

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

      4. unpow2 68.0%

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

      5. fma-undefine 68.0%

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

   7. Simplified 68.0%

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

   8. Taylor expanded in F around -inf 99.8%

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

   if -2.49999999999999994e37 < F < 1.5e8

   1. Initial program 99.4%

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

   if 1.5e8 < F

   1. Initial program 56.2%

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

   3. Taylor expanded in F around inf 99.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.5 \cdot 10^{+37}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 150000000:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.8 \cdot 10^{+44}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 920000000:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.8e+44)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 920000000.0)
     (+
      (/ -1.0 (/ (tan B) x))
      (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)))
     (+ (* x (/ -1.0 (tan B))) (/ 1.0 (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.8e+44) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 920000000.0) {
		tmp = (-1.0 / (tan(B) / x)) + ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5));
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-2.8d+44)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 920000000.0d0) then
        tmp = ((-1.0d0) / (tan(b) / x)) + ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)))
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.8e+44) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 920000000.0) {
		tmp = (-1.0 / (Math.tan(B) / x)) + ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5));
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.8e+44:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 920000000.0:
		tmp = (-1.0 / (math.tan(B) / x)) + ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5))
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.8e+44)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 920000000.0)
		tmp = Float64(Float64(-1.0 / Float64(tan(B) / x)) + Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.8e+44)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 920000000.0)
		tmp = (-1.0 / (tan(B) / x)) + ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5));
	else
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.8e+44], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 920000000.0], N[(N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.8000000000000001e44

   1. Initial program 51.8%

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

   3. Simplified 66.4%

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

   5. Taylor expanded in x around 0 66.4%

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

      1. associate-*l/ 66.4%

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

      2. *-lft-identity 66.4%

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

      3. +-commutative 66.4%

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

      4. unpow2 66.4%

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

      5. fma-undefine 66.4%

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

   7. Simplified 66.4%

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

   8. Taylor expanded in F around -inf 99.8%

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

   if -2.8000000000000001e44 < F < 9.2e8

   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. Step-by-step derivation

      1. div-inv 46.2%

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

      2. clear-num 46.2%

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

   4. Applied egg-rr 99.4%

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

   if 9.2e8 < F

   1. Initial program 56.2%

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

   3. Taylor expanded in F around inf 99.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.8 \cdot 10^{+44}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 920000000:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 2.35 \cdot 10^{-8}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{\frac{1 - \frac{1}{F} \cdot \frac{1}{F}}{F}}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.4)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 2.35e-8)
       (- (* F (/ (sqrt 0.5) (sin B))) t_0)
       (- (* F (/ (/ (- 1.0 (* (/ 1.0 F) (/ 1.0 F))) F) (sin B))) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.4) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 2.35e-8) {
		tmp = (F * (sqrt(0.5) / sin(B))) - t_0;
	} else {
		tmp = (F * (((1.0 - ((1.0 / F) * (1.0 / F))) / F) / sin(B))) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.4d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 2.35d-8) then
        tmp = (f * (sqrt(0.5d0) / sin(b))) - t_0
    else
        tmp = (f * (((1.0d0 - ((1.0d0 / f) * (1.0d0 / f))) / f) / sin(b))) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.4) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 2.35e-8) {
		tmp = (F * (Math.sqrt(0.5) / Math.sin(B))) - t_0;
	} else {
		tmp = (F * (((1.0 - ((1.0 / F) * (1.0 / F))) / F) / Math.sin(B))) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.4:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 2.35e-8:
		tmp = (F * (math.sqrt(0.5) / math.sin(B))) - t_0
	else:
		tmp = (F * (((1.0 - ((1.0 / F) * (1.0 / F))) / F) / math.sin(B))) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.4)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 2.35e-8)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / sin(B))) - t_0);
	else
		tmp = Float64(Float64(F * Float64(Float64(Float64(1.0 - Float64(Float64(1.0 / F) * Float64(1.0 / F))) / F) / sin(B))) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.4)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 2.35e-8)
		tmp = (F * (sqrt(0.5) / sin(B))) - t_0;
	else
		tmp = (F * (((1.0 - ((1.0 / F) * (1.0 / F))) / F) / sin(B))) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.4], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 2.35e-8], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(F * N[(N[(N[(1.0 - N[(N[(1.0 / F), $MachinePrecision] * N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.3999999999999999

   1. Initial program 59.0%

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

   3. Simplified 71.4%

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

   5. Taylor expanded in x around 0 71.5%

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

      1. associate-*l/ 71.4%

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

      2. *-lft-identity 71.4%

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

      3. +-commutative 71.4%

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

      4. unpow2 71.4%

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

      5. fma-undefine 71.4%

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

   7. Simplified 71.4%

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

   8. Taylor expanded in F around -inf 99.2%

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

   if -1.3999999999999999 < F < 2.3499999999999999e-8

   1. Initial program 99.4%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 99.7%

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

      1. associate-*l/ 99.7%

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

      2. *-lft-identity 99.7%

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

      3. +-commutative 99.7%

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

      4. unpow2 99.7%

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

      5. fma-undefine 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in F around 0 99.7%

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

   if 2.3499999999999999e-8 < F

   1. Initial program 58.0%

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

   3. Simplified 72.3%

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

   5. Taylor expanded in x around 0 72.2%

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

      1. associate-*l/ 72.2%

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

      2. *-lft-identity 72.2%

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

      3. +-commutative 72.2%

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

      4. unpow2 72.2%

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

      5. fma-undefine 72.2%

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

   7. Simplified 72.2%

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

   8. Taylor expanded in F around inf 98.3%

      \[\leadsto F \cdot \frac{\color{blue}{\frac{1 - \frac{1}{{F}^{2}}}{F}}}{\sin B} - \frac{x}{\tan B} \]
   9. Step-by-step derivation

      1. inv-pow 98.3%

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

      2. unpow2 98.3%

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

      3. unpow-prod-down 98.3%

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

      4. inv-pow 98.3%

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

      5. inv-pow 98.3%

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

   10. Applied egg-rr 98.3%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.35 \cdot 10^{-8}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{\frac{1 - \frac{1}{F} \cdot \frac{1}{F}}{F}}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{if}\;F \leq -0.48:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.25 \cdot 10^{-126}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 9 \cdot 10^{-169}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{elif}\;F \leq 210000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0
         (-
          (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5))
          (/ x B))))
   (if (<= F -0.48)
     (- (/ -1.0 (sin B)) (/ x (tan B)))
     (if (<= F -1.25e-126)
       t_0
       (if (<= F 9e-169)
         (/ (* x (cos B)) (- (sin B)))
         (if (<= F 210000.0)
           t_0
           (+ (* x (/ -1.0 (tan B))) (/ 1.0 (sin B)))))))))

C

double code(double F, double B, double x) {
	double t_0 = ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B);
	double tmp;
	if (F <= -0.48) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= -1.25e-126) {
		tmp = t_0;
	} else if (F <= 9e-169) {
		tmp = (x * cos(B)) / -sin(B);
	} else if (F <= 210000.0) {
		tmp = t_0;
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0))) - (x / b)
    if (f <= (-0.48d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= (-1.25d-126)) then
        tmp = t_0
    else if (f <= 9d-169) then
        tmp = (x * cos(b)) / -sin(b)
    else if (f <= 210000.0d0) then
        tmp = t_0
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B);
	double tmp;
	if (F <= -0.48) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= -1.25e-126) {
		tmp = t_0;
	} else if (F <= 9e-169) {
		tmp = (x * Math.cos(B)) / -Math.sin(B);
	} else if (F <= 210000.0) {
		tmp = t_0;
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B)
	tmp = 0
	if F <= -0.48:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= -1.25e-126:
		tmp = t_0
	elif F <= 9e-169:
		tmp = (x * math.cos(B)) / -math.sin(B)
	elif F <= 210000.0:
		tmp = t_0
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)) - Float64(x / B))
	tmp = 0.0
	if (F <= -0.48)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= -1.25e-126)
		tmp = t_0;
	elseif (F <= 9e-169)
		tmp = Float64(Float64(x * cos(B)) / Float64(-sin(B)));
	elseif (F <= 210000.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5)) - (x / B);
	tmp = 0.0;
	if (F <= -0.48)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= -1.25e-126)
		tmp = t_0;
	elseif (F <= 9e-169)
		tmp = (x * cos(B)) / -sin(B);
	elseif (F <= 210000.0)
		tmp = t_0;
	else
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.48], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.25e-126], t$95$0, If[LessEqual[F, 9e-169], N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 210000.0], t$95$0, N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.47999999999999998

   1. Initial program 59.6%

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

   3. Simplified 71.8%

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

   5. Taylor expanded in x around 0 71.8%

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

      1. associate-*l/ 71.8%

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

      2. *-lft-identity 71.8%

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

      3. +-commutative 71.8%

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

      4. unpow2 71.8%

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

      5. fma-undefine 71.8%

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

   7. Simplified 71.8%

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

   8. Taylor expanded in F around -inf 99.2%

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

   if -0.47999999999999998 < F < -1.25000000000000001e-126 or 8.9999999999999997e-169 < F < 2.1e5

   1. Initial program 99.4%

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

   3. Taylor expanded in B around 0 78.3%

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

   if -1.25000000000000001e-126 < F < 8.9999999999999997e-169

   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 52.4%

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

   4. Taylor expanded in x around inf 95.5%

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

   if 2.1e5 < F

   1. Initial program 56.2%

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

   3. Taylor expanded in F around inf 99.7%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.48:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.25 \cdot 10^{-126}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 9 \cdot 10^{-169}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{elif}\;F \leq 210000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -4.8 \cdot 10^{-41}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.48 \cdot 10^{-27}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4.8e-41)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 1.48e-27)
     (/ (* x (cos B)) (- (sin B)))
     (+ (* x (/ -1.0 (tan B))) (/ 1.0 (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.8e-41) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 1.48e-27) {
		tmp = (x * cos(B)) / -sin(B);
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-4.8d-41)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 1.48d-27) then
        tmp = (x * cos(b)) / -sin(b)
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.8e-41) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 1.48e-27) {
		tmp = (x * Math.cos(B)) / -Math.sin(B);
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -4.8e-41:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 1.48e-27:
		tmp = (x * math.cos(B)) / -math.sin(B)
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -4.8e-41)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 1.48e-27)
		tmp = Float64(Float64(x * cos(B)) / Float64(-sin(B)));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -4.8e-41)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 1.48e-27)
		tmp = (x * cos(B)) / -sin(B);
	else
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -4.8e-41], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.48e-27], N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.80000000000000044e-41

   1. Initial program 62.1%

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

   3. Simplified 73.6%

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

   5. Taylor expanded in x around 0 73.6%

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

      1. associate-*l/ 73.6%

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

      2. *-lft-identity 73.6%

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

      3. +-commutative 73.6%

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

      4. unpow2 73.6%

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

      5. fma-undefine 73.6%

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

   7. Simplified 73.6%

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

   8. Taylor expanded in F around -inf 94.6%

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

   if -4.80000000000000044e-41 < F < 1.48000000000000008e-27

   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 41.9%

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

   4. Taylor expanded in x around inf 71.1%

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

   if 1.48000000000000008e-27 < F

   1. Initial program 59.6%

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

   3. Taylor expanded in F around inf 95.9%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.8 \cdot 10^{-41}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.48 \cdot 10^{-27}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -8.8 \cdot 10^{-69}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{1}{\frac{\tan B}{x}}\\ \mathbf{elif}\;F \leq 1.25 \cdot 10^{-27}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -8.8e-69)
   (- (/ -1.0 (sin B)) (/ 1.0 (/ (tan B) x)))
   (if (<= F 1.25e-27)
     (/ (* x (cos B)) (- (sin B)))
     (+ (* x (/ -1.0 (tan B))) (/ 1.0 (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -8.8e-69) {
		tmp = (-1.0 / sin(B)) - (1.0 / (tan(B) / x));
	} else if (F <= 1.25e-27) {
		tmp = (x * cos(B)) / -sin(B);
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-8.8d-69)) then
        tmp = ((-1.0d0) / sin(b)) - (1.0d0 / (tan(b) / x))
    else if (f <= 1.25d-27) then
        tmp = (x * cos(b)) / -sin(b)
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -8.8e-69) {
		tmp = (-1.0 / Math.sin(B)) - (1.0 / (Math.tan(B) / x));
	} else if (F <= 1.25e-27) {
		tmp = (x * Math.cos(B)) / -Math.sin(B);
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -8.8e-69:
		tmp = (-1.0 / math.sin(B)) - (1.0 / (math.tan(B) / x))
	elif F <= 1.25e-27:
		tmp = (x * math.cos(B)) / -math.sin(B)
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -8.8e-69)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(1.0 / Float64(tan(B) / x)));
	elseif (F <= 1.25e-27)
		tmp = Float64(Float64(x * cos(B)) / Float64(-sin(B)));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -8.8e-69)
		tmp = (-1.0 / sin(B)) - (1.0 / (tan(B) / x));
	elseif (F <= 1.25e-27)
		tmp = (x * cos(B)) / -sin(B);
	else
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -8.8e-69], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.25e-27], N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -8.8000000000000001e-69

   1. Initial program 63.9%

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

   3. Taylor expanded in F around -inf 91.3%

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

      1. div-inv 92.5%

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

      2. clear-num 92.5%

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

   5. Applied egg-rr 92.5%

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

   if -8.8000000000000001e-69 < F < 1.25e-27

   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 41.4%

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

   4. Taylor expanded in x around inf 71.8%

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

   if 1.25e-27 < F

   1. Initial program 59.6%

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

   3. Taylor expanded in F around inf 95.9%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -8.8 \cdot 10^{-69}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{1}{\frac{\tan B}{x}}\\ \mathbf{elif}\;F \leq 1.25 \cdot 10^{-27}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 85.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -4.5 \cdot 10^{-41}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1.45 \cdot 10^{-27}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -4.5e-41)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.45e-27)
       (/ (* x (cos B)) (- (sin B)))
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -4.5e-41) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.45e-27) {
		tmp = (x * cos(B)) / -sin(B);
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-4.5d-41)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.45d-27) then
        tmp = (x * cos(b)) / -sin(b)
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -4.5e-41) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.45e-27) {
		tmp = (x * Math.cos(B)) / -Math.sin(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -4.5e-41:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.45e-27:
		tmp = (x * math.cos(B)) / -math.sin(B)
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -4.5e-41)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.45e-27)
		tmp = Float64(Float64(x * cos(B)) / Float64(-sin(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -4.5e-41)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.45e-27)
		tmp = (x * cos(B)) / -sin(B);
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -4.5e-41], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.45e-27], N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.5e-41

   1. Initial program 62.1%

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

   3. Simplified 73.6%

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

   5. Taylor expanded in x around 0 73.6%

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

      1. associate-*l/ 73.6%

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

      2. *-lft-identity 73.6%

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

      3. +-commutative 73.6%

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

      4. unpow2 73.6%

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

      5. fma-undefine 73.6%

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

   7. Simplified 73.6%

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

   8. Taylor expanded in F around -inf 94.6%

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

   if -4.5e-41 < F < 1.45000000000000002e-27

   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 41.9%

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

   4. Taylor expanded in x around inf 71.1%

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

   if 1.45000000000000002e-27 < F

   1. Initial program 59.6%

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

   3. Simplified 73.3%

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

   5. Taylor expanded in x around 0 73.3%

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

      1. associate-*l/ 73.3%

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

      2. *-lft-identity 73.3%

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

      3. +-commutative 73.3%

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

      4. unpow2 73.3%

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

      5. fma-undefine 73.3%

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

   7. Simplified 73.3%

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

   8. Taylor expanded in F around inf 95.9%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.5 \cdot 10^{-41}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.45 \cdot 10^{-27}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 71.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -9600000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.00017:\\ \;\;\;\;x \cdot \frac{\cos B}{-\sin B}\\ \mathbf{elif}\;F \leq 8 \cdot 10^{+245} \lor \neg \left(F \leq 1.28 \cdot 10^{+278}\right):\\ \;\;\;\;\frac{F}{F \cdot \sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -9600000.0)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 0.00017)
     (* x (/ (cos B) (- (sin B))))
     (if (or (<= F 8e+245) (not (<= F 1.28e+278)))
       (- (/ F (* F (sin B))) (/ x B))
       (- (/ 1.0 B) (/ x (tan B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -9600000.0) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 0.00017) {
		tmp = x * (cos(B) / -sin(B));
	} else if ((F <= 8e+245) || !(F <= 1.28e+278)) {
		tmp = (F / (F * sin(B))) - (x / B);
	} else {
		tmp = (1.0 / B) - (x / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-9600000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 0.00017d0) then
        tmp = x * (cos(b) / -sin(b))
    else if ((f <= 8d+245) .or. (.not. (f <= 1.28d+278))) then
        tmp = (f / (f * sin(b))) - (x / b)
    else
        tmp = (1.0d0 / b) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -9600000.0) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 0.00017) {
		tmp = x * (Math.cos(B) / -Math.sin(B));
	} else if ((F <= 8e+245) || !(F <= 1.28e+278)) {
		tmp = (F / (F * Math.sin(B))) - (x / B);
	} else {
		tmp = (1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -9600000.0:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 0.00017:
		tmp = x * (math.cos(B) / -math.sin(B))
	elif (F <= 8e+245) or not (F <= 1.28e+278):
		tmp = (F / (F * math.sin(B))) - (x / B)
	else:
		tmp = (1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -9600000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 0.00017)
		tmp = Float64(x * Float64(cos(B) / Float64(-sin(B))));
	elseif ((F <= 8e+245) || !(F <= 1.28e+278))
		tmp = Float64(Float64(F / Float64(F * sin(B))) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -9600000.0)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 0.00017)
		tmp = x * (cos(B) / -sin(B));
	elseif ((F <= 8e+245) || ~((F <= 1.28e+278)))
		tmp = (F / (F * sin(B))) - (x / B);
	else
		tmp = (1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -9600000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.00017], N[(x * N[(N[Cos[B], $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[F, 8e+245], N[Not[LessEqual[F, 1.28e+278]], $MachinePrecision]], N[(N[(F / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -9.6e6

   1. Initial program 57.9%

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

   3. Taylor expanded in F around -inf 98.3%

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

   4. Taylor expanded in B around 0 86.1%

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

   if -9.6e6 < F < 1.7e-4

   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 42.7%

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

   4. Taylor expanded in x around inf 68.1%

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

      1. mul-1-neg 68.1%

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

      2. associate-/l* 67.9%

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

      3. distribute-rgt-neg-in 67.9%

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

      4. distribute-frac-neg 67.9%

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

   6. Simplified 67.9%

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

   if 1.7e-4 < F < 8.00000000000000035e245 or 1.2800000000000001e278 < F

   1. Initial program 56.2%

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

   3. Taylor expanded in F around inf 78.6%

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

      1. un-div-inv 78.7%

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

      2. associate-/l/ 97.8%

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

   5. Applied egg-rr 97.8%

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

   6. Taylor expanded in B around 0 86.8%

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

   if 8.00000000000000035e245 < F < 1.2800000000000001e278

   1. Initial program 66.3%

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

   3. Simplified 66.6%

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

   5. Taylor expanded in x around 0 66.6%

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

      1. associate-*l/ 66.6%

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

      2. *-lft-identity 66.6%

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

      3. +-commutative 66.6%

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

      4. unpow2 66.6%

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

      5. fma-undefine 66.6%

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

   7. Simplified 66.6%

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

   8. Taylor expanded in F around inf 99.3%

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

   9. Taylor expanded in B around 0 99.5%

      \[\leadsto \color{blue}{\frac{1 - \frac{1}{{F}^{2}}}{B}} - \frac{x}{\tan B} \]
   10. Step-by-step derivation

       1. unpow-1 99.5%

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

       2. exp-to-pow 99.5%

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

       3. *-commutative 99.5%

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

       4. exp-prod 99.5%

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

       5. *-commutative 99.5%

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

       6. associate-*l* 99.5%

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

       7. metadata-eval 99.5%

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

       8. exp-to-pow 99.5%

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

   11. Simplified 99.5%

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

   12. Taylor expanded in F around inf 99.5%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -9600000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.00017:\\ \;\;\;\;x \cdot \frac{\cos B}{-\sin B}\\ \mathbf{elif}\;F \leq 8 \cdot 10^{+245} \lor \neg \left(F \leq 1.28 \cdot 10^{+278}\right):\\ \;\;\;\;\frac{F}{F \cdot \sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 71.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -9600000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.48:\\ \;\;\;\;\frac{\cos B}{\frac{\sin B}{-x}}\\ \mathbf{elif}\;F \leq 1.65 \cdot 10^{+246} \lor \neg \left(F \leq 1.35 \cdot 10^{+284}\right):\\ \;\;\;\;\frac{F}{F \cdot \sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -9600000.0)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 0.48)
     (/ (cos B) (/ (sin B) (- x)))
     (if (or (<= F 1.65e+246) (not (<= F 1.35e+284)))
       (- (/ F (* F (sin B))) (/ x B))
       (- (/ 1.0 B) (/ x (tan B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -9600000.0) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 0.48) {
		tmp = cos(B) / (sin(B) / -x);
	} else if ((F <= 1.65e+246) || !(F <= 1.35e+284)) {
		tmp = (F / (F * sin(B))) - (x / B);
	} else {
		tmp = (1.0 / B) - (x / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-9600000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 0.48d0) then
        tmp = cos(b) / (sin(b) / -x)
    else if ((f <= 1.65d+246) .or. (.not. (f <= 1.35d+284))) then
        tmp = (f / (f * sin(b))) - (x / b)
    else
        tmp = (1.0d0 / b) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -9600000.0) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 0.48) {
		tmp = Math.cos(B) / (Math.sin(B) / -x);
	} else if ((F <= 1.65e+246) || !(F <= 1.35e+284)) {
		tmp = (F / (F * Math.sin(B))) - (x / B);
	} else {
		tmp = (1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -9600000.0:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 0.48:
		tmp = math.cos(B) / (math.sin(B) / -x)
	elif (F <= 1.65e+246) or not (F <= 1.35e+284):
		tmp = (F / (F * math.sin(B))) - (x / B)
	else:
		tmp = (1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -9600000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 0.48)
		tmp = Float64(cos(B) / Float64(sin(B) / Float64(-x)));
	elseif ((F <= 1.65e+246) || !(F <= 1.35e+284))
		tmp = Float64(Float64(F / Float64(F * sin(B))) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -9600000.0)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 0.48)
		tmp = cos(B) / (sin(B) / -x);
	elseif ((F <= 1.65e+246) || ~((F <= 1.35e+284)))
		tmp = (F / (F * sin(B))) - (x / B);
	else
		tmp = (1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -9600000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.48], N[(N[Cos[B], $MachinePrecision] / N[(N[Sin[B], $MachinePrecision] / (-x)), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[F, 1.65e+246], N[Not[LessEqual[F, 1.35e+284]], $MachinePrecision]], N[(N[(F / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -9.6e6

   1. Initial program 57.9%

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

   3. Taylor expanded in F around -inf 98.3%

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

   4. Taylor expanded in B around 0 86.1%

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

   if -9.6e6 < F < 0.47999999999999998

   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 42.7%

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

   4. Taylor expanded in x around inf 68.1%

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

      1. clear-num 67.9%

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

      2. inv-pow 67.9%

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

      3. associate-/r* 67.9%

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

   6. Applied egg-rr 67.9%

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

      1. unpow-1 67.9%

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

      2. clear-num 67.9%

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

   8. Applied egg-rr 67.9%

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

   if 0.47999999999999998 < F < 1.65e246 or 1.35000000000000003e284 < F

   1. Initial program 56.2%

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

   3. Taylor expanded in F around inf 78.6%

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

      1. un-div-inv 78.7%

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

      2. associate-/l/ 97.8%

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

   5. Applied egg-rr 97.8%

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

   6. Taylor expanded in B around 0 86.8%

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

   if 1.65e246 < F < 1.35000000000000003e284

   1. Initial program 66.3%

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

   3. Simplified 66.6%

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

   5. Taylor expanded in x around 0 66.6%

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

      1. associate-*l/ 66.6%

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

      2. *-lft-identity 66.6%

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

      3. +-commutative 66.6%

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

      4. unpow2 66.6%

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

      5. fma-undefine 66.6%

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

   7. Simplified 66.6%

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

   8. Taylor expanded in F around inf 99.3%

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

   9. Taylor expanded in B around 0 99.5%

      \[\leadsto \color{blue}{\frac{1 - \frac{1}{{F}^{2}}}{B}} - \frac{x}{\tan B} \]
   10. Step-by-step derivation

       1. unpow-1 99.5%

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

       2. exp-to-pow 99.5%

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

       3. *-commutative 99.5%

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

       4. exp-prod 99.5%

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

       5. *-commutative 99.5%

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

       6. associate-*l* 99.5%

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

       7. metadata-eval 99.5%

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

       8. exp-to-pow 99.5%

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

   11. Simplified 99.5%

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

   12. Taylor expanded in F around inf 99.5%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -9600000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.48:\\ \;\;\;\;\frac{\cos B}{\frac{\sin B}{-x}}\\ \mathbf{elif}\;F \leq 1.65 \cdot 10^{+246} \lor \neg \left(F \leq 1.35 \cdot 10^{+284}\right):\\ \;\;\;\;\frac{F}{F \cdot \sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 71.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -9600000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.03:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{elif}\;F \leq 2.05 \cdot 10^{+246} \lor \neg \left(F \leq 1.5 \cdot 10^{+279}\right):\\ \;\;\;\;\frac{F}{F \cdot \sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -9600000.0)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 0.03)
     (/ (* x (cos B)) (- (sin B)))
     (if (or (<= F 2.05e+246) (not (<= F 1.5e+279)))
       (- (/ F (* F (sin B))) (/ x B))
       (- (/ 1.0 B) (/ x (tan B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -9600000.0) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 0.03) {
		tmp = (x * cos(B)) / -sin(B);
	} else if ((F <= 2.05e+246) || !(F <= 1.5e+279)) {
		tmp = (F / (F * sin(B))) - (x / B);
	} else {
		tmp = (1.0 / B) - (x / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-9600000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 0.03d0) then
        tmp = (x * cos(b)) / -sin(b)
    else if ((f <= 2.05d+246) .or. (.not. (f <= 1.5d+279))) then
        tmp = (f / (f * sin(b))) - (x / b)
    else
        tmp = (1.0d0 / b) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -9600000.0) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 0.03) {
		tmp = (x * Math.cos(B)) / -Math.sin(B);
	} else if ((F <= 2.05e+246) || !(F <= 1.5e+279)) {
		tmp = (F / (F * Math.sin(B))) - (x / B);
	} else {
		tmp = (1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -9600000.0:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 0.03:
		tmp = (x * math.cos(B)) / -math.sin(B)
	elif (F <= 2.05e+246) or not (F <= 1.5e+279):
		tmp = (F / (F * math.sin(B))) - (x / B)
	else:
		tmp = (1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -9600000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 0.03)
		tmp = Float64(Float64(x * cos(B)) / Float64(-sin(B)));
	elseif ((F <= 2.05e+246) || !(F <= 1.5e+279))
		tmp = Float64(Float64(F / Float64(F * sin(B))) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -9600000.0)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 0.03)
		tmp = (x * cos(B)) / -sin(B);
	elseif ((F <= 2.05e+246) || ~((F <= 1.5e+279)))
		tmp = (F / (F * sin(B))) - (x / B);
	else
		tmp = (1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -9600000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.03], N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], If[Or[LessEqual[F, 2.05e+246], N[Not[LessEqual[F, 1.5e+279]], $MachinePrecision]], N[(N[(F / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -9.6e6

   1. Initial program 57.9%

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

   3. Taylor expanded in F around -inf 98.3%

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

   4. Taylor expanded in B around 0 86.1%

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

   if -9.6e6 < F < 0.029999999999999999

   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 42.7%

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

   4. Taylor expanded in x around inf 68.1%

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

   if 0.029999999999999999 < F < 2.04999999999999988e246 or 1.4999999999999999e279 < F

   1. Initial program 56.2%

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

   3. Taylor expanded in F around inf 78.6%

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

      1. un-div-inv 78.7%

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

      2. associate-/l/ 97.8%

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

   5. Applied egg-rr 97.8%

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

   6. Taylor expanded in B around 0 86.8%

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

   if 2.04999999999999988e246 < F < 1.4999999999999999e279

   1. Initial program 66.3%

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

   3. Simplified 66.6%

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

   5. Taylor expanded in x around 0 66.6%

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

      1. associate-*l/ 66.6%

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

      2. *-lft-identity 66.6%

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

      3. +-commutative 66.6%

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

      4. unpow2 66.6%

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

      5. fma-undefine 66.6%

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

   7. Simplified 66.6%

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

   8. Taylor expanded in F around inf 99.3%

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

   9. Taylor expanded in B around 0 99.5%

      \[\leadsto \color{blue}{\frac{1 - \frac{1}{{F}^{2}}}{B}} - \frac{x}{\tan B} \]
   10. Step-by-step derivation

       1. unpow-1 99.5%

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

       2. exp-to-pow 99.5%

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

       3. *-commutative 99.5%

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

       4. exp-prod 99.5%

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

       5. *-commutative 99.5%

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

       6. associate-*l* 99.5%

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

       7. metadata-eval 99.5%

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

       8. exp-to-pow 99.5%

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

   11. Simplified 99.5%

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

   12. Taylor expanded in F around inf 99.5%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -9600000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.03:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{elif}\;F \leq 2.05 \cdot 10^{+246} \lor \neg \left(F \leq 1.5 \cdot 10^{+279}\right):\\ \;\;\;\;\frac{F}{F \cdot \sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 78.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -7.5 \cdot 10^{-42}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 0.92:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{elif}\;F \leq 2.1 \cdot 10^{+246} \lor \neg \left(F \leq 5.3 \cdot 10^{+283}\right):\\ \;\;\;\;\frac{F}{F \cdot \sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -7.5e-42)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 0.92)
       (/ (* x (cos B)) (- (sin B)))
       (if (or (<= F 2.1e+246) (not (<= F 5.3e+283)))
         (- (/ F (* F (sin B))) (/ x B))
         (- (/ 1.0 B) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -7.5e-42) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 0.92) {
		tmp = (x * cos(B)) / -sin(B);
	} else if ((F <= 2.1e+246) || !(F <= 5.3e+283)) {
		tmp = (F / (F * sin(B))) - (x / B);
	} else {
		tmp = (1.0 / B) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-7.5d-42)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 0.92d0) then
        tmp = (x * cos(b)) / -sin(b)
    else if ((f <= 2.1d+246) .or. (.not. (f <= 5.3d+283))) then
        tmp = (f / (f * sin(b))) - (x / b)
    else
        tmp = (1.0d0 / b) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -7.5e-42) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 0.92) {
		tmp = (x * Math.cos(B)) / -Math.sin(B);
	} else if ((F <= 2.1e+246) || !(F <= 5.3e+283)) {
		tmp = (F / (F * Math.sin(B))) - (x / B);
	} else {
		tmp = (1.0 / B) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -7.5e-42:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 0.92:
		tmp = (x * math.cos(B)) / -math.sin(B)
	elif (F <= 2.1e+246) or not (F <= 5.3e+283):
		tmp = (F / (F * math.sin(B))) - (x / B)
	else:
		tmp = (1.0 / B) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -7.5e-42)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 0.92)
		tmp = Float64(Float64(x * cos(B)) / Float64(-sin(B)));
	elseif ((F <= 2.1e+246) || !(F <= 5.3e+283))
		tmp = Float64(Float64(F / Float64(F * sin(B))) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / B) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -7.5e-42)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 0.92)
		tmp = (x * cos(B)) / -sin(B);
	elseif ((F <= 2.1e+246) || ~((F <= 5.3e+283)))
		tmp = (F / (F * sin(B))) - (x / B);
	else
		tmp = (1.0 / B) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -7.5e-42], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.92], N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], If[Or[LessEqual[F, 2.1e+246], N[Not[LessEqual[F, 5.3e+283]], $MachinePrecision]], N[(N[(F / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -7.49999999999999972e-42

   1. Initial program 62.1%

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

   3. Simplified 73.6%

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

   5. Taylor expanded in x around 0 73.6%

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

      1. associate-*l/ 73.6%

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

      2. *-lft-identity 73.6%

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

      3. +-commutative 73.6%

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

      4. unpow2 73.6%

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

      5. fma-undefine 73.6%

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

   7. Simplified 73.6%

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

   8. Taylor expanded in F around -inf 94.6%

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

   if -7.49999999999999972e-42 < F < 0.92000000000000004

   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 42.2%

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

   4. Taylor expanded in x around inf 70.4%

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

   if 0.92000000000000004 < F < 2.1e246 or 5.3e283 < F

   1. Initial program 56.2%

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

   3. Taylor expanded in F around inf 78.6%

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

      1. un-div-inv 78.7%

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

      2. associate-/l/ 97.8%

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

   5. Applied egg-rr 97.8%

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

   6. Taylor expanded in B around 0 86.8%

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

   if 2.1e246 < F < 5.3e283

   1. Initial program 66.3%

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

   3. Simplified 66.6%

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

   5. Taylor expanded in x around 0 66.6%

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

      1. associate-*l/ 66.6%

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

      2. *-lft-identity 66.6%

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

      3. +-commutative 66.6%

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

      4. unpow2 66.6%

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

      5. fma-undefine 66.6%

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

   7. Simplified 66.6%

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

   8. Taylor expanded in F around inf 99.3%

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

   9. Taylor expanded in B around 0 99.5%

      \[\leadsto \color{blue}{\frac{1 - \frac{1}{{F}^{2}}}{B}} - \frac{x}{\tan B} \]
   10. Step-by-step derivation

       1. unpow-1 99.5%

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

       2. exp-to-pow 99.5%

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

       3. *-commutative 99.5%

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

       4. exp-prod 99.5%

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

       5. *-commutative 99.5%

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

       6. associate-*l* 99.5%

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

       7. metadata-eval 99.5%

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

       8. exp-to-pow 99.5%

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

   11. Simplified 99.5%

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

   12. Taylor expanded in F around inf 99.5%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7.5 \cdot 10^{-42}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.92:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{elif}\;F \leq 2.1 \cdot 10^{+246} \lor \neg \left(F \leq 5.3 \cdot 10^{+283}\right):\\ \;\;\;\;\frac{F}{F \cdot \sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 62.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;F \leq -1.05 \cdot 10^{+48}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -4.2 \cdot 10^{-166}:\\ \;\;\;\;\frac{-1}{B} - t\_0\\ \mathbf{elif}\;F \leq -3.1 \cdot 10^{-235}:\\ \;\;\;\;\frac{{B}^{2} \cdot \left(x \cdot \left(--0.3333333333333333\right)\right) - x}{B}\\ \mathbf{elif}\;F \leq 1.65:\\ \;\;\;\;\frac{1}{F} \cdot \frac{F}{B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{F \cdot \sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ 1.0 (tan B)))))
   (if (<= F -1.05e+48)
     (- (/ -1.0 (sin B)) (/ x B))
     (if (<= F -4.2e-166)
       (- (/ -1.0 B) t_0)
       (if (<= F -3.1e-235)
         (/ (- (* (pow B 2.0) (* x (- -0.3333333333333333))) x) B)
         (if (<= F 1.65)
           (- (* (/ 1.0 F) (/ F B)) t_0)
           (- (/ F (* F (sin B))) (/ x B))))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (1.0 / tan(B));
	double tmp;
	if (F <= -1.05e+48) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -4.2e-166) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= -3.1e-235) {
		tmp = ((pow(B, 2.0) * (x * -(-0.3333333333333333))) - x) / B;
	} else if (F <= 1.65) {
		tmp = ((1.0 / F) * (F / B)) - t_0;
	} else {
		tmp = (F / (F * sin(B))) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (1.0d0 / tan(b))
    if (f <= (-1.05d+48)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-4.2d-166)) then
        tmp = ((-1.0d0) / b) - t_0
    else if (f <= (-3.1d-235)) then
        tmp = (((b ** 2.0d0) * (x * -(-0.3333333333333333d0))) - x) / b
    else if (f <= 1.65d0) then
        tmp = ((1.0d0 / f) * (f / b)) - t_0
    else
        tmp = (f / (f * sin(b))) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x * (1.0 / Math.tan(B));
	double tmp;
	if (F <= -1.05e+48) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -4.2e-166) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= -3.1e-235) {
		tmp = ((Math.pow(B, 2.0) * (x * -(-0.3333333333333333))) - x) / B;
	} else if (F <= 1.65) {
		tmp = ((1.0 / F) * (F / B)) - t_0;
	} else {
		tmp = (F / (F * Math.sin(B))) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * (1.0 / math.tan(B))
	tmp = 0
	if F <= -1.05e+48:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -4.2e-166:
		tmp = (-1.0 / B) - t_0
	elif F <= -3.1e-235:
		tmp = ((math.pow(B, 2.0) * (x * -(-0.3333333333333333))) - x) / B
	elif F <= 1.65:
		tmp = ((1.0 / F) * (F / B)) - t_0
	else:
		tmp = (F / (F * math.sin(B))) - (x / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(1.0 / tan(B)))
	tmp = 0.0
	if (F <= -1.05e+48)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -4.2e-166)
		tmp = Float64(Float64(-1.0 / B) - t_0);
	elseif (F <= -3.1e-235)
		tmp = Float64(Float64(Float64((B ^ 2.0) * Float64(x * Float64(-(-0.3333333333333333)))) - x) / B);
	elseif (F <= 1.65)
		tmp = Float64(Float64(Float64(1.0 / F) * Float64(F / B)) - t_0);
	else
		tmp = Float64(Float64(F / Float64(F * sin(B))) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x * (1.0 / tan(B));
	tmp = 0.0;
	if (F <= -1.05e+48)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -4.2e-166)
		tmp = (-1.0 / B) - t_0;
	elseif (F <= -3.1e-235)
		tmp = (((B ^ 2.0) * (x * -(-0.3333333333333333))) - x) / B;
	elseif (F <= 1.65)
		tmp = ((1.0 / F) * (F / B)) - t_0;
	else
		tmp = (F / (F * sin(B))) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.05e+48], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -4.2e-166], N[(N[(-1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -3.1e-235], N[(N[(N[(N[Power[B, 2.0], $MachinePrecision] * N[(x * (--0.3333333333333333)), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.65], N[(N[(N[(1.0 / F), $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(F / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -1.0499999999999999e48

   1. Initial program 51.0%

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

   3. Taylor expanded in F around -inf 98.1%

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

   4. Taylor expanded in B around 0 90.2%

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

   if -1.0499999999999999e48 < F < -4.1999999999999999e-166

   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 F around -inf 56.8%

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

   4. Taylor expanded in B around 0 55.3%

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

   if -4.1999999999999999e-166 < F < -3.1e-235

   1. Initial program 99.7%

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

   3. Taylor expanded in F around -inf 50.7%

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

   4. Taylor expanded in x around inf 99.7%

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

   5. Taylor expanded in B around 0 92.2%

      \[\leadsto -1 \cdot \color{blue}{\frac{x + {B}^{2} \cdot \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right)}{B}} \]
   6. Step-by-step derivation

      1. distribute-rgt-out-- 92.2%

         \[\leadsto -1 \cdot \frac{x + {B}^{2} \cdot \color{blue}{\left(x \cdot \left(-0.5 - -0.16666666666666666\right)\right)}}{B} \]

      2. metadata-eval 92.2%

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

   7. Simplified 92.2%

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

   if -3.1e-235 < F < 1.6499999999999999

   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 41.3%

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

   4. Taylor expanded in B around 0 50.4%

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

   if 1.6499999999999999 < F

   1. Initial program 57.4%

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

   3. Taylor expanded in F around inf 78.6%

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

      1. un-div-inv 78.7%

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

      2. associate-/l/ 98.0%

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

   5. Applied egg-rr 98.0%

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

   6. Taylor expanded in B around 0 81.8%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.05 \cdot 10^{+48}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -4.2 \cdot 10^{-166}:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq -3.1 \cdot 10^{-235}:\\ \;\;\;\;\frac{{B}^{2} \cdot \left(x \cdot \left(--0.3333333333333333\right)\right) - x}{B}\\ \mathbf{elif}\;F \leq 1.65:\\ \;\;\;\;\frac{1}{F} \cdot \frac{F}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{F \cdot \sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 61.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -8.5 \cdot 10^{+46}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -6.2 \cdot 10^{-166}:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq -8 \cdot 10^{-245}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;F \leq 5.3 \cdot 10^{+51}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{F \cdot \sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -8.5e+46)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F -6.2e-166)
     (- (/ -1.0 B) (* x (/ 1.0 (tan B))))
     (if (<= F -8e-245)
       (/ x (- B))
       (if (<= F 5.3e+51)
         (- (/ 1.0 B) (/ x (tan B)))
         (- (/ F (* F (sin B))) (/ x B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -8.5e+46) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -6.2e-166) {
		tmp = (-1.0 / B) - (x * (1.0 / tan(B)));
	} else if (F <= -8e-245) {
		tmp = x / -B;
	} else if (F <= 5.3e+51) {
		tmp = (1.0 / B) - (x / tan(B));
	} else {
		tmp = (F / (F * sin(B))) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-8.5d+46)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-6.2d-166)) then
        tmp = ((-1.0d0) / b) - (x * (1.0d0 / tan(b)))
    else if (f <= (-8d-245)) then
        tmp = x / -b
    else if (f <= 5.3d+51) then
        tmp = (1.0d0 / b) - (x / tan(b))
    else
        tmp = (f / (f * sin(b))) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -8.5e+46) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -6.2e-166) {
		tmp = (-1.0 / B) - (x * (1.0 / Math.tan(B)));
	} else if (F <= -8e-245) {
		tmp = x / -B;
	} else if (F <= 5.3e+51) {
		tmp = (1.0 / B) - (x / Math.tan(B));
	} else {
		tmp = (F / (F * Math.sin(B))) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -8.5e+46:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -6.2e-166:
		tmp = (-1.0 / B) - (x * (1.0 / math.tan(B)))
	elif F <= -8e-245:
		tmp = x / -B
	elif F <= 5.3e+51:
		tmp = (1.0 / B) - (x / math.tan(B))
	else:
		tmp = (F / (F * math.sin(B))) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -8.5e+46)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -6.2e-166)
		tmp = Float64(Float64(-1.0 / B) - Float64(x * Float64(1.0 / tan(B))));
	elseif (F <= -8e-245)
		tmp = Float64(x / Float64(-B));
	elseif (F <= 5.3e+51)
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(F / Float64(F * sin(B))) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -8.5e+46)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -6.2e-166)
		tmp = (-1.0 / B) - (x * (1.0 / tan(B)));
	elseif (F <= -8e-245)
		tmp = x / -B;
	elseif (F <= 5.3e+51)
		tmp = (1.0 / B) - (x / tan(B));
	else
		tmp = (F / (F * sin(B))) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -8.5e+46], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -6.2e-166], N[(N[(-1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -8e-245], N[(x / (-B)), $MachinePrecision], If[LessEqual[F, 5.3e+51], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(F / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -8.4999999999999996e46

   1. Initial program 51.0%

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

   3. Taylor expanded in F around -inf 98.1%

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

   4. Taylor expanded in B around 0 90.2%

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

   if -8.4999999999999996e46 < F < -6.19999999999999968e-166

   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 F around -inf 56.8%

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

   4. Taylor expanded in B around 0 55.3%

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

   if -6.19999999999999968e-166 < F < -7.9999999999999994e-245

   1. Initial program 99.7%

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

   3. Taylor expanded in F around -inf 43.7%

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

   4. Taylor expanded in B around 0 44.5%

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

      1. mul-1-neg 44.5%

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

      2. distribute-neg-frac2 44.5%

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

   6. Simplified 44.5%

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

   7. Taylor expanded in x around inf 86.2%

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

      1. associate-*r/ 86.2%

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

      2. neg-mul-1 86.2%

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

   9. Simplified 86.2%

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

   if -7.9999999999999994e-245 < F < 5.2999999999999997e51

   1. Initial program 98.3%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 99.6%

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

      1. associate-*l/ 99.7%

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

      2. *-lft-identity 99.7%

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

      3. +-commutative 99.7%

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

      4. unpow2 99.7%

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

      5. fma-undefine 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in F around inf 33.0%

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

   9. Taylor expanded in B around 0 33.1%

      \[\leadsto \color{blue}{\frac{1 - \frac{1}{{F}^{2}}}{B}} - \frac{x}{\tan B} \]
   10. Step-by-step derivation

       1. unpow-1 33.1%

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

       2. exp-to-pow 30.3%

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

       3. *-commutative 30.3%

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

       4. exp-prod 30.3%

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

       5. *-commutative 30.3%

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

       6. associate-*l* 30.3%

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

       7. metadata-eval 30.3%

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

       8. exp-to-pow 33.1%

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

   11. Simplified 33.1%

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

   12. Taylor expanded in F around inf 54.2%

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

   if 5.2999999999999997e51 < F

   1. Initial program 47.3%

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

   3. Taylor expanded in F around inf 76.3%

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

      1. un-div-inv 76.4%

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

      2. associate-/l/ 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in B around 0 83.8%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -8.5 \cdot 10^{+46}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -6.2 \cdot 10^{-166}:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq -8 \cdot 10^{-245}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;F \leq 5.3 \cdot 10^{+51}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{F \cdot \sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 54.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -8 \cdot 10^{+46}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq -5.1 \cdot 10^{-166} \lor \neg \left(F \leq -1.25 \cdot 10^{-245}\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -8e+46)
   (/ (- -1.0 x) B)
   (if (or (<= F -5.1e-166) (not (<= F -1.25e-245)))
     (- (/ 1.0 B) (/ x (tan B)))
     (/ x (- B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -8e+46) {
		tmp = (-1.0 - x) / B;
	} else if ((F <= -5.1e-166) || !(F <= -1.25e-245)) {
		tmp = (1.0 / B) - (x / tan(B));
	} else {
		tmp = x / -B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-8d+46)) then
        tmp = ((-1.0d0) - x) / b
    else if ((f <= (-5.1d-166)) .or. (.not. (f <= (-1.25d-245)))) then
        tmp = (1.0d0 / b) - (x / tan(b))
    else
        tmp = x / -b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -8e+46) {
		tmp = (-1.0 - x) / B;
	} else if ((F <= -5.1e-166) || !(F <= -1.25e-245)) {
		tmp = (1.0 / B) - (x / Math.tan(B));
	} else {
		tmp = x / -B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -8e+46:
		tmp = (-1.0 - x) / B
	elif (F <= -5.1e-166) or not (F <= -1.25e-245):
		tmp = (1.0 / B) - (x / math.tan(B))
	else:
		tmp = x / -B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -8e+46)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif ((F <= -5.1e-166) || !(F <= -1.25e-245))
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	else
		tmp = Float64(x / Float64(-B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -8e+46)
		tmp = (-1.0 - x) / B;
	elseif ((F <= -5.1e-166) || ~((F <= -1.25e-245)))
		tmp = (1.0 / B) - (x / tan(B));
	else
		tmp = x / -B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -8e+46], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[Or[LessEqual[F, -5.1e-166], N[Not[LessEqual[F, -1.25e-245]], $MachinePrecision]], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / (-B)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -7.9999999999999999e46

   1. Initial program 51.0%

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

   3. Taylor expanded in F around -inf 98.1%

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

   4. Taylor expanded in B around 0 67.9%

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

      1. mul-1-neg 67.9%

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

      2. distribute-neg-frac2 67.9%

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

   6. Simplified 67.9%

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

   if -7.9999999999999999e46 < F < -5.1000000000000002e-166 or -1.2499999999999999e-245 < F

   1. Initial program 82.6%

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

   3. Simplified 88.5%

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

   5. Taylor expanded in x around 0 88.5%

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

      1. associate-*l/ 88.5%

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

      2. *-lft-identity 88.5%

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

      3. +-commutative 88.5%

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

      4. unpow2 88.5%

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

      5. fma-undefine 88.5%

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

   7. Simplified 88.5%

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

   8. Taylor expanded in F around inf 53.0%

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

   9. Taylor expanded in B around 0 45.0%

      \[\leadsto \color{blue}{\frac{1 - \frac{1}{{F}^{2}}}{B}} - \frac{x}{\tan B} \]
   10. Step-by-step derivation

       1. unpow-1 45.0%

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

       2. exp-to-pow 32.7%

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

       3. *-commutative 32.7%

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

       4. exp-prod 32.7%

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

       5. *-commutative 32.7%

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

       6. associate-*l* 32.7%

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

       7. metadata-eval 32.7%

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

       8. exp-to-pow 45.0%

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

   11. Simplified 45.0%

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

   12. Taylor expanded in F around inf 56.0%

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

   if -5.1000000000000002e-166 < F < -1.2499999999999999e-245

   1. Initial program 99.7%

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

   3. Taylor expanded in F around -inf 43.7%

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

   4. Taylor expanded in B around 0 44.5%

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

      1. mul-1-neg 44.5%

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

      2. distribute-neg-frac2 44.5%

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

   6. Simplified 44.5%

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

   7. Taylor expanded in x around inf 86.2%

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

      1. associate-*r/ 86.2%

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

      2. neg-mul-1 86.2%

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

   9. Simplified 86.2%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -8 \cdot 10^{+46}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq -5.1 \cdot 10^{-166} \lor \neg \left(F \leq -1.25 \cdot 10^{-245}\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 61.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -9600000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -4.9 \cdot 10^{-166} \lor \neg \left(F \leq -1.12 \cdot 10^{-243}\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -9600000.0)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (or (<= F -4.9e-166) (not (<= F -1.12e-243)))
     (- (/ 1.0 B) (/ x (tan B)))
     (/ x (- B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -9600000.0) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if ((F <= -4.9e-166) || !(F <= -1.12e-243)) {
		tmp = (1.0 / B) - (x / tan(B));
	} else {
		tmp = x / -B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-9600000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if ((f <= (-4.9d-166)) .or. (.not. (f <= (-1.12d-243)))) then
        tmp = (1.0d0 / b) - (x / tan(b))
    else
        tmp = x / -b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -9600000.0) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if ((F <= -4.9e-166) || !(F <= -1.12e-243)) {
		tmp = (1.0 / B) - (x / Math.tan(B));
	} else {
		tmp = x / -B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -9600000.0:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif (F <= -4.9e-166) or not (F <= -1.12e-243):
		tmp = (1.0 / B) - (x / math.tan(B))
	else:
		tmp = x / -B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -9600000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif ((F <= -4.9e-166) || !(F <= -1.12e-243))
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	else
		tmp = Float64(x / Float64(-B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -9600000.0)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif ((F <= -4.9e-166) || ~((F <= -1.12e-243)))
		tmp = (1.0 / B) - (x / tan(B));
	else
		tmp = x / -B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -9600000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[F, -4.9e-166], N[Not[LessEqual[F, -1.12e-243]], $MachinePrecision]], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / (-B)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -9.6e6

   1. Initial program 57.9%

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

   3. Taylor expanded in F around -inf 98.3%

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

   4. Taylor expanded in B around 0 86.1%

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

   if -9.6e6 < F < -4.8999999999999999e-166 or -1.12000000000000005e-243 < F

   1. Initial program 81.6%

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

   3. Simplified 87.9%

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

   5. Taylor expanded in x around 0 87.9%

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

      1. associate-*l/ 87.9%

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

      2. *-lft-identity 87.9%

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

      3. +-commutative 87.9%

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

      4. unpow2 87.9%

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

      5. fma-undefine 87.9%

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

   7. Simplified 87.9%

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

   8. Taylor expanded in F around inf 53.1%

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

   9. Taylor expanded in B around 0 44.6%

      \[\leadsto \color{blue}{\frac{1 - \frac{1}{{F}^{2}}}{B}} - \frac{x}{\tan B} \]
   10. Step-by-step derivation

       1. unpow-1 44.6%

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

       2. exp-to-pow 34.6%

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

       3. *-commutative 34.6%

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

       4. exp-prod 34.6%

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

       5. *-commutative 34.6%

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

       6. associate-*l* 34.6%

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

       7. metadata-eval 34.6%

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

       8. exp-to-pow 44.6%

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

   11. Simplified 44.6%

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

   12. Taylor expanded in F around inf 56.2%

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

   if -4.8999999999999999e-166 < F < -1.12000000000000005e-243

   1. Initial program 99.7%

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

   3. Taylor expanded in F around -inf 43.7%

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

   4. Taylor expanded in B around 0 44.5%

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

      1. mul-1-neg 44.5%

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

      2. distribute-neg-frac2 44.5%

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

   6. Simplified 44.5%

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

   7. Taylor expanded in x around inf 86.2%

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

      1. associate-*r/ 86.2%

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

      2. neg-mul-1 86.2%

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

   9. Simplified 86.2%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -9600000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -4.9 \cdot 10^{-166} \lor \neg \left(F \leq -1.12 \cdot 10^{-243}\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 61.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -5.4 \cdot 10^{+47}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -5.2 \cdot 10^{-166}:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq -6.4 \cdot 10^{-244}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -5.4e+47)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F -5.2e-166)
     (- (/ -1.0 B) (* x (/ 1.0 (tan B))))
     (if (<= F -6.4e-244) (/ x (- B)) (- (/ 1.0 B) (/ x (tan B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -5.4e+47) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -5.2e-166) {
		tmp = (-1.0 / B) - (x * (1.0 / tan(B)));
	} else if (F <= -6.4e-244) {
		tmp = x / -B;
	} else {
		tmp = (1.0 / B) - (x / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-5.4d+47)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-5.2d-166)) then
        tmp = ((-1.0d0) / b) - (x * (1.0d0 / tan(b)))
    else if (f <= (-6.4d-244)) then
        tmp = x / -b
    else
        tmp = (1.0d0 / b) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -5.4e+47) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -5.2e-166) {
		tmp = (-1.0 / B) - (x * (1.0 / Math.tan(B)));
	} else if (F <= -6.4e-244) {
		tmp = x / -B;
	} else {
		tmp = (1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -5.4e+47:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -5.2e-166:
		tmp = (-1.0 / B) - (x * (1.0 / math.tan(B)))
	elif F <= -6.4e-244:
		tmp = x / -B
	else:
		tmp = (1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -5.4e+47)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -5.2e-166)
		tmp = Float64(Float64(-1.0 / B) - Float64(x * Float64(1.0 / tan(B))));
	elseif (F <= -6.4e-244)
		tmp = Float64(x / Float64(-B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -5.4e+47)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -5.2e-166)
		tmp = (-1.0 / B) - (x * (1.0 / tan(B)));
	elseif (F <= -6.4e-244)
		tmp = x / -B;
	else
		tmp = (1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -5.4e+47], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -5.2e-166], N[(N[(-1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -6.4e-244], N[(x / (-B)), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -5.39999999999999991e47

   1. Initial program 51.0%

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

   3. Taylor expanded in F around -inf 98.1%

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

   4. Taylor expanded in B around 0 90.2%

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

   if -5.39999999999999991e47 < F < -5.19999999999999979e-166

   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 F around -inf 56.8%

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

   4. Taylor expanded in B around 0 55.3%

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

   if -5.19999999999999979e-166 < F < -6.3999999999999996e-244

   1. Initial program 99.7%

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

   3. Taylor expanded in F around -inf 43.7%

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

   4. Taylor expanded in B around 0 44.5%

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

      1. mul-1-neg 44.5%

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

      2. distribute-neg-frac2 44.5%

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

   6. Simplified 44.5%

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

   7. Taylor expanded in x around inf 86.2%

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

      1. associate-*r/ 86.2%

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

      2. neg-mul-1 86.2%

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

   9. Simplified 86.2%

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

   if -6.3999999999999996e-244 < F

   1. Initial program 76.7%

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

   3. Simplified 84.7%

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

   5. Taylor expanded in x around 0 84.6%

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

      1. associate-*l/ 84.7%

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

      2. *-lft-identity 84.7%

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

      3. +-commutative 84.7%

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

      4. unpow2 84.7%

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

      5. fma-undefine 84.7%

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

   7. Simplified 84.7%

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

   8. Taylor expanded in F around inf 61.0%

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

   9. Taylor expanded in B around 0 45.7%

      \[\leadsto \color{blue}{\frac{1 - \frac{1}{{F}^{2}}}{B}} - \frac{x}{\tan B} \]
   10. Step-by-step derivation

       1. unpow-1 45.7%

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

       2. exp-to-pow 44.1%

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

       3. *-commutative 44.1%

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

       4. exp-prod 44.1%

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

       5. *-commutative 44.1%

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

       6. associate-*l* 44.1%

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

       7. metadata-eval 44.1%

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

       8. exp-to-pow 45.7%

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

   11. Simplified 45.7%

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

   12. Taylor expanded in F around inf 57.9%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5.4 \cdot 10^{+47}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -5.2 \cdot 10^{-166}:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq -6.4 \cdot 10^{-244}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 38.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -7.2 \cdot 10^{-50}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 66000:\\ \;\;\;\;\frac{x}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{B}{x + 1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -7.2e-50)
   (/ (- -1.0 x) B)
   (if (<= F 66000.0) (/ x (- (sin B))) (/ 1.0 (/ B (+ x 1.0))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -7.2e-50) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 66000.0) {
		tmp = x / -sin(B);
	} else {
		tmp = 1.0 / (B / (x + 1.0));
	}
	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) :: tmp
    if (f <= (-7.2d-50)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 66000.0d0) then
        tmp = x / -sin(b)
    else
        tmp = 1.0d0 / (b / (x + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -7.2e-50) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 66000.0) {
		tmp = x / -Math.sin(B);
	} else {
		tmp = 1.0 / (B / (x + 1.0));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -7.2e-50:
		tmp = (-1.0 - x) / B
	elif F <= 66000.0:
		tmp = x / -math.sin(B)
	else:
		tmp = 1.0 / (B / (x + 1.0))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -7.2e-50)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 66000.0)
		tmp = Float64(x / Float64(-sin(B)));
	else
		tmp = Float64(1.0 / Float64(B / Float64(x + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -7.2e-50)
		tmp = (-1.0 - x) / B;
	elseif (F <= 66000.0)
		tmp = x / -sin(B);
	else
		tmp = 1.0 / (B / (x + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -7.2e-50], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 66000.0], N[(x / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], N[(1.0 / N[(B / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -7.19999999999999958e-50

   1. Initial program 63.0%

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

   3. Taylor expanded in F around -inf 92.3%

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

   4. Taylor expanded in B around 0 58.2%

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

      1. mul-1-neg 58.2%

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

      2. distribute-neg-frac2 58.2%

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

   6. Simplified 58.2%

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

   if -7.19999999999999958e-50 < F < 66000

   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 41.2%

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

   4. Taylor expanded in x around inf 69.4%

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

   5. Taylor expanded in B around 0 41.9%

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

   if 66000 < F

   1. Initial program 56.2%

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

   3. Taylor expanded in F around -inf 34.1%

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

   4. Taylor expanded in B around 0 17.9%

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

      1. mul-1-neg 17.9%

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

      2. distribute-neg-frac2 17.9%

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

   6. Simplified 17.9%

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

      1. clear-num 17.9%

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

      2. inv-pow 17.9%

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

      3. add-sqr-sqrt 10.1%

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

      4. sqrt-unprod 14.8%

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

      5. sqr-neg 14.8%

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

      6. sqrt-unprod 15.1%

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

      7. add-sqr-sqrt 33.7%

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

   8. Applied egg-rr 33.7%

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

      1. unpow-1 33.7%

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

   10. Simplified 33.7%

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

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7.2 \cdot 10^{-50}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 66000:\\ \;\;\;\;\frac{x}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{B}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 37.1% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.7 \cdot 10^{-54}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.35 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{B}{x + 1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.7e-54)
   (/ (- -1.0 x) B)
   (if (<= F 2.35e-8) (/ x (- B)) (/ 1.0 (/ B (+ x 1.0))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.7e-54) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 2.35e-8) {
		tmp = x / -B;
	} else {
		tmp = 1.0 / (B / (x + 1.0));
	}
	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) :: tmp
    if (f <= (-1.7d-54)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 2.35d-8) then
        tmp = x / -b
    else
        tmp = 1.0d0 / (b / (x + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.7e-54) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 2.35e-8) {
		tmp = x / -B;
	} else {
		tmp = 1.0 / (B / (x + 1.0));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.7e-54:
		tmp = (-1.0 - x) / B
	elif F <= 2.35e-8:
		tmp = x / -B
	else:
		tmp = 1.0 / (B / (x + 1.0))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.7e-54)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 2.35e-8)
		tmp = Float64(x / Float64(-B));
	else
		tmp = Float64(1.0 / Float64(B / Float64(x + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.7e-54)
		tmp = (-1.0 - x) / B;
	elseif (F <= 2.35e-8)
		tmp = x / -B;
	else
		tmp = 1.0 / (B / (x + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.7e-54], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 2.35e-8], N[(x / (-B)), $MachinePrecision], N[(1.0 / N[(B / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.69999999999999994e-54

   1. Initial program 63.5%

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

   3. Taylor expanded in F around -inf 92.3%

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

   4. Taylor expanded in B around 0 57.5%

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

      1. mul-1-neg 57.5%

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

      2. distribute-neg-frac2 57.5%

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

   6. Simplified 57.5%

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

   if -1.69999999999999994e-54 < F < 2.3499999999999999e-8

   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 40.8%

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

   4. Taylor expanded in B around 0 24.8%

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

      1. mul-1-neg 24.8%

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

      2. distribute-neg-frac2 24.8%

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

   6. Simplified 24.8%

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

   7. Taylor expanded in x around inf 41.6%

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

      1. associate-*r/ 41.6%

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

      2. neg-mul-1 41.6%

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

   9. Simplified 41.6%

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

   if 2.3499999999999999e-8 < F

   1. Initial program 58.0%

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

   3. Taylor expanded in F around -inf 34.1%

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

   4. Taylor expanded in B around 0 17.3%

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

      1. mul-1-neg 17.3%

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

      2. distribute-neg-frac2 17.3%

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

   6. Simplified 17.3%

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

      1. clear-num 17.2%

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

      2. inv-pow 17.2%

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

      3. add-sqr-sqrt 9.7%

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

      4. sqrt-unprod 14.3%

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

      5. sqr-neg 14.3%

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

      6. sqrt-unprod 14.6%

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

      7. add-sqr-sqrt 32.5%

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

   8. Applied egg-rr 32.5%

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

      1. unpow-1 32.5%

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

   10. Simplified 32.5%

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

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.7 \cdot 10^{-54}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.35 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{B}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 30.8% accurate, 21.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.02 \cdot 10^{-10}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq 2.35 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.02e-10)
   (/ -1.0 B)
   (if (<= F 2.35e-8) (/ x (- B)) (/ (+ x 1.0) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.02e-10) {
		tmp = -1.0 / B;
	} else if (F <= 2.35e-8) {
		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 <= (-1.02d-10)) then
        tmp = (-1.0d0) / b
    else if (f <= 2.35d-8) 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 <= -1.02e-10) {
		tmp = -1.0 / B;
	} else if (F <= 2.35e-8) {
		tmp = x / -B;
	} else {
		tmp = (x + 1.0) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.02e-10:
		tmp = -1.0 / B
	elif F <= 2.35e-8:
		tmp = x / -B
	else:
		tmp = (x + 1.0) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.02e-10)
		tmp = Float64(-1.0 / B);
	elseif (F <= 2.35e-8)
		tmp = Float64(x / Float64(-B));
	else
		tmp = Float64(Float64(x + 1.0) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.02e-10)
		tmp = -1.0 / B;
	elseif (F <= 2.35e-8)
		tmp = x / -B;
	else
		tmp = (x + 1.0) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.02e-10], N[(-1.0 / B), $MachinePrecision], If[LessEqual[F, 2.35e-8], N[(x / (-B)), $MachinePrecision], N[(N[(x + 1.0), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.01999999999999997e-10

   1. Initial program 60.1%

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

   3. Taylor expanded in F around -inf 96.6%

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

   4. Taylor expanded in B around 0 59.7%

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

      1. mul-1-neg 59.7%

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

      2. distribute-neg-frac2 59.7%

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

   6. Simplified 59.7%

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

   7. Taylor expanded in x around 0 40.2%

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

   if -1.01999999999999997e-10 < F < 2.3499999999999999e-8

   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 41.2%

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

   4. Taylor expanded in B around 0 25.4%

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

      1. mul-1-neg 25.4%

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

      2. distribute-neg-frac2 25.4%

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

   6. Simplified 25.4%

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

   7. Taylor expanded in x around inf 40.9%

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

      1. associate-*r/ 40.9%

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

      2. neg-mul-1 40.9%

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

   9. Simplified 40.9%

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

   if 2.3499999999999999e-8 < F

   1. Initial program 58.0%

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

   3. Taylor expanded in F around -inf 34.1%

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

   4. Taylor expanded in B around 0 17.3%

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

      1. mul-1-neg 17.3%

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

      2. distribute-neg-frac2 17.3%

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

   6. Simplified 17.3%

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

      1. add-sqr-sqrt 9.7%

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

      2. sqrt-unprod 14.4%

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

      3. sqr-neg 14.4%

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

      4. sqrt-unprod 14.6%

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

      5. add-sqr-sqrt 32.5%

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

      6. *-un-lft-identity 32.5%

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

   8. Applied egg-rr 32.5%

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

      1. *-lft-identity 32.5%

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

   10. Simplified 32.5%

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

4. Final simplification 38.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.02 \cdot 10^{-10}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq 2.35 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 37.1% accurate, 21.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{-54}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.35 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -5e-54)
   (/ (- -1.0 x) B)
   (if (<= F 2.35e-8) (/ x (- B)) (/ (+ x 1.0) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -5e-54) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 2.35e-8) {
		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 <= (-5d-54)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 2.35d-8) 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 <= -5e-54) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 2.35e-8) {
		tmp = x / -B;
	} else {
		tmp = (x + 1.0) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -5e-54:
		tmp = (-1.0 - x) / B
	elif F <= 2.35e-8:
		tmp = x / -B
	else:
		tmp = (x + 1.0) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -5e-54)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 2.35e-8)
		tmp = Float64(x / Float64(-B));
	else
		tmp = Float64(Float64(x + 1.0) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -5e-54)
		tmp = (-1.0 - x) / B;
	elseif (F <= 2.35e-8)
		tmp = x / -B;
	else
		tmp = (x + 1.0) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -5e-54], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 2.35e-8], N[(x / (-B)), $MachinePrecision], N[(N[(x + 1.0), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.00000000000000015e-54

   1. Initial program 63.5%

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

   3. Taylor expanded in F around -inf 92.3%

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

   4. Taylor expanded in B around 0 57.5%

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

      1. mul-1-neg 57.5%

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

      2. distribute-neg-frac2 57.5%

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

   6. Simplified 57.5%

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

   if -5.00000000000000015e-54 < F < 2.3499999999999999e-8

   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 40.8%

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

   4. Taylor expanded in B around 0 24.8%

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

      1. mul-1-neg 24.8%

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

      2. distribute-neg-frac2 24.8%

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

   6. Simplified 24.8%

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

   7. Taylor expanded in x around inf 41.6%

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

      1. associate-*r/ 41.6%

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

      2. neg-mul-1 41.6%

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

   9. Simplified 41.6%

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

   if 2.3499999999999999e-8 < F

   1. Initial program 58.0%

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

   3. Taylor expanded in F around -inf 34.1%

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

   4. Taylor expanded in B around 0 17.3%

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

      1. mul-1-neg 17.3%

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

      2. distribute-neg-frac2 17.3%

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

   6. Simplified 17.3%

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

      1. add-sqr-sqrt 9.7%

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

      2. sqrt-unprod 14.4%

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

      3. sqr-neg 14.4%

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

      4. sqrt-unprod 14.6%

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

      5. add-sqr-sqrt 32.5%

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

      6. *-un-lft-identity 32.5%

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

   8. Applied egg-rr 32.5%

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

      1. *-lft-identity 32.5%

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

   10. Simplified 32.5%

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

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{-54}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.35 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + 1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 30.8% accurate, 23.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-92} \lor \neg \left(x \leq 8.3 \cdot 10^{-88}\right):\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (or (<= x -1.3e-92) (not (<= x 8.3e-88))) (/ x (- B)) (/ -1.0 B)))

C

double code(double F, double B, double x) {
	double tmp;
	if ((x <= -1.3e-92) || !(x <= 8.3e-88)) {
		tmp = x / -B;
	} else {
		tmp = -1.0 / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.3d-92)) .or. (.not. (x <= 8.3d-88))) then
        tmp = x / -b
    else
        tmp = (-1.0d0) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if ((x <= -1.3e-92) || !(x <= 8.3e-88)) {
		tmp = x / -B;
	} else {
		tmp = -1.0 / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if (x <= -1.3e-92) or not (x <= 8.3e-88):
		tmp = x / -B
	else:
		tmp = -1.0 / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if ((x <= -1.3e-92) || !(x <= 8.3e-88))
		tmp = Float64(x / Float64(-B));
	else
		tmp = Float64(-1.0 / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if ((x <= -1.3e-92) || ~((x <= 8.3e-88)))
		tmp = x / -B;
	else
		tmp = -1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[Or[LessEqual[x, -1.3e-92], N[Not[LessEqual[x, 8.3e-88]], $MachinePrecision]], N[(x / (-B)), $MachinePrecision], N[(-1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3e-92 or 8.3e-88 < x

   1. Initial program 81.7%

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

   3. Taylor expanded in F around -inf 73.2%

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

   4. Taylor expanded in B around 0 40.5%

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

      1. mul-1-neg 40.5%

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

      2. distribute-neg-frac2 40.5%

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

   6. Simplified 40.5%

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

   7. Taylor expanded in x around inf 46.5%

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

      1. associate-*r/ 46.5%

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

      2. neg-mul-1 46.5%

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

   9. Simplified 46.5%

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

   if -1.3e-92 < x < 8.3e-88

   1. Initial program 69.1%

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

   3. Taylor expanded in F around -inf 32.8%

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

   4. Taylor expanded in B around 0 23.7%

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

      1. mul-1-neg 23.7%

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

      2. distribute-neg-frac2 23.7%

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

   6. Simplified 23.7%

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

   7. Taylor expanded in x around 0 23.7%

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

4. Final simplification 36.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-92} \lor \neg \left(x \leq 8.3 \cdot 10^{-88}\right):\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 10.2% accurate, 108.0× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (/ -1.0 B))

C

double code(double F, double B, double x) {
	return -1.0 / B;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = (-1.0d0) / b
end function

Java

public static double code(double F, double B, double x) {
	return -1.0 / B;
}

Python

def code(F, B, x):
	return -1.0 / B

Julia

function code(F, B, x)
	return Float64(-1.0 / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = -1.0 / B;
end

Wolfram

code[F_, B_, x_] := N[(-1.0 / B), $MachinePrecision]

Derivation

1. Initial program 76.1%

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

3. Taylor expanded in F around -inf 55.2%

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

4. Taylor expanded in B around 0 33.0%

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

   1. mul-1-neg 33.0%

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

   2. distribute-neg-frac2 33.0%

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

6. Simplified 33.0%

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

7. Taylor expanded in x around 0 14.1%

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

8. Final simplification 14.1%

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

Reproduce

herbie shell --seed 2024080 
(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))))))