VandenBroeck and Keller, Equation (23)

Percentage Accurate: 77.1% → 99.7%

Time: 25.0s

Alternatives: 27

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: 77.1% 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, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -10000000:\\ \;\;\;\;\frac{\frac{1}{{F}^{2}} + -1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 10^{+107}:\\ \;\;\;\;F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -10000000.0)
     (- (/ (+ (/ 1.0 (pow F 2.0)) -1.0) (sin B)) t_0)
     (if (<= F 1e+107)
       (- (* F (/ (pow (fma x 2.0 (fma F F 2.0)) -0.5) (sin B))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -10000000.0) {
		tmp = (((1.0 / pow(F, 2.0)) + -1.0) / sin(B)) - t_0;
	} else if (F <= 1e+107) {
		tmp = (F * (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) / sin(B))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -10000000.0)
		tmp = Float64(Float64(Float64(Float64(1.0 / (F ^ 2.0)) + -1.0) / sin(B)) - t_0);
	elseif (F <= 1e+107)
		tmp = Float64(Float64(F * Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) / sin(B))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -10000000.0], N[(N[(N[(N[(1.0 / N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1e+107], N[(N[(F * N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1e7

   1. Initial program 52.5%

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

   3. Simplified 71.3%

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

   5. Taylor expanded in x around 0 71.4%

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

      1. associate-*l/ 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. Step-by-step derivation

      1. associate-*r/ 71.5%

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

      2. pow1/2 71.5%

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

      3. fma-define 71.5%

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

      4. inv-pow 71.5%

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

      5. metadata-eval 71.5%

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

      6. pow-pow 71.5%

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

      7. fma-define 71.5%

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

      8. metadata-eval 71.5%

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

      9. metadata-eval 71.5%

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

   9. Applied egg-rr 71.5%

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

   10. Taylor expanded in F around -inf 99.8%

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

   if -1e7 < F < 9.9999999999999997e106

   1. Initial program 97.5%

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

   3. Simplified 99.6%

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

   if 9.9999999999999997e106 < F

   1. Initial program 38.3%

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

   3. Simplified 67.1%

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

   5. Taylor expanded in x around 0 67.1%

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

      1. associate-*l/ 67.1%

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

      2. *-lft-identity 67.1%

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

      3. +-commutative 67.1%

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

      4. unpow2 67.1%

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

      5. fma-undefine 67.1%

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

   7. Simplified 67.1%

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

   8. Taylor expanded in F around inf 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -10000000:\\ \;\;\;\;\frac{\frac{1}{{F}^{2}} + -1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 10^{+107}:\\ \;\;\;\;F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -10000000:\\ \;\;\;\;\frac{\frac{1}{{F}^{2}} + -1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 155000000:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -10000000.0)
     (- (/ (+ (/ 1.0 (pow F 2.0)) -1.0) (sin B)) t_0)
     (if (<= F 155000000.0)
       (- (* F (/ (sqrt (/ 1.0 (fma F F 2.0))) (sin B))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -10000000.0) {
		tmp = (((1.0 / pow(F, 2.0)) + -1.0) / sin(B)) - t_0;
	} else if (F <= 155000000.0) {
		tmp = (F * (sqrt((1.0 / fma(F, F, 2.0))) / sin(B))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -10000000.0)
		tmp = Float64(Float64(Float64(Float64(1.0 / (F ^ 2.0)) + -1.0) / sin(B)) - t_0);
	elseif (F <= 155000000.0)
		tmp = Float64(Float64(F * Float64(sqrt(Float64(1.0 / fma(F, F, 2.0))) / sin(B))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -10000000.0], N[(N[(N[(N[(1.0 / N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 155000000.0], N[(N[(F * N[(N[Sqrt[N[(1.0 / N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1e7

   1. Initial program 52.5%

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

   3. Simplified 71.3%

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

   5. Taylor expanded in x around 0 71.4%

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

      1. associate-*l/ 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. Step-by-step derivation

      1. associate-*r/ 71.5%

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

      2. pow1/2 71.5%

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

      3. fma-define 71.5%

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

      4. inv-pow 71.5%

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

      5. metadata-eval 71.5%

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

      6. pow-pow 71.5%

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

      7. fma-define 71.5%

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

      8. metadata-eval 71.5%

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

      9. metadata-eval 71.5%

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

   9. Applied egg-rr 71.5%

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

   10. Taylor expanded in F around -inf 99.8%

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

   if -1e7 < F < 1.55e8

   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 99.5%

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

      1. associate-*l/ 99.6%

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

      2. *-lft-identity 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. fma-undefine 99.6%

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

   7. Simplified 99.6%

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

   if 1.55e8 < F

   1. Initial program 54.7%

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

   3. Simplified 78.1%

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

   5. Taylor expanded in x around 0 78.0%

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

      1. associate-*l/ 78.0%

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

      2. *-lft-identity 78.0%

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

      3. +-commutative 78.0%

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

      4. unpow2 78.0%

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

      5. fma-undefine 78.0%

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

   7. Simplified 78.0%

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

   8. Taylor expanded in F around inf 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -10000000:\\ \;\;\;\;\frac{\frac{1}{{F}^{2}} + -1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 155000000:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1 \cdot 10^{+130}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 10^{+107}:\\ \;\;\;\;\frac{F \cdot {\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1e+130)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1e+107)
       (- (/ (* F (pow (fma F F 2.0) -0.5)) (sin B)) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1e+130) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1e+107) {
		tmp = ((F * pow(fma(F, F, 2.0), -0.5)) / sin(B)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1e+130)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1e+107)
		tmp = Float64(Float64(Float64(F * (fma(F, F, 2.0) ^ -0.5)) / sin(B)) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1e+130], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1e+107], N[(N[(N[(F * N[Power[N[(F * F + 2.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.0000000000000001e130

   1. Initial program 38.8%

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

   3. Simplified 58.1%

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

   5. Taylor expanded in x around 0 58.1%

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

      1. associate-*l/ 58.1%

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

      2. *-lft-identity 58.1%

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

      3. +-commutative 58.1%

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

      4. unpow2 58.1%

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

      5. fma-undefine 58.1%

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

   7. Simplified 58.1%

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

   8. Taylor expanded in F around -inf 99.8%

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

   if -1.0000000000000001e130 < F < 9.9999999999999997e106

   1. Initial program 95.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 99.5%

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

      1. associate-*l/ 99.6%

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

      2. *-lft-identity 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. fma-undefine 99.6%

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

   7. Simplified 99.6%

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

      1. associate-*r/ 99.6%

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

      2. pow1/2 99.6%

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

      3. fma-define 99.6%

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

      4. inv-pow 99.6%

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

      5. metadata-eval 99.6%

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

      6. pow-pow 99.6%

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

      7. fma-define 99.6%

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

      8. metadata-eval 99.6%

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

      9. metadata-eval 99.6%

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

   9. Applied egg-rr 99.6%

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

   if 9.9999999999999997e106 < F

   1. Initial program 38.3%

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

   3. Simplified 67.1%

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

   5. Taylor expanded in x around 0 67.1%

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

      1. associate-*l/ 67.1%

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

      2. *-lft-identity 67.1%

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

      3. +-commutative 67.1%

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

      4. unpow2 67.1%

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

      5. fma-undefine 67.1%

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

   7. Simplified 67.1%

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

   8. Taylor expanded in F around inf 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{+130}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 10^{+107}:\\ \;\;\;\;\frac{F \cdot {\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -10000000:\\ \;\;\;\;\frac{\frac{1}{{F}^{2}} + -1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 2000:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -10000000.0)
     (- (/ (+ (/ 1.0 (pow F 2.0)) -1.0) (sin B)) t_0)
     (if (<= F 2000.0)
       (- (/ (/ F (sqrt (fma F F 2.0))) (sin B)) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -10000000.0) {
		tmp = (((1.0 / pow(F, 2.0)) + -1.0) / sin(B)) - t_0;
	} else if (F <= 2000.0) {
		tmp = ((F / sqrt(fma(F, F, 2.0))) / sin(B)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -10000000.0)
		tmp = Float64(Float64(Float64(Float64(1.0 / (F ^ 2.0)) + -1.0) / sin(B)) - t_0);
	elseif (F <= 2000.0)
		tmp = Float64(Float64(Float64(F / sqrt(fma(F, F, 2.0))) / sin(B)) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -10000000.0], N[(N[(N[(N[(1.0 / N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 2000.0], 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[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1e7

   1. Initial program 52.5%

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

   3. Simplified 71.3%

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

   5. Taylor expanded in x around 0 71.4%

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

      1. associate-*l/ 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. Step-by-step derivation

      1. associate-*r/ 71.5%

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

      2. pow1/2 71.5%

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

      3. fma-define 71.5%

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

      4. inv-pow 71.5%

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

      5. metadata-eval 71.5%

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

      6. pow-pow 71.5%

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

      7. fma-define 71.5%

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

      8. metadata-eval 71.5%

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

      9. metadata-eval 71.5%

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

   9. Applied egg-rr 71.5%

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

   10. Taylor expanded in F around -inf 99.8%

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

   if -1e7 < F < 2e3

   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 99.5%

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

      1. associate-*l/ 99.6%

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

      2. *-lft-identity 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. fma-undefine 99.6%

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

   7. Simplified 99.6%

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

      1. associate-*r/ 99.6%

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

      2. pow1/2 99.6%

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

      3. fma-define 99.6%

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

      4. inv-pow 99.6%

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

      5. metadata-eval 99.6%

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

      6. pow-pow 99.6%

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

      7. fma-define 99.6%

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

      8. metadata-eval 99.6%

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

      9. metadata-eval 99.6%

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

   9. Applied egg-rr 99.6%

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

       1. add-sqr-sqrt 99.5%

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

       2. unpow-prod-down 99.5%

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

   11. Applied egg-rr 99.5%

       \[\leadsto \frac{F \cdot \color{blue}{\left({\left(\sqrt{\mathsf{fma}\left(F, F, 2\right)}\right)}^{-0.5} \cdot {\left(\sqrt{\mathsf{fma}\left(F, F, 2\right)}\right)}^{-0.5}\right)}}{\sin B} - \frac{x}{\tan B} \]
   12. Step-by-step derivation

       1. pow-sqr 99.5%

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

       2. metadata-eval 99.5%

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

       3. unpow-1 99.5%

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

   13. Simplified 99.5%

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

       1. un-div-inv 99.6%

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

   15. Applied egg-rr 99.6%

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

   if 2e3 < F

   1. Initial program 54.7%

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

   3. Simplified 78.1%

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

   5. Taylor expanded in x around 0 78.0%

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

      1. associate-*l/ 78.0%

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

      2. *-lft-identity 78.0%

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

      3. +-commutative 78.0%

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

      4. unpow2 78.0%

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

      5. fma-undefine 78.0%

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

   7. Simplified 78.0%

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

   8. Taylor expanded in F around inf 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -10000000:\\ \;\;\;\;\frac{\frac{1}{{F}^{2}} + -1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2000:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -12000:\\ \;\;\;\;\frac{\frac{1}{{F}^{2}} + -1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 41000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + 2 \cdot x\right)}^{-0.5} + x \cdot \frac{-1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -12000.0)
     (- (/ (+ (/ 1.0 (pow F 2.0)) -1.0) (sin B)) t_0)
     (if (<= F 41000000.0)
       (+
        (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* 2.0 x)) -0.5))
        (* x (/ -1.0 (tan B))))
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -12000.0) {
		tmp = (((1.0 / pow(F, 2.0)) + -1.0) / sin(B)) - t_0;
	} else if (F <= 41000000.0) {
		tmp = ((F / sin(B)) * pow(((2.0 + (F * F)) + (2.0 * x)), -0.5)) + (x * (-1.0 / tan(B)));
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-12000.0d0)) then
        tmp = (((1.0d0 / (f ** 2.0d0)) + (-1.0d0)) / sin(b)) - t_0
    else if (f <= 41000000.0d0) then
        tmp = ((f / sin(b)) * (((2.0d0 + (f * f)) + (2.0d0 * x)) ** (-0.5d0))) + (x * ((-1.0d0) / tan(b)))
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -12000.0) {
		tmp = (((1.0 / Math.pow(F, 2.0)) + -1.0) / Math.sin(B)) - t_0;
	} else if (F <= 41000000.0) {
		tmp = ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (2.0 * x)), -0.5)) + (x * (-1.0 / Math.tan(B)));
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -12000.0:
		tmp = (((1.0 / math.pow(F, 2.0)) + -1.0) / math.sin(B)) - t_0
	elif F <= 41000000.0:
		tmp = ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (2.0 * x)), -0.5)) + (x * (-1.0 / math.tan(B)))
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -12000.0)
		tmp = Float64(Float64(Float64(Float64(1.0 / (F ^ 2.0)) + -1.0) / sin(B)) - t_0);
	elseif (F <= 41000000.0)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(2.0 * x)) ^ -0.5)) + Float64(x * Float64(-1.0 / tan(B))));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -12000.0)
		tmp = (((1.0 / (F ^ 2.0)) + -1.0) / sin(B)) - t_0;
	elseif (F <= 41000000.0)
		tmp = ((F / sin(B)) * (((2.0 + (F * F)) + (2.0 * x)) ^ -0.5)) + (x * (-1.0 / tan(B)));
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -12000.0], N[(N[(N[(N[(1.0 / N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 41000000.0], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] + N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -12000

   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 72.2%

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

   5. Taylor expanded in x around 0 72.3%

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

      1. associate-*l/ 72.3%

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

      2. *-lft-identity 72.3%

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

      3. +-commutative 72.3%

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

      4. unpow2 72.3%

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

      5. fma-undefine 72.3%

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

   7. Simplified 72.3%

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

      1. associate-*r/ 72.4%

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

      2. pow1/2 72.4%

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

      3. fma-define 72.4%

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

      4. inv-pow 72.4%

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

      5. metadata-eval 72.4%

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

      6. pow-pow 72.4%

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

      7. fma-define 72.4%

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

      8. metadata-eval 72.4%

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

      9. metadata-eval 72.4%

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

   9. Applied egg-rr 72.4%

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

   10. Taylor expanded in F around -inf 99.8%

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

   if -12000 < F < 4.1e7

   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

   if 4.1e7 < F

   1. Initial program 54.7%

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

   3. Simplified 78.1%

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

   5. Taylor expanded in x around 0 78.0%

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

      1. associate-*l/ 78.0%

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

      2. *-lft-identity 78.0%

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

      3. +-commutative 78.0%

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

      4. unpow2 78.0%

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

      5. fma-undefine 78.0%

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

   7. Simplified 78.0%

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

   8. Taylor expanded in F around inf 99.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -12000:\\ \;\;\;\;\frac{\frac{1}{{F}^{2}} + -1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 41000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + 2 \cdot x\right)}^{-0.5} + x \cdot \frac{-1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -55000:\\ \;\;\;\;\frac{\frac{1}{{F}^{2}} + -1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 100000000:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + 2 \cdot x\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -55000.0)
     (- (/ (+ (/ 1.0 (pow F 2.0)) -1.0) (sin B)) t_0)
     (if (<= F 100000000.0)
       (+
        (/ -1.0 (/ (tan B) x))
        (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* 2.0 x)) -0.5)))
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -55000.0) {
		tmp = (((1.0 / pow(F, 2.0)) + -1.0) / sin(B)) - t_0;
	} else if (F <= 100000000.0) {
		tmp = (-1.0 / (tan(B) / x)) + ((F / sin(B)) * pow(((2.0 + (F * F)) + (2.0 * x)), -0.5));
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-55000.0d0)) then
        tmp = (((1.0d0 / (f ** 2.0d0)) + (-1.0d0)) / sin(b)) - t_0
    else if (f <= 100000000.0d0) then
        tmp = ((-1.0d0) / (tan(b) / x)) + ((f / sin(b)) * (((2.0d0 + (f * f)) + (2.0d0 * x)) ** (-0.5d0)))
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -55000.0) {
		tmp = (((1.0 / Math.pow(F, 2.0)) + -1.0) / Math.sin(B)) - t_0;
	} else if (F <= 100000000.0) {
		tmp = (-1.0 / (Math.tan(B) / x)) + ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (2.0 * x)), -0.5));
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -55000.0:
		tmp = (((1.0 / math.pow(F, 2.0)) + -1.0) / math.sin(B)) - t_0
	elif F <= 100000000.0:
		tmp = (-1.0 / (math.tan(B) / x)) + ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (2.0 * x)), -0.5))
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -55000.0)
		tmp = Float64(Float64(Float64(Float64(1.0 / (F ^ 2.0)) + -1.0) / sin(B)) - t_0);
	elseif (F <= 100000000.0)
		tmp = Float64(Float64(-1.0 / Float64(tan(B) / x)) + Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(2.0 * x)) ^ -0.5)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -55000.0)
		tmp = (((1.0 / (F ^ 2.0)) + -1.0) / sin(B)) - t_0;
	elseif (F <= 100000000.0)
		tmp = (-1.0 / (tan(B) / x)) + ((F / sin(B)) * (((2.0 + (F * F)) + (2.0 * x)) ^ -0.5));
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -55000.0], N[(N[(N[(N[(1.0 / N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 100000000.0], 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[(2.0 * x), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -55000

   1. Initial program 53.3%

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

   3. Simplified 71.8%

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

   5. Taylor expanded in x around 0 71.9%

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

      1. associate-*l/ 71.8%

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

      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. Step-by-step derivation

      1. associate-*r/ 72.0%

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

      2. pow1/2 72.0%

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

      3. fma-define 72.0%

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

      4. inv-pow 72.0%

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

      5. metadata-eval 72.0%

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

      6. pow-pow 72.0%

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

      7. fma-define 72.0%

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

      8. metadata-eval 72.0%

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

      9. metadata-eval 72.0%

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

   9. Applied egg-rr 72.0%

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

   10. Taylor expanded in F around -inf 99.8%

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

   if -55000 < F < 1e8

   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 99.6%

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

      2. clear-num 99.5%

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

   4. Applied egg-rr 99.5%

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

   if 1e8 < F

   1. Initial program 54.7%

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

   3. Simplified 78.1%

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

   5. Taylor expanded in x around 0 78.0%

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

      1. associate-*l/ 78.0%

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

      2. *-lft-identity 78.0%

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

      3. +-commutative 78.0%

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

      4. unpow2 78.0%

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

      5. fma-undefine 78.0%

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

   7. Simplified 78.0%

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

   8. Taylor expanded in F around inf 99.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -55000:\\ \;\;\;\;\frac{\frac{1}{{F}^{2}} + -1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 100000000:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + 2 \cdot x\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 0.045:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.4)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 0.045)
       (- (* F (/ (sqrt 0.5) (sin B))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.4) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 0.045) {
		tmp = (F * (sqrt(0.5) / sin(B))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.4d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 0.045d0) then
        tmp = (f * (sqrt(0.5d0) / sin(b))) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.4) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 0.045) {
		tmp = (F * (Math.sqrt(0.5) / Math.sin(B))) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.4:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 0.045:
		tmp = (F * (math.sqrt(0.5) / math.sin(B))) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.4)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 0.045)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / sin(B))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.4)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 0.045)
		tmp = (F * (sqrt(0.5) / sin(B))) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.4], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.045], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.3999999999999999

   1. Initial program 54.7%

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

   3. Simplified 72.6%

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

   5. Taylor expanded in x around 0 72.7%

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

      1. associate-*l/ 72.7%

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

      2. *-lft-identity 72.7%

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

      3. +-commutative 72.7%

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

      4. unpow2 72.7%

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

      5. fma-undefine 72.7%

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

   7. Simplified 72.7%

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

   8. Taylor expanded in F around -inf 99.1%

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

   if -1.3999999999999999 < F < 0.044999999999999998

   1. Initial program 99.4%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 99.6%

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

      1. associate-*l/ 99.7%

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

      2. *-lft-identity 99.7%

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

      3. +-commutative 99.7%

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

      4. unpow2 99.7%

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

      5. fma-undefine 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in F around 0 97.6%

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

   if 0.044999999999999998 < F

   1. Initial program 57.2%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in x around 0 79.2%

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

      1. associate-*l/ 79.2%

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

      2. *-lft-identity 79.2%

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

      3. +-commutative 79.2%

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

      4. unpow2 79.2%

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

      5. fma-undefine 79.2%

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

   7. Simplified 79.2%

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

   8. Taylor expanded in F around inf 97.2%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.045:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{\frac{1}{{F}^{2}} + -1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 0.045:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.4)
     (- (/ (+ (/ 1.0 (pow F 2.0)) -1.0) (sin B)) t_0)
     (if (<= F 0.045)
       (- (* F (/ (sqrt 0.5) (sin B))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.4) {
		tmp = (((1.0 / pow(F, 2.0)) + -1.0) / sin(B)) - t_0;
	} else if (F <= 0.045) {
		tmp = (F * (sqrt(0.5) / sin(B))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.4d0)) then
        tmp = (((1.0d0 / (f ** 2.0d0)) + (-1.0d0)) / sin(b)) - t_0
    else if (f <= 0.045d0) then
        tmp = (f * (sqrt(0.5d0) / sin(b))) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.4) {
		tmp = (((1.0 / Math.pow(F, 2.0)) + -1.0) / Math.sin(B)) - t_0;
	} else if (F <= 0.045) {
		tmp = (F * (Math.sqrt(0.5) / Math.sin(B))) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.4:
		tmp = (((1.0 / math.pow(F, 2.0)) + -1.0) / math.sin(B)) - t_0
	elif F <= 0.045:
		tmp = (F * (math.sqrt(0.5) / math.sin(B))) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.4)
		tmp = Float64(Float64(Float64(Float64(1.0 / (F ^ 2.0)) + -1.0) / sin(B)) - t_0);
	elseif (F <= 0.045)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / sin(B))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.4)
		tmp = (((1.0 / (F ^ 2.0)) + -1.0) / sin(B)) - t_0;
	elseif (F <= 0.045)
		tmp = (F * (sqrt(0.5) / sin(B))) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.4], N[(N[(N[(N[(1.0 / N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.045], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.3999999999999999

   1. Initial program 54.7%

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

   3. Simplified 72.6%

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

   5. Taylor expanded in x around 0 72.7%

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

      1. associate-*l/ 72.7%

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

      2. *-lft-identity 72.7%

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

      3. +-commutative 72.7%

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

      4. unpow2 72.7%

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

      5. fma-undefine 72.7%

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

   7. Simplified 72.7%

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

      1. associate-*r/ 72.8%

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

      2. pow1/2 72.8%

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

      3. fma-define 72.8%

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

      4. inv-pow 72.8%

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

      5. metadata-eval 72.8%

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

      6. pow-pow 72.8%

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

      7. fma-define 72.8%

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

      8. metadata-eval 72.8%

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

      9. metadata-eval 72.8%

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

   9. Applied egg-rr 72.8%

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

   10. Taylor expanded in F around -inf 99.7%

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

   if -1.3999999999999999 < F < 0.044999999999999998

   1. Initial program 99.4%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 99.6%

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

      1. associate-*l/ 99.7%

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

      2. *-lft-identity 99.7%

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

      3. +-commutative 99.7%

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

      4. unpow2 99.7%

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

      5. fma-undefine 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in F around 0 97.6%

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

   if 0.044999999999999998 < F

   1. Initial program 57.2%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in x around 0 79.2%

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

      1. associate-*l/ 79.2%

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

      2. *-lft-identity 79.2%

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

      3. +-commutative 79.2%

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

      4. unpow2 79.2%

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

      5. fma-undefine 79.2%

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

   7. Simplified 79.2%

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

   8. Taylor expanded in F around inf 97.2%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{\frac{1}{{F}^{2}} + -1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.045:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 89.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + 2 \cdot x\right)}^{-0.5} - \frac{x}{B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -15500:\\ \;\;\;\;\frac{-1}{\sin B} - t\_1\\ \mathbf{elif}\;F \leq -4 \cdot 10^{-123}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 9.5 \cdot 10^{-111}:\\ \;\;\;\;\frac{x \cdot \left(-\cos B\right)}{\sin B}\\ \mathbf{elif}\;F \leq 890000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0
         (-
          (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* 2.0 x)) -0.5))
          (/ x B)))
        (t_1 (/ x (tan B))))
   (if (<= F -15500.0)
     (- (/ -1.0 (sin B)) t_1)
     (if (<= F -4e-123)
       t_0
       (if (<= F 9.5e-111)
         (/ (* x (- (cos B))) (sin B))
         (if (<= F 890000.0) t_0 (- (/ 1.0 (sin B)) t_1)))))))

C

double code(double F, double B, double x) {
	double t_0 = ((F / sin(B)) * pow(((2.0 + (F * F)) + (2.0 * x)), -0.5)) - (x / B);
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -15500.0) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= -4e-123) {
		tmp = t_0;
	} else if (F <= 9.5e-111) {
		tmp = (x * -cos(B)) / sin(B);
	} else if (F <= 890000.0) {
		tmp = t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_1;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((f / sin(b)) * (((2.0d0 + (f * f)) + (2.0d0 * x)) ** (-0.5d0))) - (x / b)
    t_1 = x / tan(b)
    if (f <= (-15500.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_1
    else if (f <= (-4d-123)) then
        tmp = t_0
    else if (f <= 9.5d-111) then
        tmp = (x * -cos(b)) / sin(b)
    else if (f <= 890000.0d0) then
        tmp = t_0
    else
        tmp = (1.0d0 / sin(b)) - t_1
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (2.0 * x)), -0.5)) - (x / B);
	double t_1 = x / Math.tan(B);
	double tmp;
	if (F <= -15500.0) {
		tmp = (-1.0 / Math.sin(B)) - t_1;
	} else if (F <= -4e-123) {
		tmp = t_0;
	} else if (F <= 9.5e-111) {
		tmp = (x * -Math.cos(B)) / Math.sin(B);
	} else if (F <= 890000.0) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (2.0 * x)), -0.5)) - (x / B)
	t_1 = x / math.tan(B)
	tmp = 0
	if F <= -15500.0:
		tmp = (-1.0 / math.sin(B)) - t_1
	elif F <= -4e-123:
		tmp = t_0
	elif F <= 9.5e-111:
		tmp = (x * -math.cos(B)) / math.sin(B)
	elif F <= 890000.0:
		tmp = t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(2.0 * x)) ^ -0.5)) - Float64(x / B))
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -15500.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= -4e-123)
		tmp = t_0;
	elseif (F <= 9.5e-111)
		tmp = Float64(Float64(x * Float64(-cos(B))) / sin(B));
	elseif (F <= 890000.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = ((F / sin(B)) * (((2.0 + (F * F)) + (2.0 * x)) ^ -0.5)) - (x / B);
	t_1 = x / tan(B);
	tmp = 0.0;
	if (F <= -15500.0)
		tmp = (-1.0 / sin(B)) - t_1;
	elseif (F <= -4e-123)
		tmp = t_0;
	elseif (F <= 9.5e-111)
		tmp = (x * -cos(B)) / sin(B);
	elseif (F <= 890000.0)
		tmp = t_0;
	else
		tmp = (1.0 / sin(B)) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -15500.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, -4e-123], t$95$0, If[LessEqual[F, 9.5e-111], N[(N[(x * (-N[Cos[B], $MachinePrecision])), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 890000.0], t$95$0, N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -15500

   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 72.2%

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

   5. Taylor expanded in x around 0 72.3%

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

      1. associate-*l/ 72.3%

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

      2. *-lft-identity 72.3%

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

      3. +-commutative 72.3%

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

      4. unpow2 72.3%

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

      5. fma-undefine 72.3%

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

   7. Simplified 72.3%

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

   8. Taylor expanded in F around -inf 99.7%

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

   if -15500 < F < -4.0000000000000002e-123 or 9.4999999999999995e-111 < F < 8.9e5

   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 86.5%

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

   if -4.0000000000000002e-123 < F < 9.4999999999999995e-111

   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 34.8%

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

   4. Taylor expanded in x around inf 88.2%

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

      1. associate-*r/ 88.2%

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

      2. neg-mul-1 88.2%

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

      3. distribute-rgt-neg-in 88.2%

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

   6. Simplified 88.2%

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

   if 8.9e5 < F

   1. Initial program 54.7%

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

   3. Simplified 78.1%

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

   5. Taylor expanded in x around 0 78.0%

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

      1. associate-*l/ 78.0%

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

      2. *-lft-identity 78.0%

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

      3. +-commutative 78.0%

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

      4. unpow2 78.0%

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

      5. fma-undefine 78.0%

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

   7. Simplified 78.0%

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

   8. Taylor expanded in F around inf 99.7%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -15500:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -4 \cdot 10^{-123}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + 2 \cdot x\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 9.5 \cdot 10^{-111}:\\ \;\;\;\;\frac{x \cdot \left(-\cos B\right)}{\sin B}\\ \mathbf{elif}\;F \leq 890000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + 2 \cdot x\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 89.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left(2 + F \cdot F\right) + 2 \cdot x\right)}^{-0.5}\\ t_1 := \frac{x}{\tan B}\\ t_2 := \frac{1}{\sin B}\\ \mathbf{if}\;F \leq -8200:\\ \;\;\;\;\frac{-1}{\sin B} - t\_1\\ \mathbf{elif}\;F \leq -3.9 \cdot 10^{-121}:\\ \;\;\;\;t\_0 \cdot \left(F \cdot t\_2\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.2 \cdot 10^{-111}:\\ \;\;\;\;\frac{x \cdot \left(-\cos B\right)}{\sin B}\\ \mathbf{elif}\;F \leq 22500:\\ \;\;\;\;\frac{F}{\sin B} \cdot t\_0 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_2 - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (pow (+ (+ 2.0 (* F F)) (* 2.0 x)) -0.5))
        (t_1 (/ x (tan B)))
        (t_2 (/ 1.0 (sin B))))
   (if (<= F -8200.0)
     (- (/ -1.0 (sin B)) t_1)
     (if (<= F -3.9e-121)
       (- (* t_0 (* F t_2)) (/ x B))
       (if (<= F 2.2e-111)
         (/ (* x (- (cos B))) (sin B))
         (if (<= F 22500.0) (- (* (/ F (sin B)) t_0) (/ x B)) (- t_2 t_1)))))))

C

double code(double F, double B, double x) {
	double t_0 = pow(((2.0 + (F * F)) + (2.0 * x)), -0.5);
	double t_1 = x / tan(B);
	double t_2 = 1.0 / sin(B);
	double tmp;
	if (F <= -8200.0) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= -3.9e-121) {
		tmp = (t_0 * (F * t_2)) - (x / B);
	} else if (F <= 2.2e-111) {
		tmp = (x * -cos(B)) / sin(B);
	} else if (F <= 22500.0) {
		tmp = ((F / sin(B)) * t_0) - (x / B);
	} else {
		tmp = t_2 - t_1;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = ((2.0d0 + (f * f)) + (2.0d0 * x)) ** (-0.5d0)
    t_1 = x / tan(b)
    t_2 = 1.0d0 / sin(b)
    if (f <= (-8200.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_1
    else if (f <= (-3.9d-121)) then
        tmp = (t_0 * (f * t_2)) - (x / b)
    else if (f <= 2.2d-111) then
        tmp = (x * -cos(b)) / sin(b)
    else if (f <= 22500.0d0) then
        tmp = ((f / sin(b)) * t_0) - (x / b)
    else
        tmp = t_2 - t_1
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = Math.pow(((2.0 + (F * F)) + (2.0 * x)), -0.5);
	double t_1 = x / Math.tan(B);
	double t_2 = 1.0 / Math.sin(B);
	double tmp;
	if (F <= -8200.0) {
		tmp = (-1.0 / Math.sin(B)) - t_1;
	} else if (F <= -3.9e-121) {
		tmp = (t_0 * (F * t_2)) - (x / B);
	} else if (F <= 2.2e-111) {
		tmp = (x * -Math.cos(B)) / Math.sin(B);
	} else if (F <= 22500.0) {
		tmp = ((F / Math.sin(B)) * t_0) - (x / B);
	} else {
		tmp = t_2 - t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = math.pow(((2.0 + (F * F)) + (2.0 * x)), -0.5)
	t_1 = x / math.tan(B)
	t_2 = 1.0 / math.sin(B)
	tmp = 0
	if F <= -8200.0:
		tmp = (-1.0 / math.sin(B)) - t_1
	elif F <= -3.9e-121:
		tmp = (t_0 * (F * t_2)) - (x / B)
	elif F <= 2.2e-111:
		tmp = (x * -math.cos(B)) / math.sin(B)
	elif F <= 22500.0:
		tmp = ((F / math.sin(B)) * t_0) - (x / B)
	else:
		tmp = t_2 - t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(2.0 + Float64(F * F)) + Float64(2.0 * x)) ^ -0.5
	t_1 = Float64(x / tan(B))
	t_2 = Float64(1.0 / sin(B))
	tmp = 0.0
	if (F <= -8200.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= -3.9e-121)
		tmp = Float64(Float64(t_0 * Float64(F * t_2)) - Float64(x / B));
	elseif (F <= 2.2e-111)
		tmp = Float64(Float64(x * Float64(-cos(B))) / sin(B));
	elseif (F <= 22500.0)
		tmp = Float64(Float64(Float64(F / sin(B)) * t_0) - Float64(x / B));
	else
		tmp = Float64(t_2 - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = ((2.0 + (F * F)) + (2.0 * x)) ^ -0.5;
	t_1 = x / tan(B);
	t_2 = 1.0 / sin(B);
	tmp = 0.0;
	if (F <= -8200.0)
		tmp = (-1.0 / sin(B)) - t_1;
	elseif (F <= -3.9e-121)
		tmp = (t_0 * (F * t_2)) - (x / B);
	elseif (F <= 2.2e-111)
		tmp = (x * -cos(B)) / sin(B);
	elseif (F <= 22500.0)
		tmp = ((F / sin(B)) * t_0) - (x / B);
	else
		tmp = t_2 - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -8200.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, -3.9e-121], N[(N[(t$95$0 * N[(F * t$95$2), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.2e-111], N[(N[(x * (-N[Cos[B], $MachinePrecision])), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 22500.0], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(t$95$2 - t$95$1), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -8200

   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 72.2%

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

   5. Taylor expanded in x around 0 72.3%

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

      1. associate-*l/ 72.3%

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

      2. *-lft-identity 72.3%

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

      3. +-commutative 72.3%

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

      4. unpow2 72.3%

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

      5. fma-undefine 72.3%

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

   7. Simplified 72.3%

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

   8. Taylor expanded in F around -inf 99.7%

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

   if -8200 < F < -3.9e-121

   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 88.1%

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

      1. div-inv 88.1%

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

   5. Applied egg-rr 88.1%

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

   if -3.9e-121 < F < 2.2e-111

   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 34.8%

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

   4. Taylor expanded in x around inf 88.2%

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

      1. associate-*r/ 88.2%

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

      2. neg-mul-1 88.2%

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

      3. distribute-rgt-neg-in 88.2%

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

   6. Simplified 88.2%

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

   if 2.2e-111 < F < 22500

   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 84.6%

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

   if 22500 < F

   1. Initial program 54.7%

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

   3. Simplified 78.1%

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

   5. Taylor expanded in x around 0 78.0%

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

      1. associate-*l/ 78.0%

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

      2. *-lft-identity 78.0%

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

      3. +-commutative 78.0%

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

      4. unpow2 78.0%

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

      5. fma-undefine 78.0%

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

   7. Simplified 78.0%

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

   8. Taylor expanded in F around inf 99.7%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -8200:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -3.9 \cdot 10^{-121}:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + 2 \cdot x\right)}^{-0.5} \cdot \left(F \cdot \frac{1}{\sin B}\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.2 \cdot 10^{-111}:\\ \;\;\;\;\frac{x \cdot \left(-\cos B\right)}{\sin B}\\ \mathbf{elif}\;F \leq 22500:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + 2 \cdot x\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 89.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + 2 \cdot x}} - \frac{x}{B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.0058:\\ \;\;\;\;\frac{-1}{\sin B} - t\_1\\ \mathbf{elif}\;F \leq -1.4 \cdot 10^{-119}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 2.9 \cdot 10^{-111}:\\ \;\;\;\;\frac{x \cdot \left(-\cos B\right)}{\sin B}\\ \mathbf{elif}\;F \leq 0.003:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (* (/ F (sin B)) (sqrt (/ 1.0 (+ 2.0 (* 2.0 x))))) (/ x B)))
        (t_1 (/ x (tan B))))
   (if (<= F -0.0058)
     (- (/ -1.0 (sin B)) t_1)
     (if (<= F -1.4e-119)
       t_0
       (if (<= F 2.9e-111)
         (/ (* x (- (cos B))) (sin B))
         (if (<= F 0.003) t_0 (- (/ 1.0 (sin B)) t_1)))))))

C

double code(double F, double B, double x) {
	double t_0 = ((F / sin(B)) * sqrt((1.0 / (2.0 + (2.0 * x))))) - (x / B);
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -0.0058) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= -1.4e-119) {
		tmp = t_0;
	} else if (F <= 2.9e-111) {
		tmp = (x * -cos(B)) / sin(B);
	} else if (F <= 0.003) {
		tmp = t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_1;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((f / sin(b)) * sqrt((1.0d0 / (2.0d0 + (2.0d0 * x))))) - (x / b)
    t_1 = x / tan(b)
    if (f <= (-0.0058d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_1
    else if (f <= (-1.4d-119)) then
        tmp = t_0
    else if (f <= 2.9d-111) then
        tmp = (x * -cos(b)) / sin(b)
    else if (f <= 0.003d0) then
        tmp = t_0
    else
        tmp = (1.0d0 / sin(b)) - t_1
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = ((F / Math.sin(B)) * Math.sqrt((1.0 / (2.0 + (2.0 * x))))) - (x / B);
	double t_1 = x / Math.tan(B);
	double tmp;
	if (F <= -0.0058) {
		tmp = (-1.0 / Math.sin(B)) - t_1;
	} else if (F <= -1.4e-119) {
		tmp = t_0;
	} else if (F <= 2.9e-111) {
		tmp = (x * -Math.cos(B)) / Math.sin(B);
	} else if (F <= 0.003) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = ((F / math.sin(B)) * math.sqrt((1.0 / (2.0 + (2.0 * x))))) - (x / B)
	t_1 = x / math.tan(B)
	tmp = 0
	if F <= -0.0058:
		tmp = (-1.0 / math.sin(B)) - t_1
	elif F <= -1.4e-119:
		tmp = t_0
	elif F <= 2.9e-111:
		tmp = (x * -math.cos(B)) / math.sin(B)
	elif F <= 0.003:
		tmp = t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(Float64(F / sin(B)) * sqrt(Float64(1.0 / Float64(2.0 + Float64(2.0 * x))))) - Float64(x / B))
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -0.0058)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= -1.4e-119)
		tmp = t_0;
	elseif (F <= 2.9e-111)
		tmp = Float64(Float64(x * Float64(-cos(B))) / sin(B));
	elseif (F <= 0.003)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = ((F / sin(B)) * sqrt((1.0 / (2.0 + (2.0 * x))))) - (x / B);
	t_1 = x / tan(B);
	tmp = 0.0;
	if (F <= -0.0058)
		tmp = (-1.0 / sin(B)) - t_1;
	elseif (F <= -1.4e-119)
		tmp = t_0;
	elseif (F <= 2.9e-111)
		tmp = (x * -cos(B)) / sin(B);
	elseif (F <= 0.003)
		tmp = t_0;
	else
		tmp = (1.0 / sin(B)) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 + N[(2.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.0058], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, -1.4e-119], t$95$0, If[LessEqual[F, 2.9e-111], N[(N[(x * (-N[Cos[B], $MachinePrecision])), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.003], t$95$0, N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.0058

   1. Initial program 56.0%

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

   3. Simplified 73.5%

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

   5. Taylor expanded in x around 0 73.5%

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

      1. associate-*l/ 73.5%

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

      2. *-lft-identity 73.5%

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

      3. +-commutative 73.5%

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

      4. unpow2 73.5%

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

      5. fma-undefine 73.5%

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

   7. Simplified 73.5%

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

   8. Taylor expanded in F around -inf 97.8%

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

   if -0.0058 < F < -1.4e-119 or 2.90000000000000002e-111 < F < 0.0030000000000000001

   1. Initial program 99.3%

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

   3. Taylor expanded in B around 0 86.6%

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

   4. Taylor expanded in F around 0 83.7%

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

   if -1.4e-119 < F < 2.90000000000000002e-111

   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 34.8%

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

   4. Taylor expanded in x around inf 88.2%

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

      1. associate-*r/ 88.2%

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

      2. neg-mul-1 88.2%

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

      3. distribute-rgt-neg-in 88.2%

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

   6. Simplified 88.2%

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

   if 0.0030000000000000001 < F

   1. Initial program 57.2%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in x around 0 79.2%

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

      1. associate-*l/ 79.2%

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

      2. *-lft-identity 79.2%

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

      3. +-commutative 79.2%

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

      4. unpow2 79.2%

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

      5. fma-undefine 79.2%

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

   7. Simplified 79.2%

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

   8. Taylor expanded in F around inf 97.2%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.0058:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.4 \cdot 10^{-119}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + 2 \cdot x}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.9 \cdot 10^{-111}:\\ \;\;\;\;\frac{x \cdot \left(-\cos B\right)}{\sin B}\\ \mathbf{elif}\;F \leq 0.003:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + 2 \cdot x}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 72.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left(2 + F \cdot F\right) + 2 \cdot x\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{if}\;F \leq -2.8 \cdot 10^{+132}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq -18000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -15600:\\ \;\;\;\;F \cdot \frac{-1}{F \cdot B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -4 \cdot 10^{-120}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 2.9 \cdot 10^{-67}:\\ \;\;\;\;x \cdot \frac{\cos B}{-\sin B}\\ \mathbf{elif}\;F \leq 0.045:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (* (pow (+ (+ 2.0 (* F F)) (* 2.0 x)) -0.5) (/ F B)) (/ x B))))
   (if (<= F -2.8e+132)
     (+ (* x (/ -1.0 (tan B))) (/ -1.0 B))
     (if (<= F -18000000.0)
       (- (/ -1.0 (sin B)) (/ x B))
       (if (<= F -15600.0)
         (- (* F (/ -1.0 (* F B))) (/ x (tan B)))
         (if (<= F -4e-120)
           t_0
           (if (<= F 2.9e-67)
             (* x (/ (cos B) (- (sin B))))
             (if (<= F 0.045) t_0 (- (/ 1.0 (sin B)) (/ x B))))))))))

C

double code(double F, double B, double x) {
	double t_0 = (pow(((2.0 + (F * F)) + (2.0 * x)), -0.5) * (F / B)) - (x / B);
	double tmp;
	if (F <= -2.8e+132) {
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / B);
	} else if (F <= -18000000.0) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -15600.0) {
		tmp = (F * (-1.0 / (F * B))) - (x / tan(B));
	} else if (F <= -4e-120) {
		tmp = t_0;
	} else if (F <= 2.9e-67) {
		tmp = x * (cos(B) / -sin(B));
	} else if (F <= 0.045) {
		tmp = t_0;
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((((2.0d0 + (f * f)) + (2.0d0 * x)) ** (-0.5d0)) * (f / b)) - (x / b)
    if (f <= (-2.8d+132)) then
        tmp = (x * ((-1.0d0) / tan(b))) + ((-1.0d0) / b)
    else if (f <= (-18000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-15600.0d0)) then
        tmp = (f * ((-1.0d0) / (f * b))) - (x / tan(b))
    else if (f <= (-4d-120)) then
        tmp = t_0
    else if (f <= 2.9d-67) then
        tmp = x * (cos(b) / -sin(b))
    else if (f <= 0.045d0) then
        tmp = t_0
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (Math.pow(((2.0 + (F * F)) + (2.0 * x)), -0.5) * (F / B)) - (x / B);
	double tmp;
	if (F <= -2.8e+132) {
		tmp = (x * (-1.0 / Math.tan(B))) + (-1.0 / B);
	} else if (F <= -18000000.0) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -15600.0) {
		tmp = (F * (-1.0 / (F * B))) - (x / Math.tan(B));
	} else if (F <= -4e-120) {
		tmp = t_0;
	} else if (F <= 2.9e-67) {
		tmp = x * (Math.cos(B) / -Math.sin(B));
	} else if (F <= 0.045) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (math.pow(((2.0 + (F * F)) + (2.0 * x)), -0.5) * (F / B)) - (x / B)
	tmp = 0
	if F <= -2.8e+132:
		tmp = (x * (-1.0 / math.tan(B))) + (-1.0 / B)
	elif F <= -18000000.0:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -15600.0:
		tmp = (F * (-1.0 / (F * B))) - (x / math.tan(B))
	elif F <= -4e-120:
		tmp = t_0
	elif F <= 2.9e-67:
		tmp = x * (math.cos(B) / -math.sin(B))
	elif F <= 0.045:
		tmp = t_0
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64((Float64(Float64(2.0 + Float64(F * F)) + Float64(2.0 * x)) ^ -0.5) * Float64(F / B)) - Float64(x / B))
	tmp = 0.0
	if (F <= -2.8e+132)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(-1.0 / B));
	elseif (F <= -18000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -15600.0)
		tmp = Float64(Float64(F * Float64(-1.0 / Float64(F * B))) - Float64(x / tan(B)));
	elseif (F <= -4e-120)
		tmp = t_0;
	elseif (F <= 2.9e-67)
		tmp = Float64(x * Float64(cos(B) / Float64(-sin(B))));
	elseif (F <= 0.045)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = ((((2.0 + (F * F)) + (2.0 * x)) ^ -0.5) * (F / B)) - (x / B);
	tmp = 0.0;
	if (F <= -2.8e+132)
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / B);
	elseif (F <= -18000000.0)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -15600.0)
		tmp = (F * (-1.0 / (F * B))) - (x / tan(B));
	elseif (F <= -4e-120)
		tmp = t_0;
	elseif (F <= 2.9e-67)
		tmp = x * (cos(B) / -sin(B));
	elseif (F <= 0.045)
		tmp = t_0;
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.8e+132], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -18000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -15600.0], N[(N[(F * N[(-1.0 / N[(F * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -4e-120], t$95$0, If[LessEqual[F, 2.9e-67], N[(x * N[(N[Cos[B], $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.045], t$95$0, N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if F < -2.7999999999999999e132

   1. Initial program 39.6%

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

   3. Taylor expanded in F around -inf 99.7%

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

   4. Taylor expanded in B around 0 77.1%

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

   if -2.7999999999999999e132 < F < -1.8e7

   1. Initial program 77.8%

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

   3. Taylor expanded in B around 0 72.6%

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

   4. Taylor expanded in F around -inf 94.2%

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

      1. mul-1-neg 94.2%

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

      2. distribute-neg-in 94.2%

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

      3. distribute-neg-frac 94.2%

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

      4. metadata-eval 94.2%

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

      5. unsub-neg 94.2%

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

   6. Simplified 94.2%

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

   if -1.8e7 < F < -15600

   1. Initial program 99.2%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

      1. associate-*l/ 100.0%

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

      2. *-lft-identity 100.0%

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

      3. +-commutative 100.0%

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

      4. unpow2 100.0%

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

      5. fma-undefine 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in F around -inf 100.0%

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

   9. Taylor expanded in B around 0 100.0%

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

       1. *-commutative 100.0%

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

   11. Simplified 100.0%

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

   if -15600 < F < -3.99999999999999991e-120 or 2.90000000000000005e-67 < F < 0.044999999999999998

   1. Initial program 99.3%

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

   3. Taylor expanded in B around 0 90.3%

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

   4. Taylor expanded in B around 0 53.3%

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

   if -3.99999999999999991e-120 < F < 2.90000000000000005e-67

   1. Initial program 99.5%

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

   3. Taylor expanded in F around -inf 36.2%

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

   4. Taylor expanded in x around inf 83.0%

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

      1. mul-1-neg 83.0%

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

      2. associate-/l* 82.8%

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

      3. distribute-lft-neg-in 82.8%

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

   6. Simplified 82.8%

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

   if 0.044999999999999998 < F

   1. Initial program 57.2%

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

   3. Taylor expanded in B around 0 37.9%

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

   4. Taylor expanded in F around inf 77.5%

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{B}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.8 \cdot 10^{+132}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq -18000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -15600:\\ \;\;\;\;F \cdot \frac{-1}{F \cdot B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -4 \cdot 10^{-120}:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + 2 \cdot x\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.9 \cdot 10^{-67}:\\ \;\;\;\;x \cdot \frac{\cos B}{-\sin B}\\ \mathbf{elif}\;F \leq 0.045:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + 2 \cdot x\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 72.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left(2 + F \cdot F\right) + 2 \cdot x\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{if}\;F \leq -5 \cdot 10^{+131}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq -4700000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -6700:\\ \;\;\;\;F \cdot \frac{-1}{F \cdot B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -5.3 \cdot 10^{-120}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 1.75 \cdot 10^{-47}:\\ \;\;\;\;\frac{x \cdot \left(-\cos B\right)}{\sin B}\\ \mathbf{elif}\;F \leq 0.035:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (* (pow (+ (+ 2.0 (* F F)) (* 2.0 x)) -0.5) (/ F B)) (/ x B))))
   (if (<= F -5e+131)
     (+ (* x (/ -1.0 (tan B))) (/ -1.0 B))
     (if (<= F -4700000.0)
       (- (/ -1.0 (sin B)) (/ x B))
       (if (<= F -6700.0)
         (- (* F (/ -1.0 (* F B))) (/ x (tan B)))
         (if (<= F -5.3e-120)
           t_0
           (if (<= F 1.75e-47)
             (/ (* x (- (cos B))) (sin B))
             (if (<= F 0.035) t_0 (- (/ 1.0 (sin B)) (/ x B))))))))))

C

double code(double F, double B, double x) {
	double t_0 = (pow(((2.0 + (F * F)) + (2.0 * x)), -0.5) * (F / B)) - (x / B);
	double tmp;
	if (F <= -5e+131) {
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / B);
	} else if (F <= -4700000.0) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -6700.0) {
		tmp = (F * (-1.0 / (F * B))) - (x / tan(B));
	} else if (F <= -5.3e-120) {
		tmp = t_0;
	} else if (F <= 1.75e-47) {
		tmp = (x * -cos(B)) / sin(B);
	} else if (F <= 0.035) {
		tmp = t_0;
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((((2.0d0 + (f * f)) + (2.0d0 * x)) ** (-0.5d0)) * (f / b)) - (x / b)
    if (f <= (-5d+131)) then
        tmp = (x * ((-1.0d0) / tan(b))) + ((-1.0d0) / b)
    else if (f <= (-4700000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-6700.0d0)) then
        tmp = (f * ((-1.0d0) / (f * b))) - (x / tan(b))
    else if (f <= (-5.3d-120)) then
        tmp = t_0
    else if (f <= 1.75d-47) then
        tmp = (x * -cos(b)) / sin(b)
    else if (f <= 0.035d0) then
        tmp = t_0
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (Math.pow(((2.0 + (F * F)) + (2.0 * x)), -0.5) * (F / B)) - (x / B);
	double tmp;
	if (F <= -5e+131) {
		tmp = (x * (-1.0 / Math.tan(B))) + (-1.0 / B);
	} else if (F <= -4700000.0) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -6700.0) {
		tmp = (F * (-1.0 / (F * B))) - (x / Math.tan(B));
	} else if (F <= -5.3e-120) {
		tmp = t_0;
	} else if (F <= 1.75e-47) {
		tmp = (x * -Math.cos(B)) / Math.sin(B);
	} else if (F <= 0.035) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (math.pow(((2.0 + (F * F)) + (2.0 * x)), -0.5) * (F / B)) - (x / B)
	tmp = 0
	if F <= -5e+131:
		tmp = (x * (-1.0 / math.tan(B))) + (-1.0 / B)
	elif F <= -4700000.0:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -6700.0:
		tmp = (F * (-1.0 / (F * B))) - (x / math.tan(B))
	elif F <= -5.3e-120:
		tmp = t_0
	elif F <= 1.75e-47:
		tmp = (x * -math.cos(B)) / math.sin(B)
	elif F <= 0.035:
		tmp = t_0
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64((Float64(Float64(2.0 + Float64(F * F)) + Float64(2.0 * x)) ^ -0.5) * Float64(F / B)) - Float64(x / B))
	tmp = 0.0
	if (F <= -5e+131)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(-1.0 / B));
	elseif (F <= -4700000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -6700.0)
		tmp = Float64(Float64(F * Float64(-1.0 / Float64(F * B))) - Float64(x / tan(B)));
	elseif (F <= -5.3e-120)
		tmp = t_0;
	elseif (F <= 1.75e-47)
		tmp = Float64(Float64(x * Float64(-cos(B))) / sin(B));
	elseif (F <= 0.035)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = ((((2.0 + (F * F)) + (2.0 * x)) ^ -0.5) * (F / B)) - (x / B);
	tmp = 0.0;
	if (F <= -5e+131)
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / B);
	elseif (F <= -4700000.0)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -6700.0)
		tmp = (F * (-1.0 / (F * B))) - (x / tan(B));
	elseif (F <= -5.3e-120)
		tmp = t_0;
	elseif (F <= 1.75e-47)
		tmp = (x * -cos(B)) / sin(B);
	elseif (F <= 0.035)
		tmp = t_0;
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -5e+131], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -4700000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -6700.0], N[(N[(F * N[(-1.0 / N[(F * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -5.3e-120], t$95$0, If[LessEqual[F, 1.75e-47], N[(N[(x * (-N[Cos[B], $MachinePrecision])), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.035], t$95$0, N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if F < -4.99999999999999995e131

   1. Initial program 39.6%

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

   3. Taylor expanded in F around -inf 99.7%

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

   4. Taylor expanded in B around 0 77.1%

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

   if -4.99999999999999995e131 < F < -4.7e6

   1. Initial program 77.8%

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

   3. Taylor expanded in B around 0 72.6%

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

   4. Taylor expanded in F around -inf 94.2%

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

      1. mul-1-neg 94.2%

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

      2. distribute-neg-in 94.2%

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

      3. distribute-neg-frac 94.2%

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

      4. metadata-eval 94.2%

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

      5. unsub-neg 94.2%

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

   6. Simplified 94.2%

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

   if -4.7e6 < F < -6700

   1. Initial program 99.2%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

      1. associate-*l/ 100.0%

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

      2. *-lft-identity 100.0%

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

      3. +-commutative 100.0%

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

      4. unpow2 100.0%

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

      5. fma-undefine 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in F around -inf 100.0%

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

   9. Taylor expanded in B around 0 100.0%

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

       1. *-commutative 100.0%

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

   11. Simplified 100.0%

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

   if -6700 < F < -5.29999999999999997e-120 or 1.7499999999999999e-47 < F < 0.035000000000000003

   1. Initial program 99.3%

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

   3. Taylor expanded in B around 0 89.7%

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

   4. Taylor expanded in B around 0 52.4%

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

   if -5.29999999999999997e-120 < F < 1.7499999999999999e-47

   1. Initial program 99.5%

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

   3. Taylor expanded in F around inf 35.6%

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

   4. Taylor expanded in x around inf 82.4%

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

      1. associate-*r/ 82.4%

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

      2. neg-mul-1 82.4%

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

      3. distribute-rgt-neg-in 82.4%

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

   6. Simplified 82.4%

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

   if 0.035000000000000003 < F

   1. Initial program 57.2%

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

   3. Taylor expanded in B around 0 37.9%

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

   4. Taylor expanded in F around inf 77.5%

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{B}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{+131}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq -4700000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -6700:\\ \;\;\;\;F \cdot \frac{-1}{F \cdot B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -5.3 \cdot 10^{-120}:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + 2 \cdot x\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.75 \cdot 10^{-47}:\\ \;\;\;\;\frac{x \cdot \left(-\cos B\right)}{\sin B}\\ \mathbf{elif}\;F \leq 0.035:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + 2 \cdot x\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 85.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left(2 + F \cdot F\right) + 2 \cdot x\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -6000:\\ \;\;\;\;\frac{-1}{\sin B} - t\_1\\ \mathbf{elif}\;F \leq -5.3 \cdot 10^{-120}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 1.2 \cdot 10^{-47}:\\ \;\;\;\;\frac{x \cdot \left(-\cos B\right)}{\sin B}\\ \mathbf{elif}\;F \leq 0.0122:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (* (pow (+ (+ 2.0 (* F F)) (* 2.0 x)) -0.5) (/ F B)) (/ x B)))
        (t_1 (/ x (tan B))))
   (if (<= F -6000.0)
     (- (/ -1.0 (sin B)) t_1)
     (if (<= F -5.3e-120)
       t_0
       (if (<= F 1.2e-47)
         (/ (* x (- (cos B))) (sin B))
         (if (<= F 0.0122) t_0 (- (/ 1.0 (sin B)) t_1)))))))

C

double code(double F, double B, double x) {
	double t_0 = (pow(((2.0 + (F * F)) + (2.0 * x)), -0.5) * (F / B)) - (x / B);
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -6000.0) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= -5.3e-120) {
		tmp = t_0;
	} else if (F <= 1.2e-47) {
		tmp = (x * -cos(B)) / sin(B);
	} else if (F <= 0.0122) {
		tmp = t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_1;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((((2.0d0 + (f * f)) + (2.0d0 * x)) ** (-0.5d0)) * (f / b)) - (x / b)
    t_1 = x / tan(b)
    if (f <= (-6000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_1
    else if (f <= (-5.3d-120)) then
        tmp = t_0
    else if (f <= 1.2d-47) then
        tmp = (x * -cos(b)) / sin(b)
    else if (f <= 0.0122d0) then
        tmp = t_0
    else
        tmp = (1.0d0 / sin(b)) - t_1
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (Math.pow(((2.0 + (F * F)) + (2.0 * x)), -0.5) * (F / B)) - (x / B);
	double t_1 = x / Math.tan(B);
	double tmp;
	if (F <= -6000.0) {
		tmp = (-1.0 / Math.sin(B)) - t_1;
	} else if (F <= -5.3e-120) {
		tmp = t_0;
	} else if (F <= 1.2e-47) {
		tmp = (x * -Math.cos(B)) / Math.sin(B);
	} else if (F <= 0.0122) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (math.pow(((2.0 + (F * F)) + (2.0 * x)), -0.5) * (F / B)) - (x / B)
	t_1 = x / math.tan(B)
	tmp = 0
	if F <= -6000.0:
		tmp = (-1.0 / math.sin(B)) - t_1
	elif F <= -5.3e-120:
		tmp = t_0
	elif F <= 1.2e-47:
		tmp = (x * -math.cos(B)) / math.sin(B)
	elif F <= 0.0122:
		tmp = t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64((Float64(Float64(2.0 + Float64(F * F)) + Float64(2.0 * x)) ^ -0.5) * Float64(F / B)) - Float64(x / B))
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -6000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= -5.3e-120)
		tmp = t_0;
	elseif (F <= 1.2e-47)
		tmp = Float64(Float64(x * Float64(-cos(B))) / sin(B));
	elseif (F <= 0.0122)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = ((((2.0 + (F * F)) + (2.0 * x)) ^ -0.5) * (F / B)) - (x / B);
	t_1 = x / tan(B);
	tmp = 0.0;
	if (F <= -6000.0)
		tmp = (-1.0 / sin(B)) - t_1;
	elseif (F <= -5.3e-120)
		tmp = t_0;
	elseif (F <= 1.2e-47)
		tmp = (x * -cos(B)) / sin(B);
	elseif (F <= 0.0122)
		tmp = t_0;
	else
		tmp = (1.0 / sin(B)) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -6000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, -5.3e-120], t$95$0, If[LessEqual[F, 1.2e-47], N[(N[(x * (-N[Cos[B], $MachinePrecision])), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.0122], t$95$0, N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -6e3

   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 72.2%

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

   5. Taylor expanded in x around 0 72.3%

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

      1. associate-*l/ 72.3%

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

      2. *-lft-identity 72.3%

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

      3. +-commutative 72.3%

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

      4. unpow2 72.3%

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

      5. fma-undefine 72.3%

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

   7. Simplified 72.3%

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

   8. Taylor expanded in F around -inf 99.7%

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

   if -6e3 < F < -5.29999999999999997e-120 or 1.2e-47 < F < 0.0122000000000000008

   1. Initial program 99.3%

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

   3. Taylor expanded in B around 0 89.7%

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

   4. Taylor expanded in B around 0 52.4%

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

   if -5.29999999999999997e-120 < F < 1.2e-47

   1. Initial program 99.5%

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

   3. Taylor expanded in F around inf 35.6%

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

   4. Taylor expanded in x around inf 82.4%

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

      1. associate-*r/ 82.4%

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

      2. neg-mul-1 82.4%

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

      3. distribute-rgt-neg-in 82.4%

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

   6. Simplified 82.4%

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

   if 0.0122000000000000008 < F

   1. Initial program 57.2%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in x around 0 79.2%

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

      1. associate-*l/ 79.2%

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

      2. *-lft-identity 79.2%

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

      3. +-commutative 79.2%

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

      4. unpow2 79.2%

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

      5. fma-undefine 79.2%

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

   7. Simplified 79.2%

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

   8. Taylor expanded in F around inf 97.2%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -5.3 \cdot 10^{-120}:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + 2 \cdot x\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.2 \cdot 10^{-47}:\\ \;\;\;\;\frac{x \cdot \left(-\cos B\right)}{\sin B}\\ \mathbf{elif}\;F \leq 0.0122:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + 2 \cdot x\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 79.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left(2 + F \cdot F\right) + 2 \cdot x\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{if}\;F \leq -6000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1 \cdot 10^{-119}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 4 \cdot 10^{-48}:\\ \;\;\;\;\frac{x \cdot \left(-\cos B\right)}{\sin B}\\ \mathbf{elif}\;F \leq 0.045:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (* (pow (+ (+ 2.0 (* F F)) (* 2.0 x)) -0.5) (/ F B)) (/ x B))))
   (if (<= F -6000.0)
     (- (/ -1.0 (sin B)) (/ x (tan B)))
     (if (<= F -1e-119)
       t_0
       (if (<= F 4e-48)
         (/ (* x (- (cos B))) (sin B))
         (if (<= F 0.045) t_0 (- (/ 1.0 (sin B)) (/ x B))))))))

C

double code(double F, double B, double x) {
	double t_0 = (pow(((2.0 + (F * F)) + (2.0 * x)), -0.5) * (F / B)) - (x / B);
	double tmp;
	if (F <= -6000.0) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= -1e-119) {
		tmp = t_0;
	} else if (F <= 4e-48) {
		tmp = (x * -cos(B)) / sin(B);
	} else if (F <= 0.045) {
		tmp = t_0;
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((((2.0d0 + (f * f)) + (2.0d0 * x)) ** (-0.5d0)) * (f / b)) - (x / b)
    if (f <= (-6000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= (-1d-119)) then
        tmp = t_0
    else if (f <= 4d-48) then
        tmp = (x * -cos(b)) / sin(b)
    else if (f <= 0.045d0) then
        tmp = t_0
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (Math.pow(((2.0 + (F * F)) + (2.0 * x)), -0.5) * (F / B)) - (x / B);
	double tmp;
	if (F <= -6000.0) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= -1e-119) {
		tmp = t_0;
	} else if (F <= 4e-48) {
		tmp = (x * -Math.cos(B)) / Math.sin(B);
	} else if (F <= 0.045) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (math.pow(((2.0 + (F * F)) + (2.0 * x)), -0.5) * (F / B)) - (x / B)
	tmp = 0
	if F <= -6000.0:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= -1e-119:
		tmp = t_0
	elif F <= 4e-48:
		tmp = (x * -math.cos(B)) / math.sin(B)
	elif F <= 0.045:
		tmp = t_0
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64((Float64(Float64(2.0 + Float64(F * F)) + Float64(2.0 * x)) ^ -0.5) * Float64(F / B)) - Float64(x / B))
	tmp = 0.0
	if (F <= -6000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= -1e-119)
		tmp = t_0;
	elseif (F <= 4e-48)
		tmp = Float64(Float64(x * Float64(-cos(B))) / sin(B));
	elseif (F <= 0.045)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = ((((2.0 + (F * F)) + (2.0 * x)) ^ -0.5) * (F / B)) - (x / B);
	tmp = 0.0;
	if (F <= -6000.0)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= -1e-119)
		tmp = t_0;
	elseif (F <= 4e-48)
		tmp = (x * -cos(B)) / sin(B);
	elseif (F <= 0.045)
		tmp = t_0;
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -6000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1e-119], t$95$0, If[LessEqual[F, 4e-48], N[(N[(x * (-N[Cos[B], $MachinePrecision])), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.045], t$95$0, N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -6e3

   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 72.2%

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

   5. Taylor expanded in x around 0 72.3%

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

      1. associate-*l/ 72.3%

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

      2. *-lft-identity 72.3%

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

      3. +-commutative 72.3%

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

      4. unpow2 72.3%

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

      5. fma-undefine 72.3%

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

   7. Simplified 72.3%

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

   8. Taylor expanded in F around -inf 99.7%

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

   if -6e3 < F < -1.00000000000000001e-119 or 3.9999999999999999e-48 < F < 0.044999999999999998

   1. Initial program 99.3%

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

   3. Taylor expanded in B around 0 89.7%

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

   4. Taylor expanded in B around 0 52.4%

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

   if -1.00000000000000001e-119 < F < 3.9999999999999999e-48

   1. Initial program 99.5%

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

   3. Taylor expanded in F around inf 35.6%

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

   4. Taylor expanded in x around inf 82.4%

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

      1. associate-*r/ 82.4%

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

      2. neg-mul-1 82.4%

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

      3. distribute-rgt-neg-in 82.4%

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

   6. Simplified 82.4%

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

   if 0.044999999999999998 < F

   1. Initial program 57.2%

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

   3. Taylor expanded in B around 0 37.9%

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

   4. Taylor expanded in F around inf 77.5%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1 \cdot 10^{-119}:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + 2 \cdot x\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4 \cdot 10^{-48}:\\ \;\;\;\;\frac{x \cdot \left(-\cos B\right)}{\sin B}\\ \mathbf{elif}\;F \leq 0.045:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + 2 \cdot x\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 64.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left(2 + F \cdot F\right) + 2 \cdot x\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ t_1 := x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ t_2 := F \cdot \frac{-1}{F \cdot B} - \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -8.5 \cdot 10^{+131}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;F \leq -12000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -8000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;F \leq -1.7 \cdot 10^{-136}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq -2.5 \cdot 10^{-256}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;F \leq 4.9 \cdot 10^{-287}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;F \leq 9.8 \cdot 10^{-71}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;F \leq 0.045:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (* (pow (+ (+ 2.0 (* F F)) (* 2.0 x)) -0.5) (/ F B)) (/ x B)))
        (t_1 (+ (* x (/ -1.0 (tan B))) (/ -1.0 B)))
        (t_2 (- (* F (/ -1.0 (* F B))) (/ x (tan B)))))
   (if (<= F -8.5e+131)
     t_1
     (if (<= F -12000000.0)
       (- (/ -1.0 (sin B)) (/ x B))
       (if (<= F -8000.0)
         t_2
         (if (<= F -1.7e-136)
           t_0
           (if (<= F -2.5e-256)
             t_2
             (if (<= F 4.9e-287)
               (/ x (- B))
               (if (<= F 9.8e-71)
                 t_1
                 (if (<= F 0.045) t_0 (- (/ 1.0 (sin B)) (/ x B))))))))))))

C

double code(double F, double B, double x) {
	double t_0 = (pow(((2.0 + (F * F)) + (2.0 * x)), -0.5) * (F / B)) - (x / B);
	double t_1 = (x * (-1.0 / tan(B))) + (-1.0 / B);
	double t_2 = (F * (-1.0 / (F * B))) - (x / tan(B));
	double tmp;
	if (F <= -8.5e+131) {
		tmp = t_1;
	} else if (F <= -12000000.0) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -8000.0) {
		tmp = t_2;
	} else if (F <= -1.7e-136) {
		tmp = t_0;
	} else if (F <= -2.5e-256) {
		tmp = t_2;
	} else if (F <= 4.9e-287) {
		tmp = x / -B;
	} else if (F <= 9.8e-71) {
		tmp = t_1;
	} else if (F <= 0.045) {
		tmp = t_0;
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = ((((2.0d0 + (f * f)) + (2.0d0 * x)) ** (-0.5d0)) * (f / b)) - (x / b)
    t_1 = (x * ((-1.0d0) / tan(b))) + ((-1.0d0) / b)
    t_2 = (f * ((-1.0d0) / (f * b))) - (x / tan(b))
    if (f <= (-8.5d+131)) then
        tmp = t_1
    else if (f <= (-12000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-8000.0d0)) then
        tmp = t_2
    else if (f <= (-1.7d-136)) then
        tmp = t_0
    else if (f <= (-2.5d-256)) then
        tmp = t_2
    else if (f <= 4.9d-287) then
        tmp = x / -b
    else if (f <= 9.8d-71) then
        tmp = t_1
    else if (f <= 0.045d0) then
        tmp = t_0
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (Math.pow(((2.0 + (F * F)) + (2.0 * x)), -0.5) * (F / B)) - (x / B);
	double t_1 = (x * (-1.0 / Math.tan(B))) + (-1.0 / B);
	double t_2 = (F * (-1.0 / (F * B))) - (x / Math.tan(B));
	double tmp;
	if (F <= -8.5e+131) {
		tmp = t_1;
	} else if (F <= -12000000.0) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -8000.0) {
		tmp = t_2;
	} else if (F <= -1.7e-136) {
		tmp = t_0;
	} else if (F <= -2.5e-256) {
		tmp = t_2;
	} else if (F <= 4.9e-287) {
		tmp = x / -B;
	} else if (F <= 9.8e-71) {
		tmp = t_1;
	} else if (F <= 0.045) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (math.pow(((2.0 + (F * F)) + (2.0 * x)), -0.5) * (F / B)) - (x / B)
	t_1 = (x * (-1.0 / math.tan(B))) + (-1.0 / B)
	t_2 = (F * (-1.0 / (F * B))) - (x / math.tan(B))
	tmp = 0
	if F <= -8.5e+131:
		tmp = t_1
	elif F <= -12000000.0:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -8000.0:
		tmp = t_2
	elif F <= -1.7e-136:
		tmp = t_0
	elif F <= -2.5e-256:
		tmp = t_2
	elif F <= 4.9e-287:
		tmp = x / -B
	elif F <= 9.8e-71:
		tmp = t_1
	elif F <= 0.045:
		tmp = t_0
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64((Float64(Float64(2.0 + Float64(F * F)) + Float64(2.0 * x)) ^ -0.5) * Float64(F / B)) - Float64(x / B))
	t_1 = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(-1.0 / B))
	t_2 = Float64(Float64(F * Float64(-1.0 / Float64(F * B))) - Float64(x / tan(B)))
	tmp = 0.0
	if (F <= -8.5e+131)
		tmp = t_1;
	elseif (F <= -12000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -8000.0)
		tmp = t_2;
	elseif (F <= -1.7e-136)
		tmp = t_0;
	elseif (F <= -2.5e-256)
		tmp = t_2;
	elseif (F <= 4.9e-287)
		tmp = Float64(x / Float64(-B));
	elseif (F <= 9.8e-71)
		tmp = t_1;
	elseif (F <= 0.045)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = ((((2.0 + (F * F)) + (2.0 * x)) ^ -0.5) * (F / B)) - (x / B);
	t_1 = (x * (-1.0 / tan(B))) + (-1.0 / B);
	t_2 = (F * (-1.0 / (F * B))) - (x / tan(B));
	tmp = 0.0;
	if (F <= -8.5e+131)
		tmp = t_1;
	elseif (F <= -12000000.0)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -8000.0)
		tmp = t_2;
	elseif (F <= -1.7e-136)
		tmp = t_0;
	elseif (F <= -2.5e-256)
		tmp = t_2;
	elseif (F <= 4.9e-287)
		tmp = x / -B;
	elseif (F <= 9.8e-71)
		tmp = t_1;
	elseif (F <= 0.045)
		tmp = t_0;
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(F * N[(-1.0 / N[(F * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -8.5e+131], t$95$1, If[LessEqual[F, -12000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -8000.0], t$95$2, If[LessEqual[F, -1.7e-136], t$95$0, If[LessEqual[F, -2.5e-256], t$95$2, If[LessEqual[F, 4.9e-287], N[(x / (-B)), $MachinePrecision], If[LessEqual[F, 9.8e-71], t$95$1, If[LessEqual[F, 0.045], t$95$0, N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if F < -8.50000000000000063e131 or 4.9000000000000001e-287 < F < 9.7999999999999994e-71

   1. Initial program 66.8%

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

   3. Taylor expanded in F around -inf 72.8%

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

   4. Taylor expanded in B around 0 70.9%

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

   if -8.50000000000000063e131 < F < -1.2e7

   1. Initial program 77.8%

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

   3. Taylor expanded in B around 0 72.6%

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

   4. Taylor expanded in F around -inf 94.2%

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

      1. mul-1-neg 94.2%

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

      2. distribute-neg-in 94.2%

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

      3. distribute-neg-frac 94.2%

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

      4. metadata-eval 94.2%

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

      5. unsub-neg 94.2%

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

   6. Simplified 94.2%

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

   if -1.2e7 < F < -8e3 or -1.7e-136 < F < -2.5e-256

   1. Initial program 99.3%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 99.7%

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

      1. associate-*l/ 99.7%

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

      2. *-lft-identity 99.7%

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

      3. +-commutative 99.7%

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

      4. unpow2 99.7%

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

      5. fma-undefine 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in F around -inf 46.8%

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

   9. Taylor expanded in B around 0 64.8%

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

       1. *-commutative 64.8%

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

   11. Simplified 64.8%

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

   if -8e3 < F < -1.7e-136 or 9.7999999999999994e-71 < F < 0.044999999999999998

   1. Initial program 99.3%

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

   3. Taylor expanded in B around 0 88.8%

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

   4. Taylor expanded in B around 0 53.2%

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

   if -2.5e-256 < F < 4.9000000000000001e-287

   1. Initial program 99.6%

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

   3. Taylor expanded in B around 0 71.9%

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

   4. Taylor expanded in x around inf 71.9%

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

      1. mul-1-neg 71.9%

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

      2. distribute-frac-neg 71.9%

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

   6. Simplified 71.9%

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

   if 0.044999999999999998 < F

   1. Initial program 57.2%

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

   3. Taylor expanded in B around 0 37.9%

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

   4. Taylor expanded in F around inf 77.5%

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{B}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -8.5 \cdot 10^{+131}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq -12000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -8000:\\ \;\;\;\;F \cdot \frac{-1}{F \cdot B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.7 \cdot 10^{-136}:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + 2 \cdot x\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -2.5 \cdot 10^{-256}:\\ \;\;\;\;F \cdot \frac{-1}{F \cdot B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 4.9 \cdot 10^{-287}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;F \leq 9.8 \cdot 10^{-71}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq 0.045:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + 2 \cdot x\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 64.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{2 + 2 \cdot x}} \cdot \frac{F}{B} - \frac{x}{B}\\ t_1 := x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ t_2 := F \cdot \frac{-1}{F \cdot B} - \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -6.9 \cdot 10^{+131}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;F \leq -10200000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.02 \cdot 10^{-5}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;F \leq -1.65 \cdot 10^{-136}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq -1.45 \cdot 10^{-262}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;F \leq 8 \cdot 10^{-287}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{-59}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;F \leq 0.04:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (* (sqrt (/ 1.0 (+ 2.0 (* 2.0 x)))) (/ F B)) (/ x B)))
        (t_1 (+ (* x (/ -1.0 (tan B))) (/ -1.0 B)))
        (t_2 (- (* F (/ -1.0 (* F B))) (/ x (tan B)))))
   (if (<= F -6.9e+131)
     t_1
     (if (<= F -10200000.0)
       (- (/ -1.0 (sin B)) (/ x B))
       (if (<= F -1.02e-5)
         t_2
         (if (<= F -1.65e-136)
           t_0
           (if (<= F -1.45e-262)
             t_2
             (if (<= F 8e-287)
               (/ x (- B))
               (if (<= F 5e-59)
                 t_1
                 (if (<= F 0.04) t_0 (- (/ 1.0 (sin B)) (/ x B))))))))))))

C

double code(double F, double B, double x) {
	double t_0 = (sqrt((1.0 / (2.0 + (2.0 * x)))) * (F / B)) - (x / B);
	double t_1 = (x * (-1.0 / tan(B))) + (-1.0 / B);
	double t_2 = (F * (-1.0 / (F * B))) - (x / tan(B));
	double tmp;
	if (F <= -6.9e+131) {
		tmp = t_1;
	} else if (F <= -10200000.0) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -1.02e-5) {
		tmp = t_2;
	} else if (F <= -1.65e-136) {
		tmp = t_0;
	} else if (F <= -1.45e-262) {
		tmp = t_2;
	} else if (F <= 8e-287) {
		tmp = x / -B;
	} else if (F <= 5e-59) {
		tmp = t_1;
	} else if (F <= 0.04) {
		tmp = t_0;
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (sqrt((1.0d0 / (2.0d0 + (2.0d0 * x)))) * (f / b)) - (x / b)
    t_1 = (x * ((-1.0d0) / tan(b))) + ((-1.0d0) / b)
    t_2 = (f * ((-1.0d0) / (f * b))) - (x / tan(b))
    if (f <= (-6.9d+131)) then
        tmp = t_1
    else if (f <= (-10200000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-1.02d-5)) then
        tmp = t_2
    else if (f <= (-1.65d-136)) then
        tmp = t_0
    else if (f <= (-1.45d-262)) then
        tmp = t_2
    else if (f <= 8d-287) then
        tmp = x / -b
    else if (f <= 5d-59) then
        tmp = t_1
    else if (f <= 0.04d0) then
        tmp = t_0
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (Math.sqrt((1.0 / (2.0 + (2.0 * x)))) * (F / B)) - (x / B);
	double t_1 = (x * (-1.0 / Math.tan(B))) + (-1.0 / B);
	double t_2 = (F * (-1.0 / (F * B))) - (x / Math.tan(B));
	double tmp;
	if (F <= -6.9e+131) {
		tmp = t_1;
	} else if (F <= -10200000.0) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -1.02e-5) {
		tmp = t_2;
	} else if (F <= -1.65e-136) {
		tmp = t_0;
	} else if (F <= -1.45e-262) {
		tmp = t_2;
	} else if (F <= 8e-287) {
		tmp = x / -B;
	} else if (F <= 5e-59) {
		tmp = t_1;
	} else if (F <= 0.04) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (math.sqrt((1.0 / (2.0 + (2.0 * x)))) * (F / B)) - (x / B)
	t_1 = (x * (-1.0 / math.tan(B))) + (-1.0 / B)
	t_2 = (F * (-1.0 / (F * B))) - (x / math.tan(B))
	tmp = 0
	if F <= -6.9e+131:
		tmp = t_1
	elif F <= -10200000.0:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -1.02e-5:
		tmp = t_2
	elif F <= -1.65e-136:
		tmp = t_0
	elif F <= -1.45e-262:
		tmp = t_2
	elif F <= 8e-287:
		tmp = x / -B
	elif F <= 5e-59:
		tmp = t_1
	elif F <= 0.04:
		tmp = t_0
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(2.0 * x)))) * Float64(F / B)) - Float64(x / B))
	t_1 = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(-1.0 / B))
	t_2 = Float64(Float64(F * Float64(-1.0 / Float64(F * B))) - Float64(x / tan(B)))
	tmp = 0.0
	if (F <= -6.9e+131)
		tmp = t_1;
	elseif (F <= -10200000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -1.02e-5)
		tmp = t_2;
	elseif (F <= -1.65e-136)
		tmp = t_0;
	elseif (F <= -1.45e-262)
		tmp = t_2;
	elseif (F <= 8e-287)
		tmp = Float64(x / Float64(-B));
	elseif (F <= 5e-59)
		tmp = t_1;
	elseif (F <= 0.04)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (sqrt((1.0 / (2.0 + (2.0 * x)))) * (F / B)) - (x / B);
	t_1 = (x * (-1.0 / tan(B))) + (-1.0 / B);
	t_2 = (F * (-1.0 / (F * B))) - (x / tan(B));
	tmp = 0.0;
	if (F <= -6.9e+131)
		tmp = t_1;
	elseif (F <= -10200000.0)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -1.02e-5)
		tmp = t_2;
	elseif (F <= -1.65e-136)
		tmp = t_0;
	elseif (F <= -1.45e-262)
		tmp = t_2;
	elseif (F <= 8e-287)
		tmp = x / -B;
	elseif (F <= 5e-59)
		tmp = t_1;
	elseif (F <= 0.04)
		tmp = t_0;
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(2.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(F * N[(-1.0 / N[(F * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -6.9e+131], t$95$1, If[LessEqual[F, -10200000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.02e-5], t$95$2, If[LessEqual[F, -1.65e-136], t$95$0, If[LessEqual[F, -1.45e-262], t$95$2, If[LessEqual[F, 8e-287], N[(x / (-B)), $MachinePrecision], If[LessEqual[F, 5e-59], t$95$1, If[LessEqual[F, 0.04], t$95$0, N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if F < -6.9000000000000001e131 or 8.00000000000000017e-287 < F < 5.0000000000000001e-59

   1. Initial program 66.8%

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

   3. Taylor expanded in F around -inf 72.8%

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

   4. Taylor expanded in B around 0 70.9%

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

   if -6.9000000000000001e131 < F < -1.02e7

   1. Initial program 77.8%

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

   3. Taylor expanded in B around 0 72.6%

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

   4. Taylor expanded in F around -inf 94.2%

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

      1. mul-1-neg 94.2%

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

      2. distribute-neg-in 94.2%

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

      3. distribute-neg-frac 94.2%

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

      4. metadata-eval 94.2%

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

      5. unsub-neg 94.2%

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

   6. Simplified 94.2%

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

   if -1.02e7 < F < -1.0200000000000001e-5 or -1.65000000000000009e-136 < F < -1.44999999999999998e-262

   1. Initial program 99.3%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 99.5%

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

      1. associate-*l/ 99.6%

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

      2. *-lft-identity 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. fma-undefine 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in F around -inf 46.7%

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

   9. Taylor expanded in B around 0 61.2%

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

       1. *-commutative 61.2%

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

   11. Simplified 61.2%

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

   if -1.0200000000000001e-5 < F < -1.65000000000000009e-136 or 5.0000000000000001e-59 < F < 0.0400000000000000008

   1. Initial program 99.3%

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

   3. Taylor expanded in B around 0 89.9%

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

   4. Taylor expanded in B around 0 53.4%

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

   5. Taylor expanded in F around 0 51.2%

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

   if -1.44999999999999998e-262 < F < 8.00000000000000017e-287

   1. Initial program 99.6%

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

   3. Taylor expanded in B around 0 71.9%

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

   4. Taylor expanded in x around inf 71.9%

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

      1. mul-1-neg 71.9%

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

      2. distribute-frac-neg 71.9%

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

   6. Simplified 71.9%

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

   if 0.0400000000000000008 < F

   1. Initial program 57.2%

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

   3. Taylor expanded in B around 0 37.9%

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

   4. Taylor expanded in F around inf 77.5%

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{B}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6.9 \cdot 10^{+131}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq -10200000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.02 \cdot 10^{-5}:\\ \;\;\;\;F \cdot \frac{-1}{F \cdot B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.65 \cdot 10^{-136}:\\ \;\;\;\;\sqrt{\frac{1}{2 + 2 \cdot x}} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.45 \cdot 10^{-262}:\\ \;\;\;\;F \cdot \frac{-1}{F \cdot B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 8 \cdot 10^{-287}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{-59}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq 0.04:\\ \;\;\;\;\sqrt{\frac{1}{2 + 2 \cdot x}} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 62.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{if}\;F \leq -2.8 \cdot 10^{+132}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq -5800000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -2.1 \cdot 10^{-256}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 3.5 \cdot 10^{-287}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;F \leq 10^{-55}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (+ (* x (/ -1.0 (tan B))) (/ -1.0 B))))
   (if (<= F -2.8e+132)
     t_0
     (if (<= F -5800000.0)
       (- (/ -1.0 (sin B)) (/ x B))
       (if (<= F -2.1e-256)
         t_0
         (if (<= F 3.5e-287)
           (/ x (- B))
           (if (<= F 1e-55) t_0 (- (/ 1.0 (sin B)) (/ x B)))))))))

C

double code(double F, double B, double x) {
	double t_0 = (x * (-1.0 / tan(B))) + (-1.0 / B);
	double tmp;
	if (F <= -2.8e+132) {
		tmp = t_0;
	} else if (F <= -5800000.0) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -2.1e-256) {
		tmp = t_0;
	} else if (F <= 3.5e-287) {
		tmp = x / -B;
	} else if (F <= 1e-55) {
		tmp = t_0;
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * ((-1.0d0) / tan(b))) + ((-1.0d0) / b)
    if (f <= (-2.8d+132)) then
        tmp = t_0
    else if (f <= (-5800000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-2.1d-256)) then
        tmp = t_0
    else if (f <= 3.5d-287) then
        tmp = x / -b
    else if (f <= 1d-55) then
        tmp = t_0
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (x * (-1.0 / Math.tan(B))) + (-1.0 / B);
	double tmp;
	if (F <= -2.8e+132) {
		tmp = t_0;
	} else if (F <= -5800000.0) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -2.1e-256) {
		tmp = t_0;
	} else if (F <= 3.5e-287) {
		tmp = x / -B;
	} else if (F <= 1e-55) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (x * (-1.0 / math.tan(B))) + (-1.0 / B)
	tmp = 0
	if F <= -2.8e+132:
		tmp = t_0
	elif F <= -5800000.0:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -2.1e-256:
		tmp = t_0
	elif F <= 3.5e-287:
		tmp = x / -B
	elif F <= 1e-55:
		tmp = t_0
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(-1.0 / B))
	tmp = 0.0
	if (F <= -2.8e+132)
		tmp = t_0;
	elseif (F <= -5800000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -2.1e-256)
		tmp = t_0;
	elseif (F <= 3.5e-287)
		tmp = Float64(x / Float64(-B));
	elseif (F <= 1e-55)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (x * (-1.0 / tan(B))) + (-1.0 / B);
	tmp = 0.0;
	if (F <= -2.8e+132)
		tmp = t_0;
	elseif (F <= -5800000.0)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -2.1e-256)
		tmp = t_0;
	elseif (F <= 3.5e-287)
		tmp = x / -B;
	elseif (F <= 1e-55)
		tmp = t_0;
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.8e+132], t$95$0, If[LessEqual[F, -5800000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -2.1e-256], t$95$0, If[LessEqual[F, 3.5e-287], N[(x / (-B)), $MachinePrecision], If[LessEqual[F, 1e-55], t$95$0, N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.7999999999999999e132 or -5.8e6 < F < -2.10000000000000003e-256 or 3.5e-287 < F < 9.99999999999999995e-56

   1. Initial program 81.0%

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

   3. Taylor expanded in F around -inf 53.3%

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

   4. Taylor expanded in B around 0 55.5%

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

   if -2.7999999999999999e132 < F < -5.8e6

   1. Initial program 77.8%

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

   3. Taylor expanded in B around 0 72.6%

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

   4. Taylor expanded in F around -inf 94.2%

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

      1. mul-1-neg 94.2%

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

      2. distribute-neg-in 94.2%

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

      3. distribute-neg-frac 94.2%

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

      4. metadata-eval 94.2%

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

      5. unsub-neg 94.2%

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

   6. Simplified 94.2%

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

   if -2.10000000000000003e-256 < F < 3.5e-287

   1. Initial program 99.6%

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

   3. Taylor expanded in B around 0 71.9%

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

   4. Taylor expanded in x around inf 71.9%

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

      1. mul-1-neg 71.9%

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

      2. distribute-frac-neg 71.9%

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

   6. Simplified 71.9%

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

   if 9.99999999999999995e-56 < F

   1. Initial program 63.6%

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

   3. Taylor expanded in B around 0 46.8%

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

   4. Taylor expanded in F around inf 70.3%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.8 \cdot 10^{+132}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq -5800000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -2.1 \cdot 10^{-256}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq 3.5 \cdot 10^{-287}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;F \leq 10^{-55}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 44.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.85 \cdot 10^{-18}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.65:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{+160}:\\ \;\;\;\;\frac{1}{\sin B} + \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.85e-18)
   (- (/ -1.0 B) (/ x B))
   (if (<= F 1.65)
     (/ x (- B))
     (if (<= F 4.5e+160) (+ (/ 1.0 (sin B)) (/ x B)) (/ (- 1.0 x) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.85e-18) {
		tmp = (-1.0 / B) - (x / B);
	} else if (F <= 1.65) {
		tmp = x / -B;
	} else if (F <= 4.5e+160) {
		tmp = (1.0 / sin(B)) + (x / B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-2.85d-18)) then
        tmp = ((-1.0d0) / b) - (x / b)
    else if (f <= 1.65d0) then
        tmp = x / -b
    else if (f <= 4.5d+160) then
        tmp = (1.0d0 / sin(b)) + (x / b)
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.85e-18) {
		tmp = (-1.0 / B) - (x / B);
	} else if (F <= 1.65) {
		tmp = x / -B;
	} else if (F <= 4.5e+160) {
		tmp = (1.0 / Math.sin(B)) + (x / B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.85e-18:
		tmp = (-1.0 / B) - (x / B)
	elif F <= 1.65:
		tmp = x / -B
	elif F <= 4.5e+160:
		tmp = (1.0 / math.sin(B)) + (x / B)
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.85e-18)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / B));
	elseif (F <= 1.65)
		tmp = Float64(x / Float64(-B));
	elseif (F <= 4.5e+160)
		tmp = Float64(Float64(1.0 / sin(B)) + Float64(x / B));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.85e-18)
		tmp = (-1.0 / B) - (x / B);
	elseif (F <= 1.65)
		tmp = x / -B;
	elseif (F <= 4.5e+160)
		tmp = (1.0 / sin(B)) + (x / B);
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.85e-18], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.65], N[(x / (-B)), $MachinePrecision], If[LessEqual[F, 4.5e+160], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.84999999999999986e-18

   1. Initial program 57.3%

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

   3. Taylor expanded in B around 0 32.0%

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

   4. Taylor expanded in B around 0 22.4%

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

   5. Taylor expanded in F around -inf 48.1%

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

      1. distribute-lft-in 48.1%

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

      2. mul-1-neg 48.1%

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

      3. unsub-neg 48.1%

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

      4. neg-mul-1 48.1%

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

      5. distribute-neg-frac 48.1%

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

      6. metadata-eval 48.1%

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

   7. Simplified 48.1%

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

   if -2.84999999999999986e-18 < F < 1.6499999999999999

   1. Initial program 99.5%

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

   3. Taylor expanded in B around 0 66.6%

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

   4. Taylor expanded in x around inf 30.3%

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

      1. mul-1-neg 30.3%

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

      2. distribute-frac-neg 30.3%

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

   6. Simplified 30.3%

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

   if 1.6499999999999999 < F < 4.4999999999999998e160

   1. Initial program 76.9%

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

   3. Taylor expanded in B around 0 66.0%

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

   4. Taylor expanded in F around inf 63.4%

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

      1. *-un-lft-identity 63.4%

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

      2. add-sqr-sqrt 45.3%

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

      3. sqrt-unprod 54.3%

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

      4. sqr-neg 54.3%

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

      5. sqrt-unprod 34.4%

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

      6. add-sqr-sqrt 51.3%

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

      7. associate-*l/ 56.5%

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

      8. pow1 56.5%

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

      9. inv-pow 56.5%

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

      10. pow-prod-up 56.5%

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

      11. metadata-eval 56.5%

          \[\leadsto 1 \cdot \left(\frac{x}{B} + \frac{{F}^{\color{blue}{0}}}{\sin B}\right) \]

      12. metadata-eval 56.5%

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

   6. Applied egg-rr 56.5%

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

      1. *-lft-identity 56.5%

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

      2. +-commutative 56.5%

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

   8. Simplified 56.5%

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

   if 4.4999999999999998e160 < F

   1. Initial program 37.9%

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

   3. Taylor expanded in B around 0 10.5%

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

   4. Taylor expanded in F around inf 40.3%

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

   5. Taylor expanded in B around 0 58.9%

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

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.85 \cdot 10^{-18}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.65:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{+160}:\\ \;\;\;\;\frac{1}{\sin B} + \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 51.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.85 \cdot 10^{-30}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 8:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;F \leq 2.95 \cdot 10^{+162}:\\ \;\;\;\;\frac{1}{\sin B} + \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.85e-30)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 8.0)
     (/ x (- B))
     (if (<= F 2.95e+162) (+ (/ 1.0 (sin B)) (/ x B)) (/ (- 1.0 x) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.85e-30) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 8.0) {
		tmp = x / -B;
	} else if (F <= 2.95e+162) {
		tmp = (1.0 / sin(B)) + (x / B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.85d-30)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 8.0d0) then
        tmp = x / -b
    else if (f <= 2.95d+162) then
        tmp = (1.0d0 / sin(b)) + (x / b)
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.85e-30) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 8.0) {
		tmp = x / -B;
	} else if (F <= 2.95e+162) {
		tmp = (1.0 / Math.sin(B)) + (x / B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.85e-30:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 8.0:
		tmp = x / -B
	elif F <= 2.95e+162:
		tmp = (1.0 / math.sin(B)) + (x / B)
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.85e-30)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 8.0)
		tmp = Float64(x / Float64(-B));
	elseif (F <= 2.95e+162)
		tmp = Float64(Float64(1.0 / sin(B)) + Float64(x / B));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.85e-30)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 8.0)
		tmp = x / -B;
	elseif (F <= 2.95e+162)
		tmp = (1.0 / sin(B)) + (x / B);
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.85e-30], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 8.0], N[(x / (-B)), $MachinePrecision], If[LessEqual[F, 2.95e+162], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.8500000000000002e-30

   1. Initial program 59.5%

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

   3. Taylor expanded in B around 0 35.7%

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

   4. Taylor expanded in F around -inf 67.7%

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

      1. mul-1-neg 67.7%

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

      2. distribute-neg-in 67.7%

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

      3. distribute-neg-frac 67.7%

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

      4. metadata-eval 67.7%

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

      5. unsub-neg 67.7%

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

   6. Simplified 67.7%

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

   if -1.8500000000000002e-30 < F < 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 B around 0 65.4%

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

   4. Taylor expanded in x around inf 30.4%

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

      1. mul-1-neg 30.4%

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

      2. distribute-frac-neg 30.4%

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

   6. Simplified 30.4%

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

   if 8 < F < 2.95000000000000014e162

   1. Initial program 76.9%

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

   3. Taylor expanded in B around 0 66.0%

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

   4. Taylor expanded in F around inf 63.4%

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

      1. *-un-lft-identity 63.4%

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

      2. add-sqr-sqrt 45.3%

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

      3. sqrt-unprod 54.3%

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

      4. sqr-neg 54.3%

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

      5. sqrt-unprod 34.4%

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

      6. add-sqr-sqrt 51.3%

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

      7. associate-*l/ 56.5%

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

      8. pow1 56.5%

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

      9. inv-pow 56.5%

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

      10. pow-prod-up 56.5%

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

      11. metadata-eval 56.5%

          \[\leadsto 1 \cdot \left(\frac{x}{B} + \frac{{F}^{\color{blue}{0}}}{\sin B}\right) \]

      12. metadata-eval 56.5%

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

   6. Applied egg-rr 56.5%

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

      1. *-lft-identity 56.5%

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

      2. +-commutative 56.5%

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

   8. Simplified 56.5%

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

   if 2.95000000000000014e162 < F

   1. Initial program 37.9%

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

   3. Taylor expanded in B around 0 10.5%

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

   4. Taylor expanded in F around inf 40.3%

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

   5. Taylor expanded in B around 0 58.9%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.85 \cdot 10^{-30}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 8:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;F \leq 2.95 \cdot 10^{+162}:\\ \;\;\;\;\frac{1}{\sin B} + \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 58.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{-42}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.86 \cdot 10^{-56}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2e-42)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 1.86e-56) (/ x (- B)) (- (/ 1.0 (sin B)) (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2e-42) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 1.86e-56) {
		tmp = x / -B;
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-2d-42)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 1.86d-56) then
        tmp = x / -b
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2e-42) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 1.86e-56) {
		tmp = x / -B;
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2e-42:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 1.86e-56:
		tmp = x / -B
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2e-42)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 1.86e-56)
		tmp = Float64(x / Float64(-B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2e-42)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 1.86e-56)
		tmp = x / -B;
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2e-42], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.86e-56], N[(x / (-B)), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.00000000000000008e-42

   1. Initial program 59.5%

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

   3. Taylor expanded in B around 0 35.7%

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

   4. Taylor expanded in F around -inf 67.7%

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

      1. mul-1-neg 67.7%

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

      2. distribute-neg-in 67.7%

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

      3. distribute-neg-frac 67.7%

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

      4. metadata-eval 67.7%

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

      5. unsub-neg 67.7%

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

   6. Simplified 67.7%

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

   if -2.00000000000000008e-42 < F < 1.85999999999999997e-56

   1. Initial program 99.5%

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

   3. Taylor expanded in B around 0 61.0%

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

   4. Taylor expanded in x around inf 30.3%

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

      1. mul-1-neg 30.3%

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

      2. distribute-frac-neg 30.3%

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

   6. Simplified 30.3%

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

   if 1.85999999999999997e-56 < F

   1. Initial program 63.6%

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

   3. Taylor expanded in B around 0 46.8%

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

   4. Taylor expanded in F around inf 70.3%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{-42}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.86 \cdot 10^{-56}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 30.2% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-B}\\ \mathbf{if}\;x \leq -1.9 \cdot 10^{-54}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-271}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-194} \lor \neg \left(x \leq 1.48 \cdot 10^{-36}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (- B))))
   (if (<= x -1.9e-54)
     t_0
     (if (<= x 4.5e-271)
       (/ -1.0 B)
       (if (or (<= x 3.9e-194) (not (<= x 1.48e-36))) t_0 (/ (+ 1.0 x) B))))))

C

double code(double F, double B, double x) {
	double t_0 = x / -B;
	double tmp;
	if (x <= -1.9e-54) {
		tmp = t_0;
	} else if (x <= 4.5e-271) {
		tmp = -1.0 / B;
	} else if ((x <= 3.9e-194) || !(x <= 1.48e-36)) {
		tmp = t_0;
	} else {
		tmp = (1.0 + x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / -b
    if (x <= (-1.9d-54)) then
        tmp = t_0
    else if (x <= 4.5d-271) then
        tmp = (-1.0d0) / b
    else if ((x <= 3.9d-194) .or. (.not. (x <= 1.48d-36))) then
        tmp = t_0
    else
        tmp = (1.0d0 + x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / -B;
	double tmp;
	if (x <= -1.9e-54) {
		tmp = t_0;
	} else if (x <= 4.5e-271) {
		tmp = -1.0 / B;
	} else if ((x <= 3.9e-194) || !(x <= 1.48e-36)) {
		tmp = t_0;
	} else {
		tmp = (1.0 + x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / -B
	tmp = 0
	if x <= -1.9e-54:
		tmp = t_0
	elif x <= 4.5e-271:
		tmp = -1.0 / B
	elif (x <= 3.9e-194) or not (x <= 1.48e-36):
		tmp = t_0
	else:
		tmp = (1.0 + x) / B
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / Float64(-B))
	tmp = 0.0
	if (x <= -1.9e-54)
		tmp = t_0;
	elseif (x <= 4.5e-271)
		tmp = Float64(-1.0 / B);
	elseif ((x <= 3.9e-194) || !(x <= 1.48e-36))
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 + x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / -B;
	tmp = 0.0;
	if (x <= -1.9e-54)
		tmp = t_0;
	elseif (x <= 4.5e-271)
		tmp = -1.0 / B;
	elseif ((x <= 3.9e-194) || ~((x <= 1.48e-36)))
		tmp = t_0;
	else
		tmp = (1.0 + x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / (-B)), $MachinePrecision]}, If[LessEqual[x, -1.9e-54], t$95$0, If[LessEqual[x, 4.5e-271], N[(-1.0 / B), $MachinePrecision], If[Or[LessEqual[x, 3.9e-194], N[Not[LessEqual[x, 1.48e-36]], $MachinePrecision]], t$95$0, N[(N[(1.0 + x), $MachinePrecision] / B), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.9000000000000001e-54 or 4.4999999999999998e-271 < x < 3.8999999999999999e-194 or 1.48e-36 < x

   1. Initial program 80.8%

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

   3. Taylor expanded in B around 0 42.4%

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

   4. Taylor expanded in x around inf 45.9%

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

      1. mul-1-neg 45.9%

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

      2. distribute-frac-neg 45.9%

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

   6. Simplified 45.9%

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

   if -1.9000000000000001e-54 < x < 4.4999999999999998e-271

   1. Initial program 66.0%

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

   3. Taylor expanded in F around -inf 36.4%

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

   4. Taylor expanded in B around 0 23.9%

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

      1. mul-1-neg 23.9%

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

      2. distribute-neg-frac2 23.9%

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

   6. Simplified 23.9%

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

   7. Taylor expanded in x around 0 23.9%

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

   if 3.8999999999999999e-194 < x < 1.48e-36

   1. Initial program 75.7%

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

   3. Taylor expanded in F around -inf 18.9%

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

   4. Taylor expanded in B around 0 13.4%

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

      1. mul-1-neg 13.4%

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

      2. distribute-neg-frac2 13.4%

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

   6. Simplified 13.4%

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

      1. add-sqr-sqrt 10.0%

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

      2. sqrt-unprod 11.6%

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

      3. sqr-neg 11.6%

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

      4. sqrt-unprod 7.2%

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

      5. add-sqr-sqrt 18.1%

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

      6. *-un-lft-identity 18.1%

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

   8. Applied egg-rr 18.1%

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

      1. *-lft-identity 18.1%

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

   10. Simplified 18.1%

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

4. Final simplification 35.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-54}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-271}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-194} \lor \neg \left(x \leq 1.48 \cdot 10^{-36}\right):\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 36.8% accurate, 16.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1 - x}{B}\\ \mathbf{if}\;F \leq -1.2 \cdot 10^{-18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 1.05 \cdot 10^{+189}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;F \leq 2.7 \cdot 10^{+222}:\\ \;\;\;\;\frac{1 + x}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- -1.0 x) B)))
   (if (<= F -1.2e-18)
     t_0
     (if (<= F 1.05e+189)
       (/ x (- B))
       (if (<= F 2.7e+222) (/ (+ 1.0 x) B) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 - x) / B;
	double tmp;
	if (F <= -1.2e-18) {
		tmp = t_0;
	} else if (F <= 1.05e+189) {
		tmp = x / -B;
	} else if (F <= 2.7e+222) {
		tmp = (1.0 + x) / B;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-1.0d0) - x) / b
    if (f <= (-1.2d-18)) then
        tmp = t_0
    else if (f <= 1.05d+189) then
        tmp = x / -b
    else if (f <= 2.7d+222) then
        tmp = (1.0d0 + x) / b
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (-1.0 - x) / B;
	double tmp;
	if (F <= -1.2e-18) {
		tmp = t_0;
	} else if (F <= 1.05e+189) {
		tmp = x / -B;
	} else if (F <= 2.7e+222) {
		tmp = (1.0 + x) / B;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (-1.0 - x) / B
	tmp = 0
	if F <= -1.2e-18:
		tmp = t_0
	elif F <= 1.05e+189:
		tmp = x / -B
	elif F <= 2.7e+222:
		tmp = (1.0 + x) / B
	else:
		tmp = t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 - x) / B)
	tmp = 0.0
	if (F <= -1.2e-18)
		tmp = t_0;
	elseif (F <= 1.05e+189)
		tmp = Float64(x / Float64(-B));
	elseif (F <= 2.7e+222)
		tmp = Float64(Float64(1.0 + x) / B);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (-1.0 - x) / B;
	tmp = 0.0;
	if (F <= -1.2e-18)
		tmp = t_0;
	elseif (F <= 1.05e+189)
		tmp = x / -B;
	elseif (F <= 2.7e+222)
		tmp = (1.0 + x) / B;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision]}, If[LessEqual[F, -1.2e-18], t$95$0, If[LessEqual[F, 1.05e+189], N[(x / (-B)), $MachinePrecision], If[LessEqual[F, 2.7e+222], N[(N[(1.0 + x), $MachinePrecision] / B), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.19999999999999997e-18 or 2.70000000000000013e222 < F

   1. Initial program 51.1%

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

   3. Taylor expanded in F around -inf 87.8%

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

   4. Taylor expanded in B around 0 47.3%

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

      1. mul-1-neg 47.3%

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

      2. distribute-neg-frac2 47.3%

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

   6. Simplified 47.3%

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

   7. Taylor expanded in B around 0 47.3%

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

      1. associate-*r/ 47.3%

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

      2. neg-mul-1 47.3%

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

      3. distribute-neg-in 47.3%

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

      4. metadata-eval 47.3%

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

      5. unsub-neg 47.3%

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

   9. Simplified 47.3%

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

   if -1.19999999999999997e-18 < F < 1.04999999999999996e189

   1. Initial program 93.9%

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

   3. Taylor expanded in B around 0 65.0%

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

   4. Taylor expanded in x around inf 30.3%

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

      1. mul-1-neg 30.3%

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

      2. distribute-frac-neg 30.3%

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

   6. Simplified 30.3%

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

   if 1.04999999999999996e189 < F < 2.70000000000000013e222

   1. Initial program 26.0%

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

   3. Taylor expanded in F around -inf 25.5%

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

   4. Taylor expanded in B around 0 1.4%

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

      1. mul-1-neg 1.4%

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

      2. distribute-neg-frac2 1.4%

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

   6. Simplified 1.4%

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

      1. add-sqr-sqrt 0.6%

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

      2. sqrt-unprod 14.8%

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

      3. sqr-neg 14.8%

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

      4. sqrt-unprod 25.9%

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

      5. add-sqr-sqrt 63.8%

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

      6. *-un-lft-identity 63.8%

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

   8. Applied egg-rr 63.8%

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

      1. *-lft-identity 63.8%

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

   10. Simplified 63.8%

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

4. Final simplification 37.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.2 \cdot 10^{-18}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.05 \cdot 10^{+189}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;F \leq 2.7 \cdot 10^{+222}:\\ \;\;\;\;\frac{1 + x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 44.1% accurate, 21.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -9 \cdot 10^{-19}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 8.5 \cdot 10^{-52}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -9e-19)
   (/ (- -1.0 x) B)
   (if (<= F 8.5e-52) (/ x (- B)) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -9e-19) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 8.5e-52) {
		tmp = x / -B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-9d-19)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 8.5d-52) then
        tmp = x / -b
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -9e-19) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 8.5e-52) {
		tmp = x / -B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -9e-19:
		tmp = (-1.0 - x) / B
	elif F <= 8.5e-52:
		tmp = x / -B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -9e-19)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 8.5e-52)
		tmp = Float64(x / Float64(-B));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -9e-19)
		tmp = (-1.0 - x) / B;
	elseif (F <= 8.5e-52)
		tmp = x / -B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -9e-19], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 8.5e-52], N[(x / (-B)), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -9.00000000000000026e-19

   1. Initial program 57.3%

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

   3. Taylor expanded in F around -inf 95.2%

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

   4. Taylor expanded in B around 0 48.1%

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

      1. mul-1-neg 48.1%

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

      2. distribute-neg-frac2 48.1%

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

   6. Simplified 48.1%

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

   7. Taylor expanded in B around 0 48.1%

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

      1. associate-*r/ 48.1%

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

      2. neg-mul-1 48.1%

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

      3. distribute-neg-in 48.1%

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

      4. metadata-eval 48.1%

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

      5. unsub-neg 48.1%

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

   9. Simplified 48.1%

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

   if -9.00000000000000026e-19 < F < 8.50000000000000006e-52

   1. Initial program 99.5%

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

   3. Taylor expanded in B around 0 63.3%

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

   4. Taylor expanded in x around inf 30.6%

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

      1. mul-1-neg 30.6%

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

      2. distribute-frac-neg 30.6%

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

   6. Simplified 30.6%

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

   if 8.50000000000000006e-52 < F

   1. Initial program 62.8%

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

   3. Taylor expanded in B around 0 45.5%

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

   4. Taylor expanded in F around inf 48.7%

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

   5. Taylor expanded in B around 0 48.0%

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

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -9 \cdot 10^{-19}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 8.5 \cdot 10^{-52}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 44.1% accurate, 21.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 8.2 \cdot 10^{-52}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.2e-15)
   (- (/ -1.0 B) (/ x B))
   (if (<= F 8.2e-52) (/ x (- B)) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.2e-15) {
		tmp = (-1.0 / B) - (x / B);
	} else if (F <= 8.2e-52) {
		tmp = x / -B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.2d-15)) then
        tmp = ((-1.0d0) / b) - (x / b)
    else if (f <= 8.2d-52) then
        tmp = x / -b
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.2e-15) {
		tmp = (-1.0 / B) - (x / B);
	} else if (F <= 8.2e-52) {
		tmp = x / -B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.2e-15:
		tmp = (-1.0 / B) - (x / B)
	elif F <= 8.2e-52:
		tmp = x / -B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.2e-15)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / B));
	elseif (F <= 8.2e-52)
		tmp = Float64(x / Float64(-B));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.2e-15)
		tmp = (-1.0 / B) - (x / B);
	elseif (F <= 8.2e-52)
		tmp = x / -B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.2e-15], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 8.2e-52], N[(x / (-B)), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.19999999999999997e-15

   1. Initial program 57.3%

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

   3. Taylor expanded in B around 0 32.0%

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

   4. Taylor expanded in B around 0 22.4%

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

   5. Taylor expanded in F around -inf 48.1%

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

      1. distribute-lft-in 48.1%

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

      2. mul-1-neg 48.1%

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

      3. unsub-neg 48.1%

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

      4. neg-mul-1 48.1%

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

      5. distribute-neg-frac 48.1%

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

      6. metadata-eval 48.1%

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

   7. Simplified 48.1%

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

   if -1.19999999999999997e-15 < F < 8.19999999999999977e-52

   1. Initial program 99.5%

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

   3. Taylor expanded in B around 0 63.3%

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

   4. Taylor expanded in x around inf 30.6%

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

      1. mul-1-neg 30.6%

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

      2. distribute-frac-neg 30.6%

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

   6. Simplified 30.6%

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

   if 8.19999999999999977e-52 < F

   1. Initial program 62.8%

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

   3. Taylor expanded in B around 0 45.5%

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

   4. Taylor expanded in F around inf 48.7%

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

   5. Taylor expanded in B around 0 48.0%

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

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 8.2 \cdot 10^{-52}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 30.6% accurate, 23.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-54} \lor \neg \left(x \leq 4.6 \cdot 10^{-271}\right):\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (or (<= x -4e-54) (not (<= x 4.6e-271))) (/ x (- B)) (/ -1.0 B)))

C

double code(double F, double B, double x) {
	double tmp;
	if ((x <= -4e-54) || !(x <= 4.6e-271)) {
		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 <= (-4d-54)) .or. (.not. (x <= 4.6d-271))) 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 <= -4e-54) || !(x <= 4.6e-271)) {
		tmp = x / -B;
	} else {
		tmp = -1.0 / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if (x <= -4e-54) or not (x <= 4.6e-271):
		tmp = x / -B
	else:
		tmp = -1.0 / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if ((x <= -4e-54) || !(x <= 4.6e-271))
		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 <= -4e-54) || ~((x <= 4.6e-271)))
		tmp = x / -B;
	else
		tmp = -1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[Or[LessEqual[x, -4e-54], N[Not[LessEqual[x, 4.6e-271]], $MachinePrecision]], N[(x / (-B)), $MachinePrecision], N[(-1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.0000000000000001e-54 or 4.60000000000000017e-271 < x

   1. Initial program 79.5%

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

   3. Taylor expanded in B around 0 46.2%

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

   4. Taylor expanded in x around inf 36.0%

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

      1. mul-1-neg 36.0%

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

      2. distribute-frac-neg 36.0%

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

   6. Simplified 36.0%

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

   if -4.0000000000000001e-54 < x < 4.60000000000000017e-271

   1. Initial program 66.0%

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

   3. Taylor expanded in F around -inf 36.4%

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

   4. Taylor expanded in B around 0 23.9%

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

      1. mul-1-neg 23.9%

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

      2. distribute-neg-frac2 23.9%

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

   6. Simplified 23.9%

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

   7. Taylor expanded in x around 0 23.9%

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

4. Final simplification 33.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-54} \lor \neg \left(x \leq 4.6 \cdot 10^{-271}\right):\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 9.8% 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 76.2%

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

3. Taylor expanded in F around -inf 53.4%

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

4. Taylor expanded in B around 0 29.3%

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

   1. mul-1-neg 29.3%

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

   2. distribute-neg-frac2 29.3%

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

6. Simplified 29.3%

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

7. Taylor expanded in x around 0 10.9%

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

8. Final simplification 10.9%

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

Reproduce

herbie shell --seed 2024076 
(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))))))