VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.4% → 99.5%

Time: 22.0s

Alternatives: 24

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2.3 \cdot 10^{+60}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 0.048:\\ \;\;\;\;F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -2.3e+60)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 0.048)
       (- (* 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 <= -2.3e+60) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 0.048) {
		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 <= -2.3e+60)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 0.048)
		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, -2.3e+60], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.048], 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 < -2.30000000000000017e60

   1. Initial program 52.8%

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

   3. Taylor expanded in F around -inf 99.8%

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

      1. +-commutative 99.8%

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

      2. unsub-neg 99.8%

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

      3. un-div-inv 99.9%

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

   5. Applied egg-rr 99.9%

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

   if -2.30000000000000017e60 < F < 0.048000000000000001

   1. Initial program 98.6%

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

   3. Simplified 99.7%

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

   if 0.048000000000000001 < F

   1. Initial program 56.0%

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

   3. Taylor expanded in F around -inf 47.1%

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

      1. div-inv 47.1%

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

      2. metadata-eval 47.1%

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

      3. rgt-mult-inverse 47.1%

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

      4. associate-*l/ 37.7%

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

      5. add-sqr-sqrt 16.8%

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

      6. sqrt-unprod 53.3%

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

      7. *-commutative 53.3%

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

      8. *-commutative 53.3%

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

      9. associate-*r/ 53.3%

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

      10. *-commutative 53.3%

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

      11. rgt-mult-inverse 53.3%

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

      12. associate-*r/ 56.5%

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

      13. *-commutative 56.5%

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

      14. rgt-mult-inverse 56.5%

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

      15. frac-times 56.5%

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

      16. metadata-eval 56.5%

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

      17. metadata-eval 56.5%

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

      18. frac-times 56.5%

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

      19. sqrt-unprod 49.0%

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

   5. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{0 - \left(\frac{x}{\tan B} + \frac{-1}{\sin B}\right)} \]
   6. Step-by-step derivation

      1. neg-sub0 99.8%

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

      2. distribute-neg-in 99.8%

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

      3. distribute-neg-frac 99.8%

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

      4. metadata-eval 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.3 \cdot 10^{+60}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.048:\\ \;\;\;\;F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -112000:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq -2.15 \cdot 10^{-68}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.048:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, 2\right)}}}{B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -112000.0)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F -2.15e-68)
       (- (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)) (/ x B))
       (if (<= F 0.048)
         (- (* F (/ (sqrt (/ 1.0 (fma x 2.0 2.0))) 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 <= -112000.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= -2.15e-68) {
		tmp = ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B);
	} else if (F <= 0.048) {
		tmp = (F * (sqrt((1.0 / fma(x, 2.0, 2.0))) / 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 <= -112000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= -2.15e-68)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)) - Float64(x / B));
	elseif (F <= 0.048)
		tmp = Float64(Float64(F * Float64(sqrt(Float64(1.0 / fma(x, 2.0, 2.0))) / 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, -112000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -2.15e-68], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.048], N[(N[(F * N[(N[Sqrt[N[(1.0 / N[(x * 2.0 + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -112000

   1. Initial program 58.6%

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

   3. Taylor expanded in F around -inf 99.4%

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

      1. +-commutative 99.4%

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

      2. unsub-neg 99.4%

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

      3. un-div-inv 99.5%

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

   5. Applied egg-rr 99.5%

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

   if -112000 < F < -2.15000000000000005e-68

   1. Initial program 99.0%

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

   3. Taylor expanded in B around 0 87.5%

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

   if -2.15000000000000005e-68 < F < 0.048000000000000001

   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%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in B around 0 86.9%

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

   6. Taylor expanded in F around 0 86.9%

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

      1. associate-*l/ 86.9%

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

      2. associate-*r/ 86.9%

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

      3. +-commutative 86.9%

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

      4. *-commutative 86.9%

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

      5. fma-define 86.9%

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

   8. Simplified 86.9%

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

   if 0.048000000000000001 < F

   1. Initial program 56.0%

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

   3. Taylor expanded in F around -inf 47.1%

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

      1. div-inv 47.1%

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

      2. metadata-eval 47.1%

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

      3. rgt-mult-inverse 47.1%

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

      4. associate-*l/ 37.7%

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

      5. add-sqr-sqrt 16.8%

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

      6. sqrt-unprod 53.3%

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

      7. *-commutative 53.3%

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

      8. *-commutative 53.3%

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

      9. associate-*r/ 53.3%

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

      10. *-commutative 53.3%

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

      11. rgt-mult-inverse 53.3%

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

      12. associate-*r/ 56.5%

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

      13. *-commutative 56.5%

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

      14. rgt-mult-inverse 56.5%

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

      15. frac-times 56.5%

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

      16. metadata-eval 56.5%

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

      17. metadata-eval 56.5%

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

      18. frac-times 56.5%

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

      19. sqrt-unprod 49.0%

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

   5. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{0 - \left(\frac{x}{\tan B} + \frac{-1}{\sin B}\right)} \]
   6. Step-by-step derivation

      1. neg-sub0 99.8%

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

      2. distribute-neg-in 99.8%

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

      3. distribute-neg-frac 99.8%

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

      4. metadata-eval 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -112000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -2.15 \cdot 10^{-68}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.048:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{\mathsf{fma}\left(x, 2, 2\right)}}}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -135000:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 0.048:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -135000.0)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 0.048)
       (- (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -135000.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 0.048) {
		tmp = ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-135000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 0.048d0) then
        tmp = ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0))) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -135000.0) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 0.048) {
		tmp = ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -135000.0:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 0.048:
		tmp = ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -135000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 0.048)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -135000.0)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 0.048)
		tmp = ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5)) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -135000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.048], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - 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 < -135000

   1. Initial program 58.6%

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

   3. Taylor expanded in F around -inf 99.4%

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

      1. +-commutative 99.4%

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

      2. unsub-neg 99.4%

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

      3. un-div-inv 99.5%

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

   5. Applied egg-rr 99.5%

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

   if -135000 < F < 0.048000000000000001

   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. Step-by-step derivation

      1. div-inv 99.5%

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

   4. Applied egg-rr 99.5%

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

   if 0.048000000000000001 < F

   1. Initial program 56.0%

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

   3. Taylor expanded in F around -inf 47.1%

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

      1. div-inv 47.1%

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

      2. metadata-eval 47.1%

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

      3. rgt-mult-inverse 47.1%

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

      4. associate-*l/ 37.7%

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

      5. add-sqr-sqrt 16.8%

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

      6. sqrt-unprod 53.3%

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

      7. *-commutative 53.3%

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

      8. *-commutative 53.3%

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

      9. associate-*r/ 53.3%

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

      10. *-commutative 53.3%

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

      11. rgt-mult-inverse 53.3%

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

      12. associate-*r/ 56.5%

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

      13. *-commutative 56.5%

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

      14. rgt-mult-inverse 56.5%

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

      15. frac-times 56.5%

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

      16. metadata-eval 56.5%

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

      17. metadata-eval 56.5%

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

      18. frac-times 56.5%

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

      19. sqrt-unprod 49.0%

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

   5. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{0 - \left(\frac{x}{\tan B} + \frac{-1}{\sin B}\right)} \]
   6. Step-by-step derivation

      1. neg-sub0 99.8%

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

      2. distribute-neg-in 99.8%

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

      3. distribute-neg-frac 99.8%

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

      4. metadata-eval 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -135000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.048:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.46:\\ \;\;\;\;\frac{-1}{\sin B} - t\_1\\ \mathbf{elif}\;F \leq 0.048:\\ \;\;\;\;F \cdot \left(t\_0 \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\right) - t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))) (t_1 (/ x (tan B))))
   (if (<= F -1.46)
     (- (/ -1.0 (sin B)) t_1)
     (if (<= F 0.048)
       (- (* F (* t_0 (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))))) t_1)
       (- t_0 t_1)))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / sin(B);
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -1.46) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= 0.048) {
		tmp = (F * (t_0 * sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_1;
	} else {
		tmp = t_0 - t_1;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 / sin(b)
    t_1 = x / tan(b)
    if (f <= (-1.46d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_1
    else if (f <= 0.048d0) then
        tmp = (f * (t_0 * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))))) - t_1
    else
        tmp = t_0 - t_1
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = 1.0 / Math.sin(B);
	double t_1 = x / Math.tan(B);
	double tmp;
	if (F <= -1.46) {
		tmp = (-1.0 / Math.sin(B)) - t_1;
	} else if (F <= 0.048) {
		tmp = (F * (t_0 * Math.sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_1;
	} else {
		tmp = t_0 - t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = 1.0 / math.sin(B)
	t_1 = x / math.tan(B)
	tmp = 0
	if F <= -1.46:
		tmp = (-1.0 / math.sin(B)) - t_1
	elif F <= 0.048:
		tmp = (F * (t_0 * math.sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_1
	else:
		tmp = t_0 - t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / sin(B))
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.46)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= 0.048)
		tmp = Float64(Float64(F * Float64(t_0 * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0)))))) - t_1);
	else
		tmp = Float64(t_0 - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = 1.0 / sin(B);
	t_1 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.46)
		tmp = (-1.0 / sin(B)) - t_1;
	elseif (F <= 0.048)
		tmp = (F * (t_0 * sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_1;
	else
		tmp = t_0 - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.46], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, 0.048], N[(N[(F * N[(t$95$0 * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(t$95$0 - t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.46

   1. Initial program 60.0%

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

   3. Taylor expanded in F around -inf 97.4%

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

      1. +-commutative 97.4%

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

      2. unsub-neg 97.4%

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

      3. un-div-inv 97.5%

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

   5. Applied egg-rr 97.5%

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

   if -1.46 < F < 0.048000000000000001

   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 F around 0 98.8%

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

   if 0.048000000000000001 < F

   1. Initial program 56.0%

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

   3. Taylor expanded in F around -inf 47.1%

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

      1. div-inv 47.1%

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

      2. metadata-eval 47.1%

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

      3. rgt-mult-inverse 47.1%

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

      4. associate-*l/ 37.7%

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

      5. add-sqr-sqrt 16.8%

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

      6. sqrt-unprod 53.3%

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

      7. *-commutative 53.3%

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

      8. *-commutative 53.3%

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

      9. associate-*r/ 53.3%

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

      10. *-commutative 53.3%

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

      11. rgt-mult-inverse 53.3%

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

      12. associate-*r/ 56.5%

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

      13. *-commutative 56.5%

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

      14. rgt-mult-inverse 56.5%

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

      15. frac-times 56.5%

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

      16. metadata-eval 56.5%

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

      17. metadata-eval 56.5%

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

      18. frac-times 56.5%

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

      19. sqrt-unprod 49.0%

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

   5. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{0 - \left(\frac{x}{\tan B} + \frac{-1}{\sin B}\right)} \]
   6. Step-by-step derivation

      1. neg-sub0 99.8%

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

      2. distribute-neg-in 99.8%

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

      3. distribute-neg-frac 99.8%

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

      4. metadata-eval 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.46:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.048:\\ \;\;\;\;F \cdot \left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\right) - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.0031:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 0.048:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B \cdot \sqrt{2 + x \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -0.0031)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 0.048)
       (+ (* x (/ -1.0 (tan B))) (/ F (* (sin B) (sqrt (+ 2.0 (* x 2.0))))))
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -0.0031) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 0.048) {
		tmp = (x * (-1.0 / tan(B))) + (F / (sin(B) * sqrt((2.0 + (x * 2.0)))));
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-0.0031d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 0.048d0) then
        tmp = (x * ((-1.0d0) / tan(b))) + (f / (sin(b) * sqrt((2.0d0 + (x * 2.0d0)))))
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -0.0031) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 0.048) {
		tmp = (x * (-1.0 / Math.tan(B))) + (F / (Math.sin(B) * Math.sqrt((2.0 + (x * 2.0)))));
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -0.0031:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 0.048:
		tmp = (x * (-1.0 / math.tan(B))) + (F / (math.sin(B) * math.sqrt((2.0 + (x * 2.0)))))
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -0.0031)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 0.048)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(F / Float64(sin(B) * sqrt(Float64(2.0 + Float64(x * 2.0))))));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -0.0031)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 0.048)
		tmp = (x * (-1.0 / tan(B))) + (F / (sin(B) * sqrt((2.0 + (x * 2.0)))));
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.0031], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.048], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]], $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 < -0.00309999999999999989

   1. Initial program 60.4%

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

   3. Taylor expanded in F around -inf 97.4%

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

      1. +-commutative 97.4%

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

      2. unsub-neg 97.4%

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

      3. un-div-inv 97.5%

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

   5. Applied egg-rr 97.5%

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

   if -0.00309999999999999989 < F < 0.048000000000000001

   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. Step-by-step derivation

      1. associate-*l/ 99.5%

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

      2. +-commutative 99.5%

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

      3. *-commutative 99.5%

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

      4. fma-undefine 99.5%

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

      5. fma-define 99.5%

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

      6. metadata-eval 99.5%

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

      7. metadata-eval 99.5%

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

      8. associate-*r/ 99.4%

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

      9. clear-num 99.3%

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

      10. un-div-inv 99.4%

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

      11. fma-define 99.4%

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

      12. fma-undefine 99.4%

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

      13. *-commutative 99.4%

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

      14. fma-define 99.4%

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

      15. fma-define 99.4%

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in F around 0 98.5%

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

   if 0.048000000000000001 < F

   1. Initial program 56.0%

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

   3. Taylor expanded in F around -inf 47.1%

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

      1. div-inv 47.1%

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

      2. metadata-eval 47.1%

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

      3. rgt-mult-inverse 47.1%

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

      4. associate-*l/ 37.7%

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

      5. add-sqr-sqrt 16.8%

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

      6. sqrt-unprod 53.3%

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

      7. *-commutative 53.3%

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

      8. *-commutative 53.3%

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

      9. associate-*r/ 53.3%

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

      10. *-commutative 53.3%

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

      11. rgt-mult-inverse 53.3%

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

      12. associate-*r/ 56.5%

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

      13. *-commutative 56.5%

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

      14. rgt-mult-inverse 56.5%

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

      15. frac-times 56.5%

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

      16. metadata-eval 56.5%

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

      17. metadata-eval 56.5%

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

      18. frac-times 56.5%

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

      19. sqrt-unprod 49.0%

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

   5. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{0 - \left(\frac{x}{\tan B} + \frac{-1}{\sin B}\right)} \]
   6. Step-by-step derivation

      1. neg-sub0 99.8%

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

      2. distribute-neg-in 99.8%

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

      3. distribute-neg-frac 99.8%

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

      4. metadata-eval 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.0031:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.048:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B \cdot \sqrt{2 + x \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.45:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 0.048:\\ \;\;\;\;\frac{F}{\sin B} \cdot \frac{1}{\sqrt{2 + x \cdot 2}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.45)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 0.048)
       (- (* (/ F (sin B)) (/ 1.0 (sqrt (+ 2.0 (* x 2.0))))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.45) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 0.048) {
		tmp = ((F / sin(B)) * (1.0 / sqrt((2.0 + (x * 2.0))))) - 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 <= (-1.45d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 0.048d0) then
        tmp = ((f / sin(b)) * (1.0d0 / sqrt((2.0d0 + (x * 2.0d0))))) - 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 <= -1.45) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 0.048) {
		tmp = ((F / Math.sin(B)) * (1.0 / Math.sqrt((2.0 + (x * 2.0))))) - 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 <= -1.45:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 0.048:
		tmp = ((F / math.sin(B)) * (1.0 / math.sqrt((2.0 + (x * 2.0))))) - 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 <= -1.45)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 0.048)
		tmp = Float64(Float64(Float64(F / sin(B)) * Float64(1.0 / sqrt(Float64(2.0 + Float64(x * 2.0))))) - 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 <= -1.45)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 0.048)
		tmp = ((F / sin(B)) * (1.0 / sqrt((2.0 + (x * 2.0))))) - 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, -1.45], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.048], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.44999999999999996

   1. Initial program 60.0%

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

   3. Taylor expanded in F around -inf 97.4%

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

      1. +-commutative 97.4%

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

      2. unsub-neg 97.4%

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

      3. un-div-inv 97.5%

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

   5. Applied egg-rr 97.5%

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

   if -1.44999999999999996 < F < 0.048000000000000001

   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. Step-by-step derivation

      1. div-inv 99.5%

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

   4. Applied egg-rr 99.5%

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

      1. add-sqr-sqrt 99.5%

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

      2. unpow-prod-down 99.5%

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

      3. +-commutative 99.5%

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

      4. fma-define 99.5%

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

      5. fma-define 99.5%

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

      6. metadata-eval 99.5%

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

      7. metadata-eval 99.5%

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

      8. +-commutative 99.5%

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

      9. fma-define 99.5%

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

      10. fma-define 99.5%

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

      11. metadata-eval 99.5%

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

      12. metadata-eval 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. pow-sqr 99.5%

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

      2. metadata-eval 99.5%

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

      3. unpow-1 99.5%

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

   8. Simplified 99.5%

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

   9. Taylor expanded in F around 0 98.7%

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

   if 0.048000000000000001 < F

   1. Initial program 56.0%

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

   3. Taylor expanded in F around -inf 47.1%

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

      1. div-inv 47.1%

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

      2. metadata-eval 47.1%

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

      3. rgt-mult-inverse 47.1%

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

      4. associate-*l/ 37.7%

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

      5. add-sqr-sqrt 16.8%

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

      6. sqrt-unprod 53.3%

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

      7. *-commutative 53.3%

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

      8. *-commutative 53.3%

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

      9. associate-*r/ 53.3%

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

      10. *-commutative 53.3%

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

      11. rgt-mult-inverse 53.3%

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

      12. associate-*r/ 56.5%

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

      13. *-commutative 56.5%

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

      14. rgt-mult-inverse 56.5%

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

      15. frac-times 56.5%

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

      16. metadata-eval 56.5%

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

      17. metadata-eval 56.5%

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

      18. frac-times 56.5%

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

      19. sqrt-unprod 49.0%

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

   5. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{0 - \left(\frac{x}{\tan B} + \frac{-1}{\sin B}\right)} \]
   6. Step-by-step derivation

      1. neg-sub0 99.8%

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

      2. distribute-neg-in 99.8%

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

      3. distribute-neg-frac 99.8%

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

      4. metadata-eval 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.45:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.048:\\ \;\;\;\;\frac{F}{\sin B} \cdot \frac{1}{\sqrt{2 + x \cdot 2}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -55000:\\ \;\;\;\;\frac{-1}{\sin B} - t\_1\\ \mathbf{elif}\;F \leq -4.9 \cdot 10^{-69}:\\ \;\;\;\;\frac{F}{\sin B} \cdot t\_0 - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.048:\\ \;\;\;\;t\_0 \cdot \frac{F}{B} - t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)) (t_1 (/ x (tan B))))
   (if (<= F -55000.0)
     (- (/ -1.0 (sin B)) t_1)
     (if (<= F -4.9e-69)
       (- (* (/ F (sin B)) t_0) (/ x B))
       (if (<= F 0.048) (- (* t_0 (/ F B)) t_1) (- (/ 1.0 (sin B)) t_1))))))

C

double code(double F, double B, double x) {
	double t_0 = pow(((2.0 + (F * F)) + (x * 2.0)), -0.5);
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -55000.0) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= -4.9e-69) {
		tmp = ((F / sin(B)) * t_0) - (x / B);
	} else if (F <= 0.048) {
		tmp = (t_0 * (F / B)) - t_1;
	} else {
		tmp = (1.0 / sin(B)) - t_1;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)
    t_1 = x / tan(b)
    if (f <= (-55000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_1
    else if (f <= (-4.9d-69)) then
        tmp = ((f / sin(b)) * t_0) - (x / b)
    else if (f <= 0.048d0) then
        tmp = (t_0 * (f / b)) - t_1
    else
        tmp = (1.0d0 / sin(b)) - t_1
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5);
	double t_1 = x / Math.tan(B);
	double tmp;
	if (F <= -55000.0) {
		tmp = (-1.0 / Math.sin(B)) - t_1;
	} else if (F <= -4.9e-69) {
		tmp = ((F / Math.sin(B)) * t_0) - (x / B);
	} else if (F <= 0.048) {
		tmp = (t_0 * (F / B)) - t_1;
	} 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)) + (x * 2.0)), -0.5)
	t_1 = x / math.tan(B)
	tmp = 0
	if F <= -55000.0:
		tmp = (-1.0 / math.sin(B)) - t_1
	elif F <= -4.9e-69:
		tmp = ((F / math.sin(B)) * t_0) - (x / B)
	elif F <= 0.048:
		tmp = (t_0 * (F / B)) - t_1
	else:
		tmp = (1.0 / math.sin(B)) - t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -55000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= -4.9e-69)
		tmp = Float64(Float64(Float64(F / sin(B)) * t_0) - Float64(x / B));
	elseif (F <= 0.048)
		tmp = Float64(Float64(t_0 * Float64(F / B)) - t_1);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = ((2.0 + (F * F)) + (x * 2.0)) ^ -0.5;
	t_1 = x / tan(B);
	tmp = 0.0;
	if (F <= -55000.0)
		tmp = (-1.0 / sin(B)) - t_1;
	elseif (F <= -4.9e-69)
		tmp = ((F / sin(B)) * t_0) - (x / B);
	elseif (F <= 0.048)
		tmp = (t_0 * (F / B)) - t_1;
	else
		tmp = (1.0 / sin(B)) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -55000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, -4.9e-69], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.048], N[(N[(t$95$0 * N[(F / B), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -55000

   1. Initial program 58.6%

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

   3. Taylor expanded in F around -inf 99.4%

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

      1. +-commutative 99.4%

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

      2. unsub-neg 99.4%

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

      3. un-div-inv 99.5%

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

   5. Applied egg-rr 99.5%

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

   if -55000 < F < -4.8999999999999998e-69

   1. Initial program 99.0%

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

   3. Taylor expanded in B around 0 87.5%

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

   if -4.8999999999999998e-69 < F < 0.048000000000000001

   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%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in B around 0 86.9%

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

   if 0.048000000000000001 < F

   1. Initial program 56.0%

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

   3. Taylor expanded in F around -inf 47.1%

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

      1. div-inv 47.1%

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

      2. metadata-eval 47.1%

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

      3. rgt-mult-inverse 47.1%

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

      4. associate-*l/ 37.7%

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

      5. add-sqr-sqrt 16.8%

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

      6. sqrt-unprod 53.3%

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

      7. *-commutative 53.3%

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

      8. *-commutative 53.3%

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

      9. associate-*r/ 53.3%

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

      10. *-commutative 53.3%

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

      11. rgt-mult-inverse 53.3%

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

      12. associate-*r/ 56.5%

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

      13. *-commutative 56.5%

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

      14. rgt-mult-inverse 56.5%

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

      15. frac-times 56.5%

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

      16. metadata-eval 56.5%

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

      17. metadata-eval 56.5%

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

      18. frac-times 56.5%

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

      19. sqrt-unprod 49.0%

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

   5. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{0 - \left(\frac{x}{\tan B} + \frac{-1}{\sin B}\right)} \]
   6. Step-by-step derivation

      1. neg-sub0 99.8%

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

      2. distribute-neg-in 99.8%

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

      3. distribute-neg-frac 99.8%

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

      4. metadata-eval 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -55000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -4.9 \cdot 10^{-69}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.048:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -31500:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq -9.4 \cdot 10^{-69}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.029:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -31500.0)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F -9.4e-69)
       (- (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)) (/ x B))
       (if (<= F 0.029)
         (- (* (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))) (/ F 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 <= -31500.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= -9.4e-69) {
		tmp = ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B);
	} else if (F <= 0.029) {
		tmp = (sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / 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 <= (-31500.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= (-9.4d-69)) then
        tmp = ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0))) - (x / b)
    else if (f <= 0.029d0) then
        tmp = (sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))) * (f / 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 <= -31500.0) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= -9.4e-69) {
		tmp = ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B);
	} else if (F <= 0.029) {
		tmp = (Math.sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / 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 <= -31500.0:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= -9.4e-69:
		tmp = ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B)
	elif F <= 0.029:
		tmp = (math.sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / 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 <= -31500.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= -9.4e-69)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)) - Float64(x / B));
	elseif (F <= 0.029)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0)))) * Float64(F / 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 <= -31500.0)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= -9.4e-69)
		tmp = ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5)) - (x / B);
	elseif (F <= 0.029)
		tmp = (sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / 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, -31500.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -9.4e-69], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.029], N[(N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / 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 4 regimes

2. if F < -31500

   1. Initial program 58.6%

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

   3. Taylor expanded in F around -inf 99.4%

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

      1. +-commutative 99.4%

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

      2. unsub-neg 99.4%

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

      3. un-div-inv 99.5%

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

   5. Applied egg-rr 99.5%

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

   if -31500 < F < -9.39999999999999935e-69

   1. Initial program 99.0%

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

   3. Taylor expanded in B around 0 87.5%

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

   if -9.39999999999999935e-69 < F < 0.0290000000000000015

   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%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in B around 0 86.9%

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

   6. Taylor expanded in F around 0 86.9%

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

      1. *-commutative 86.9%

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

   8. Simplified 86.9%

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

   if 0.0290000000000000015 < F

   1. Initial program 56.0%

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

   3. Taylor expanded in F around -inf 47.1%

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

      1. div-inv 47.1%

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

      2. metadata-eval 47.1%

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

      3. rgt-mult-inverse 47.1%

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

      4. associate-*l/ 37.7%

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

      5. add-sqr-sqrt 16.8%

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

      6. sqrt-unprod 53.3%

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

      7. *-commutative 53.3%

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

      8. *-commutative 53.3%

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

      9. associate-*r/ 53.3%

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

      10. *-commutative 53.3%

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

      11. rgt-mult-inverse 53.3%

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

      12. associate-*r/ 56.5%

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

      13. *-commutative 56.5%

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

      14. rgt-mult-inverse 56.5%

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

      15. frac-times 56.5%

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

      16. metadata-eval 56.5%

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

      17. metadata-eval 56.5%

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

      18. frac-times 56.5%

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

      19. sqrt-unprod 49.0%

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

   5. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{0 - \left(\frac{x}{\tan B} + \frac{-1}{\sin B}\right)} \]
   6. Step-by-step derivation

      1. neg-sub0 99.8%

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

      2. distribute-neg-in 99.8%

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

      3. distribute-neg-frac 99.8%

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

      4. metadata-eval 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -31500:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -9.4 \cdot 10^{-69}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.029:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{if}\;F \leq -1.45 \cdot 10^{+97}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq -1.15 \cdot 10^{+60}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -3.7 \cdot 10^{-35}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 1.85:\\ \;\;\;\;x \cdot \frac{-\cos B}{\sin B}\\ \mathbf{elif}\;F \leq 8.2 \cdot 10^{+108} \lor \neg \left(F \leq 5.4 \cdot 10^{+243}\right):\\ \;\;\;\;\frac{F}{F \cdot \sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ -1.0 (sin B)) (/ x B))))
   (if (<= F -1.45e+97)
     t_0
     (if (<= F -1.15e+60)
       (- (/ -1.0 B) (/ x (tan B)))
       (if (<= F -3.7e-35)
         t_0
         (if (<= F 1.85)
           (* x (/ (- (cos B)) (sin B)))
           (if (or (<= F 8.2e+108) (not (<= F 5.4e+243)))
             (- (/ F (* F (sin B))) (/ x B))
             (+ (* x (/ -1.0 (tan B))) (/ 1.0 B)))))))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 / sin(B)) - (x / B);
	double tmp;
	if (F <= -1.45e+97) {
		tmp = t_0;
	} else if (F <= -1.15e+60) {
		tmp = (-1.0 / B) - (x / tan(B));
	} else if (F <= -3.7e-35) {
		tmp = t_0;
	} else if (F <= 1.85) {
		tmp = x * (-cos(B) / sin(B));
	} else if ((F <= 8.2e+108) || !(F <= 5.4e+243)) {
		tmp = (F / (F * sin(B))) - (x / B);
	} else {
		tmp = (x * (-1.0 / tan(B))) + (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) :: t_0
    real(8) :: tmp
    t_0 = ((-1.0d0) / sin(b)) - (x / b)
    if (f <= (-1.45d+97)) then
        tmp = t_0
    else if (f <= (-1.15d+60)) then
        tmp = ((-1.0d0) / b) - (x / tan(b))
    else if (f <= (-3.7d-35)) then
        tmp = t_0
    else if (f <= 1.85d0) then
        tmp = x * (-cos(b) / sin(b))
    else if ((f <= 8.2d+108) .or. (.not. (f <= 5.4d+243))) then
        tmp = (f / (f * sin(b))) - (x / b)
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (-1.0 / Math.sin(B)) - (x / B);
	double tmp;
	if (F <= -1.45e+97) {
		tmp = t_0;
	} else if (F <= -1.15e+60) {
		tmp = (-1.0 / B) - (x / Math.tan(B));
	} else if (F <= -3.7e-35) {
		tmp = t_0;
	} else if (F <= 1.85) {
		tmp = x * (-Math.cos(B) / Math.sin(B));
	} else if ((F <= 8.2e+108) || !(F <= 5.4e+243)) {
		tmp = (F / (F * Math.sin(B))) - (x / B);
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (-1.0 / math.sin(B)) - (x / B)
	tmp = 0
	if F <= -1.45e+97:
		tmp = t_0
	elif F <= -1.15e+60:
		tmp = (-1.0 / B) - (x / math.tan(B))
	elif F <= -3.7e-35:
		tmp = t_0
	elif F <= 1.85:
		tmp = x * (-math.cos(B) / math.sin(B))
	elif (F <= 8.2e+108) or not (F <= 5.4e+243):
		tmp = (F / (F * math.sin(B))) - (x / B)
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 / sin(B)) - Float64(x / B))
	tmp = 0.0
	if (F <= -1.45e+97)
		tmp = t_0;
	elseif (F <= -1.15e+60)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / tan(B)));
	elseif (F <= -3.7e-35)
		tmp = t_0;
	elseif (F <= 1.85)
		tmp = Float64(x * Float64(Float64(-cos(B)) / sin(B)));
	elseif ((F <= 8.2e+108) || !(F <= 5.4e+243))
		tmp = Float64(Float64(F / Float64(F * sin(B))) - Float64(x / B));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (-1.0 / sin(B)) - (x / B);
	tmp = 0.0;
	if (F <= -1.45e+97)
		tmp = t_0;
	elseif (F <= -1.15e+60)
		tmp = (-1.0 / B) - (x / tan(B));
	elseif (F <= -3.7e-35)
		tmp = t_0;
	elseif (F <= 1.85)
		tmp = x * (-cos(B) / sin(B));
	elseif ((F <= 8.2e+108) || ~((F <= 5.4e+243)))
		tmp = (F / (F * sin(B))) - (x / B);
	else
		tmp = (x * (-1.0 / tan(B))) + (1.0 / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.45e+97], t$95$0, If[LessEqual[F, -1.15e+60], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -3.7e-35], t$95$0, If[LessEqual[F, 1.85], N[(x * N[((-N[Cos[B], $MachinePrecision]) / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[F, 8.2e+108], N[Not[LessEqual[F, 5.4e+243]], $MachinePrecision]], N[(N[(F / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -1.44999999999999994e97 or -1.15000000000000008e60 < F < -3.6999999999999999e-35

   1. Initial program 58.6%

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

   3. Taylor expanded in F around -inf 90.9%

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

   4. Taylor expanded in B around 0 75.7%

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

   if -1.44999999999999994e97 < F < -1.15000000000000008e60

   1. Initial program 99.1%

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

   3. Taylor expanded in F around -inf 99.4%

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

      1. +-commutative 99.4%

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

      2. unsub-neg 99.4%

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

      3. un-div-inv 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in B around 0 99.9%

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

   if -3.6999999999999999e-35 < F < 1.8500000000000001

   1. Initial program 99.3%

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

   3. Taylor expanded in F around -inf 34.0%

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

   4. Taylor expanded in x around inf 70.6%

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

      1. mul-1-neg 70.6%

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

      2. associate-/l* 70.5%

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

      3. distribute-lft-neg-in 70.5%

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

   6. Simplified 70.5%

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

   if 1.8500000000000001 < F < 8.1999999999999998e108 or 5.4000000000000003e243 < F

   1. Initial program 51.8%

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

   3. Taylor expanded in F around inf 73.7%

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

   4. Taylor expanded in B around 0 61.1%

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

      1. un-div-inv 61.1%

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

      2. associate-/l/ 86.9%

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

   6. Applied egg-rr 86.9%

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

   if 8.1999999999999998e108 < F < 5.4000000000000003e243

   1. Initial program 60.6%

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

   3. Taylor expanded in F around inf 77.4%

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

   4. Taylor expanded in B around 0 80.7%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{B}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.45 \cdot 10^{+97}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.15 \cdot 10^{+60}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -3.7 \cdot 10^{-35}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.85:\\ \;\;\;\;x \cdot \frac{-\cos B}{\sin B}\\ \mathbf{elif}\;F \leq 8.2 \cdot 10^{+108} \lor \neg \left(F \leq 5.4 \cdot 10^{+243}\right):\\ \;\;\;\;\frac{F}{F \cdot \sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{if}\;F \leq -9.5 \cdot 10^{+96}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq -6.2 \cdot 10^{+58}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -3.7 \cdot 10^{-35}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 3:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{elif}\;F \leq 1.75 \cdot 10^{+109} \lor \neg \left(F \leq 4.2 \cdot 10^{+242}\right):\\ \;\;\;\;\frac{F}{F \cdot \sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ -1.0 (sin B)) (/ x B))))
   (if (<= F -9.5e+96)
     t_0
     (if (<= F -6.2e+58)
       (- (/ -1.0 B) (/ x (tan B)))
       (if (<= F -3.7e-35)
         t_0
         (if (<= F 3.0)
           (/ (* x (cos B)) (- (sin B)))
           (if (or (<= F 1.75e+109) (not (<= F 4.2e+242)))
             (- (/ F (* F (sin B))) (/ x B))
             (+ (* x (/ -1.0 (tan B))) (/ 1.0 B)))))))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 / sin(B)) - (x / B);
	double tmp;
	if (F <= -9.5e+96) {
		tmp = t_0;
	} else if (F <= -6.2e+58) {
		tmp = (-1.0 / B) - (x / tan(B));
	} else if (F <= -3.7e-35) {
		tmp = t_0;
	} else if (F <= 3.0) {
		tmp = (x * cos(B)) / -sin(B);
	} else if ((F <= 1.75e+109) || !(F <= 4.2e+242)) {
		tmp = (F / (F * sin(B))) - (x / B);
	} else {
		tmp = (x * (-1.0 / tan(B))) + (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) :: t_0
    real(8) :: tmp
    t_0 = ((-1.0d0) / sin(b)) - (x / b)
    if (f <= (-9.5d+96)) then
        tmp = t_0
    else if (f <= (-6.2d+58)) then
        tmp = ((-1.0d0) / b) - (x / tan(b))
    else if (f <= (-3.7d-35)) then
        tmp = t_0
    else if (f <= 3.0d0) then
        tmp = (x * cos(b)) / -sin(b)
    else if ((f <= 1.75d+109) .or. (.not. (f <= 4.2d+242))) then
        tmp = (f / (f * sin(b))) - (x / b)
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (-1.0 / Math.sin(B)) - (x / B);
	double tmp;
	if (F <= -9.5e+96) {
		tmp = t_0;
	} else if (F <= -6.2e+58) {
		tmp = (-1.0 / B) - (x / Math.tan(B));
	} else if (F <= -3.7e-35) {
		tmp = t_0;
	} else if (F <= 3.0) {
		tmp = (x * Math.cos(B)) / -Math.sin(B);
	} else if ((F <= 1.75e+109) || !(F <= 4.2e+242)) {
		tmp = (F / (F * Math.sin(B))) - (x / B);
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (-1.0 / math.sin(B)) - (x / B)
	tmp = 0
	if F <= -9.5e+96:
		tmp = t_0
	elif F <= -6.2e+58:
		tmp = (-1.0 / B) - (x / math.tan(B))
	elif F <= -3.7e-35:
		tmp = t_0
	elif F <= 3.0:
		tmp = (x * math.cos(B)) / -math.sin(B)
	elif (F <= 1.75e+109) or not (F <= 4.2e+242):
		tmp = (F / (F * math.sin(B))) - (x / B)
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 / sin(B)) - Float64(x / B))
	tmp = 0.0
	if (F <= -9.5e+96)
		tmp = t_0;
	elseif (F <= -6.2e+58)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / tan(B)));
	elseif (F <= -3.7e-35)
		tmp = t_0;
	elseif (F <= 3.0)
		tmp = Float64(Float64(x * cos(B)) / Float64(-sin(B)));
	elseif ((F <= 1.75e+109) || !(F <= 4.2e+242))
		tmp = Float64(Float64(F / Float64(F * sin(B))) - Float64(x / B));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (-1.0 / sin(B)) - (x / B);
	tmp = 0.0;
	if (F <= -9.5e+96)
		tmp = t_0;
	elseif (F <= -6.2e+58)
		tmp = (-1.0 / B) - (x / tan(B));
	elseif (F <= -3.7e-35)
		tmp = t_0;
	elseif (F <= 3.0)
		tmp = (x * cos(B)) / -sin(B);
	elseif ((F <= 1.75e+109) || ~((F <= 4.2e+242)))
		tmp = (F / (F * sin(B))) - (x / B);
	else
		tmp = (x * (-1.0 / tan(B))) + (1.0 / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -9.5e+96], t$95$0, If[LessEqual[F, -6.2e+58], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -3.7e-35], t$95$0, If[LessEqual[F, 3.0], N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], If[Or[LessEqual[F, 1.75e+109], N[Not[LessEqual[F, 4.2e+242]], $MachinePrecision]], N[(N[(F / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -9.50000000000000089e96 or -6.1999999999999998e58 < F < -3.6999999999999999e-35

   1. Initial program 58.6%

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

   3. Taylor expanded in F around -inf 90.9%

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

   4. Taylor expanded in B around 0 75.7%

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

   if -9.50000000000000089e96 < F < -6.1999999999999998e58

   1. Initial program 99.1%

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

   3. Taylor expanded in F around -inf 99.4%

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

      1. +-commutative 99.4%

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

      2. unsub-neg 99.4%

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

      3. un-div-inv 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in B around 0 99.9%

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

   if -3.6999999999999999e-35 < F < 3

   1. Initial program 99.3%

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

   3. Taylor expanded in F around -inf 34.0%

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

   4. Taylor expanded in x around inf 70.6%

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

   if 3 < F < 1.74999999999999992e109 or 4.1999999999999999e242 < F

   1. Initial program 51.8%

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

   3. Taylor expanded in F around inf 73.7%

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

   4. Taylor expanded in B around 0 61.1%

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

      1. un-div-inv 61.1%

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

      2. associate-/l/ 86.9%

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

   6. Applied egg-rr 86.9%

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

   if 1.74999999999999992e109 < F < 4.1999999999999999e242

   1. Initial program 60.6%

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

   3. Taylor expanded in F around inf 77.4%

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

   4. Taylor expanded in B around 0 80.7%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{B}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -9.5 \cdot 10^{+96}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -6.2 \cdot 10^{+58}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -3.7 \cdot 10^{-35}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{elif}\;F \leq 1.75 \cdot 10^{+109} \lor \neg \left(F \leq 4.2 \cdot 10^{+242}\right):\\ \;\;\;\;\frac{F}{F \cdot \sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 91.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.0003:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 0.029:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -0.0003)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 0.029)
       (- (* (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))) (/ F B)) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -0.0003) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 0.029) {
		tmp = (sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-0.0003d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 0.029d0) then
        tmp = (sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))) * (f / b)) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -0.0003) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 0.029) {
		tmp = (Math.sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -0.0003:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 0.029:
		tmp = (math.sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -0.0003)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 0.029)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0)))) * Float64(F / B)) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -0.0003)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 0.029)
		tmp = (sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.0003], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.029], N[(N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / 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 < -2.99999999999999974e-4

   1. Initial program 60.9%

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

   3. Taylor expanded in F around -inf 96.5%

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

      1. +-commutative 96.5%

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

      2. unsub-neg 96.5%

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

      3. un-div-inv 96.5%

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

   5. Applied egg-rr 96.5%

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

   if -2.99999999999999974e-4 < F < 0.0290000000000000015

   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. Step-by-step derivation

      1. div-inv 99.5%

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in B around 0 83.7%

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

   6. Taylor expanded in F around 0 83.7%

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

      1. *-commutative 83.7%

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

   8. Simplified 83.7%

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

   if 0.0290000000000000015 < F

   1. Initial program 56.0%

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

   3. Taylor expanded in F around -inf 47.1%

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

      1. div-inv 47.1%

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

      2. metadata-eval 47.1%

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

      3. rgt-mult-inverse 47.1%

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

      4. associate-*l/ 37.7%

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

      5. add-sqr-sqrt 16.8%

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

      6. sqrt-unprod 53.3%

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

      7. *-commutative 53.3%

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

      8. *-commutative 53.3%

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

      9. associate-*r/ 53.3%

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

      10. *-commutative 53.3%

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

      11. rgt-mult-inverse 53.3%

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

      12. associate-*r/ 56.5%

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

      13. *-commutative 56.5%

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

      14. rgt-mult-inverse 56.5%

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

      15. frac-times 56.5%

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

      16. metadata-eval 56.5%

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

      17. metadata-eval 56.5%

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

      18. frac-times 56.5%

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

      19. sqrt-unprod 49.0%

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

   5. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{0 - \left(\frac{x}{\tan B} + \frac{-1}{\sin B}\right)} \]
   6. Step-by-step derivation

      1. neg-sub0 99.8%

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

      2. distribute-neg-in 99.8%

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

      3. distribute-neg-frac 99.8%

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

      4. metadata-eval 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.0003:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.029:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 84.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -7.6 \cdot 10^{-58}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 8 \cdot 10^{-34}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -7.6e-58)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 8e-34) (/ (* x (cos B)) (- (sin B))) (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -7.6e-58) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 8e-34) {
		tmp = (x * cos(B)) / -sin(B);
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-7.6d-58)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 8d-34) then
        tmp = (x * cos(b)) / -sin(b)
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -7.6e-58) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 8e-34) {
		tmp = (x * Math.cos(B)) / -Math.sin(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -7.6e-58:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 8e-34:
		tmp = (x * math.cos(B)) / -math.sin(B)
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -7.6e-58)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 8e-34)
		tmp = Float64(Float64(x * cos(B)) / Float64(-sin(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -7.6e-58)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 8e-34)
		tmp = (x * cos(B)) / -sin(B);
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -7.6e-58], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 8e-34], N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -7.5999999999999995e-58

   1. Initial program 64.0%

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

   3. Taylor expanded in F around -inf 91.1%

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

      1. +-commutative 91.1%

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

      2. unsub-neg 91.1%

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

      3. un-div-inv 91.1%

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

   5. Applied egg-rr 91.1%

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

   if -7.5999999999999995e-58 < F < 7.99999999999999942e-34

   1. Initial program 99.4%

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

   3. Taylor expanded in F around -inf 32.2%

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

   4. Taylor expanded in x around inf 72.4%

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

   if 7.99999999999999942e-34 < F

   1. Initial program 60.4%

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

   3. Taylor expanded in F around -inf 46.7%

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

      1. div-inv 46.7%

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

      2. metadata-eval 46.7%

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

      3. rgt-mult-inverse 46.7%

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

      4. associate-*l/ 38.2%

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

      5. add-sqr-sqrt 18.0%

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

      6. sqrt-unprod 52.5%

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

      7. *-commutative 52.5%

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

      8. *-commutative 52.5%

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

      9. associate-*r/ 52.5%

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

      10. *-commutative 52.5%

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

      11. rgt-mult-inverse 52.5%

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

      12. associate-*r/ 55.4%

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

      13. *-commutative 55.4%

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

      14. rgt-mult-inverse 55.4%

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

      15. frac-times 55.4%

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

      16. metadata-eval 55.4%

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

      17. metadata-eval 55.4%

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

      18. frac-times 55.4%

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

      19. sqrt-unprod 45.8%

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

   5. Applied egg-rr 94.7%

      \[\leadsto \color{blue}{0 - \left(\frac{x}{\tan B} + \frac{-1}{\sin B}\right)} \]
   6. Step-by-step derivation

      1. neg-sub0 94.7%

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

      2. distribute-neg-in 94.7%

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

      3. distribute-neg-frac 94.7%

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

      4. metadata-eval 94.7%

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

   7. Simplified 94.7%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7.6 \cdot 10^{-58}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 8 \cdot 10^{-34}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 77.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -7.6 \cdot 10^{-58}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 3850:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{elif}\;F \leq 3.7 \cdot 10^{+109} \lor \neg \left(F \leq 1.9 \cdot 10^{+244}\right):\\ \;\;\;\;\frac{F}{F \cdot \sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -7.6e-58)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 3850.0)
     (/ (* x (cos B)) (- (sin B)))
     (if (or (<= F 3.7e+109) (not (<= F 1.9e+244)))
       (- (/ F (* F (sin B))) (/ x B))
       (+ (* x (/ -1.0 (tan B))) (/ 1.0 B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -7.6e-58) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 3850.0) {
		tmp = (x * cos(B)) / -sin(B);
	} else if ((F <= 3.7e+109) || !(F <= 1.9e+244)) {
		tmp = (F / (F * sin(B))) - (x / B);
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-7.6d-58)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 3850.0d0) then
        tmp = (x * cos(b)) / -sin(b)
    else if ((f <= 3.7d+109) .or. (.not. (f <= 1.9d+244))) then
        tmp = (f / (f * sin(b))) - (x / b)
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -7.6e-58) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 3850.0) {
		tmp = (x * Math.cos(B)) / -Math.sin(B);
	} else if ((F <= 3.7e+109) || !(F <= 1.9e+244)) {
		tmp = (F / (F * Math.sin(B))) - (x / B);
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -7.6e-58:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 3850.0:
		tmp = (x * math.cos(B)) / -math.sin(B)
	elif (F <= 3.7e+109) or not (F <= 1.9e+244):
		tmp = (F / (F * math.sin(B))) - (x / B)
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -7.6e-58)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 3850.0)
		tmp = Float64(Float64(x * cos(B)) / Float64(-sin(B)));
	elseif ((F <= 3.7e+109) || !(F <= 1.9e+244))
		tmp = Float64(Float64(F / Float64(F * sin(B))) - Float64(x / B));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -7.6e-58)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 3850.0)
		tmp = (x * cos(B)) / -sin(B);
	elseif ((F <= 3.7e+109) || ~((F <= 1.9e+244)))
		tmp = (F / (F * sin(B))) - (x / B);
	else
		tmp = (x * (-1.0 / tan(B))) + (1.0 / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -7.6e-58], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3850.0], N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], If[Or[LessEqual[F, 3.7e+109], N[Not[LessEqual[F, 1.9e+244]], $MachinePrecision]], N[(N[(F / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -7.5999999999999995e-58

   1. Initial program 64.0%

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

   3. Taylor expanded in F around -inf 91.1%

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

      1. +-commutative 91.1%

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

      2. unsub-neg 91.1%

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

      3. un-div-inv 91.1%

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

   5. Applied egg-rr 91.1%

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

   if -7.5999999999999995e-58 < F < 3850

   1. Initial program 99.3%

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

   3. Taylor expanded in F around -inf 33.7%

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

   4. Taylor expanded in x around inf 71.0%

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

   if 3850 < F < 3.7000000000000002e109 or 1.89999999999999991e244 < F

   1. Initial program 51.8%

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

   3. Taylor expanded in F around inf 73.7%

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

   4. Taylor expanded in B around 0 61.1%

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

      1. un-div-inv 61.1%

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

      2. associate-/l/ 86.9%

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

   6. Applied egg-rr 86.9%

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

   if 3.7000000000000002e109 < F < 1.89999999999999991e244

   1. Initial program 60.6%

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

   3. Taylor expanded in F around inf 77.4%

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

   4. Taylor expanded in B around 0 80.7%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7.6 \cdot 10^{-58}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 3850:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{elif}\;F \leq 3.7 \cdot 10^{+109} \lor \neg \left(F \leq 1.9 \cdot 10^{+244}\right):\\ \;\;\;\;\frac{F}{F \cdot \sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{\sin B} - \frac{x}{B}\\ t_1 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -2.8 \cdot 10^{+96}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq -1.55 \cdot 10^{+60}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -3.7 \cdot 10^{-35}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 2.05 \cdot 10^{+42}:\\ \;\;\;\;t\_1 + \frac{\frac{1}{F}}{\frac{B}{F}}\\ \mathbf{elif}\;F \leq 9.2 \cdot 10^{+108} \lor \neg \left(F \leq 4.2 \cdot 10^{+242}\right):\\ \;\;\;\;\frac{F}{F \cdot \sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ -1.0 (sin B)) (/ x B))) (t_1 (* x (/ -1.0 (tan B)))))
   (if (<= F -2.8e+96)
     t_0
     (if (<= F -1.55e+60)
       (- (/ -1.0 B) (/ x (tan B)))
       (if (<= F -3.7e-35)
         t_0
         (if (<= F 2.05e+42)
           (+ t_1 (/ (/ 1.0 F) (/ B F)))
           (if (or (<= F 9.2e+108) (not (<= F 4.2e+242)))
             (- (/ F (* F (sin B))) (/ x B))
             (+ t_1 (/ 1.0 B)))))))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 / sin(B)) - (x / B);
	double t_1 = x * (-1.0 / tan(B));
	double tmp;
	if (F <= -2.8e+96) {
		tmp = t_0;
	} else if (F <= -1.55e+60) {
		tmp = (-1.0 / B) - (x / tan(B));
	} else if (F <= -3.7e-35) {
		tmp = t_0;
	} else if (F <= 2.05e+42) {
		tmp = t_1 + ((1.0 / F) / (B / F));
	} else if ((F <= 9.2e+108) || !(F <= 4.2e+242)) {
		tmp = (F / (F * sin(B))) - (x / B);
	} else {
		tmp = t_1 + (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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((-1.0d0) / sin(b)) - (x / b)
    t_1 = x * ((-1.0d0) / tan(b))
    if (f <= (-2.8d+96)) then
        tmp = t_0
    else if (f <= (-1.55d+60)) then
        tmp = ((-1.0d0) / b) - (x / tan(b))
    else if (f <= (-3.7d-35)) then
        tmp = t_0
    else if (f <= 2.05d+42) then
        tmp = t_1 + ((1.0d0 / f) / (b / f))
    else if ((f <= 9.2d+108) .or. (.not. (f <= 4.2d+242))) then
        tmp = (f / (f * sin(b))) - (x / b)
    else
        tmp = t_1 + (1.0d0 / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (-1.0 / Math.sin(B)) - (x / B);
	double t_1 = x * (-1.0 / Math.tan(B));
	double tmp;
	if (F <= -2.8e+96) {
		tmp = t_0;
	} else if (F <= -1.55e+60) {
		tmp = (-1.0 / B) - (x / Math.tan(B));
	} else if (F <= -3.7e-35) {
		tmp = t_0;
	} else if (F <= 2.05e+42) {
		tmp = t_1 + ((1.0 / F) / (B / F));
	} else if ((F <= 9.2e+108) || !(F <= 4.2e+242)) {
		tmp = (F / (F * Math.sin(B))) - (x / B);
	} else {
		tmp = t_1 + (1.0 / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (-1.0 / math.sin(B)) - (x / B)
	t_1 = x * (-1.0 / math.tan(B))
	tmp = 0
	if F <= -2.8e+96:
		tmp = t_0
	elif F <= -1.55e+60:
		tmp = (-1.0 / B) - (x / math.tan(B))
	elif F <= -3.7e-35:
		tmp = t_0
	elif F <= 2.05e+42:
		tmp = t_1 + ((1.0 / F) / (B / F))
	elif (F <= 9.2e+108) or not (F <= 4.2e+242):
		tmp = (F / (F * math.sin(B))) - (x / B)
	else:
		tmp = t_1 + (1.0 / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 / sin(B)) - Float64(x / B))
	t_1 = Float64(x * Float64(-1.0 / tan(B)))
	tmp = 0.0
	if (F <= -2.8e+96)
		tmp = t_0;
	elseif (F <= -1.55e+60)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / tan(B)));
	elseif (F <= -3.7e-35)
		tmp = t_0;
	elseif (F <= 2.05e+42)
		tmp = Float64(t_1 + Float64(Float64(1.0 / F) / Float64(B / F)));
	elseif ((F <= 9.2e+108) || !(F <= 4.2e+242))
		tmp = Float64(Float64(F / Float64(F * sin(B))) - Float64(x / B));
	else
		tmp = Float64(t_1 + Float64(1.0 / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (-1.0 / sin(B)) - (x / B);
	t_1 = x * (-1.0 / tan(B));
	tmp = 0.0;
	if (F <= -2.8e+96)
		tmp = t_0;
	elseif (F <= -1.55e+60)
		tmp = (-1.0 / B) - (x / tan(B));
	elseif (F <= -3.7e-35)
		tmp = t_0;
	elseif (F <= 2.05e+42)
		tmp = t_1 + ((1.0 / F) / (B / F));
	elseif ((F <= 9.2e+108) || ~((F <= 4.2e+242)))
		tmp = (F / (F * sin(B))) - (x / B);
	else
		tmp = t_1 + (1.0 / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.8e+96], t$95$0, If[LessEqual[F, -1.55e+60], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -3.7e-35], t$95$0, If[LessEqual[F, 2.05e+42], N[(t$95$1 + N[(N[(1.0 / F), $MachinePrecision] / N[(B / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[F, 9.2e+108], N[Not[LessEqual[F, 4.2e+242]], $MachinePrecision]], N[(N[(F / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -2.8e96 or -1.55e60 < F < -3.6999999999999999e-35

   1. Initial program 58.6%

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

   3. Taylor expanded in F around -inf 90.9%

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

   4. Taylor expanded in B around 0 75.7%

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

   if -2.8e96 < F < -1.55e60

   1. Initial program 99.1%

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

   3. Taylor expanded in F around -inf 99.4%

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

      1. +-commutative 99.4%

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

      2. unsub-neg 99.4%

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

      3. un-div-inv 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in B around 0 99.9%

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

   if -3.6999999999999999e-35 < F < 2.05e42

   1. Initial program 99.3%

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

   3. Taylor expanded in F around inf 37.5%

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

      1. *-commutative 37.5%

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

      2. clear-num 37.5%

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

      3. un-div-inv 37.5%

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

   5. Applied egg-rr 37.5%

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

   6. Taylor expanded in B around 0 49.8%

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

   if 2.05e42 < F < 9.1999999999999996e108 or 4.1999999999999999e242 < F

   1. Initial program 46.0%

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

   3. Taylor expanded in F around inf 70.6%

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

   4. Taylor expanded in B around 0 59.1%

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

      1. un-div-inv 59.1%

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

      2. associate-/l/ 88.1%

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

   6. Applied egg-rr 88.1%

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

   if 9.1999999999999996e108 < F < 4.1999999999999999e242

   1. Initial program 60.6%

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

   3. Taylor expanded in F around inf 77.4%

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

   4. Taylor expanded in B around 0 80.7%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{B}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.8 \cdot 10^{+96}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.55 \cdot 10^{+60}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -3.7 \cdot 10^{-35}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.05 \cdot 10^{+42}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{\frac{1}{F}}{\frac{B}{F}}\\ \mathbf{elif}\;F \leq 9.2 \cdot 10^{+108} \lor \neg \left(F \leq 4.2 \cdot 10^{+242}\right):\\ \;\;\;\;\frac{F}{F \cdot \sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 60.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ t_1 := \frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{if}\;F \leq -3 \cdot 10^{+96}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;F \leq -9.5 \cdot 10^{+59}:\\ \;\;\;\;\frac{-1}{B} - t\_0\\ \mathbf{elif}\;F \leq -1.25 \cdot 10^{-35}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;F \leq -4.5 \cdot 10^{-270}:\\ \;\;\;\;\frac{F}{B} \cdot \frac{-1}{F} - t\_0\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{-177}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;F \leq 2.55 \cdot 10^{+39}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{F \cdot \sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))) (t_1 (- (/ -1.0 (sin B)) (/ x B))))
   (if (<= F -3e+96)
     t_1
     (if (<= F -9.5e+59)
       (- (/ -1.0 B) t_0)
       (if (<= F -1.25e-35)
         t_1
         (if (<= F -4.5e-270)
           (- (* (/ F B) (/ -1.0 F)) t_0)
           (if (<= F 1.4e-177)
             (/ x (- B))
             (if (<= F 2.55e+39)
               (+ (* x (/ -1.0 (tan B))) (/ 1.0 B))
               (- (/ F (* F (sin B))) (/ x B))))))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double t_1 = (-1.0 / sin(B)) - (x / B);
	double tmp;
	if (F <= -3e+96) {
		tmp = t_1;
	} else if (F <= -9.5e+59) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= -1.25e-35) {
		tmp = t_1;
	} else if (F <= -4.5e-270) {
		tmp = ((F / B) * (-1.0 / F)) - t_0;
	} else if (F <= 1.4e-177) {
		tmp = x / -B;
	} else if (F <= 2.55e+39) {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / B);
	} else {
		tmp = (F / (F * sin(B))) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x / tan(b)
    t_1 = ((-1.0d0) / sin(b)) - (x / b)
    if (f <= (-3d+96)) then
        tmp = t_1
    else if (f <= (-9.5d+59)) then
        tmp = ((-1.0d0) / b) - t_0
    else if (f <= (-1.25d-35)) then
        tmp = t_1
    else if (f <= (-4.5d-270)) then
        tmp = ((f / b) * ((-1.0d0) / f)) - t_0
    else if (f <= 1.4d-177) then
        tmp = x / -b
    else if (f <= 2.55d+39) then
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / b)
    else
        tmp = (f / (f * sin(b))) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double t_1 = (-1.0 / Math.sin(B)) - (x / B);
	double tmp;
	if (F <= -3e+96) {
		tmp = t_1;
	} else if (F <= -9.5e+59) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= -1.25e-35) {
		tmp = t_1;
	} else if (F <= -4.5e-270) {
		tmp = ((F / B) * (-1.0 / F)) - t_0;
	} else if (F <= 1.4e-177) {
		tmp = x / -B;
	} else if (F <= 2.55e+39) {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / B);
	} else {
		tmp = (F / (F * Math.sin(B))) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	t_1 = (-1.0 / math.sin(B)) - (x / B)
	tmp = 0
	if F <= -3e+96:
		tmp = t_1
	elif F <= -9.5e+59:
		tmp = (-1.0 / B) - t_0
	elif F <= -1.25e-35:
		tmp = t_1
	elif F <= -4.5e-270:
		tmp = ((F / B) * (-1.0 / F)) - t_0
	elif F <= 1.4e-177:
		tmp = x / -B
	elif F <= 2.55e+39:
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / B)
	else:
		tmp = (F / (F * math.sin(B))) - (x / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	t_1 = Float64(Float64(-1.0 / sin(B)) - Float64(x / B))
	tmp = 0.0
	if (F <= -3e+96)
		tmp = t_1;
	elseif (F <= -9.5e+59)
		tmp = Float64(Float64(-1.0 / B) - t_0);
	elseif (F <= -1.25e-35)
		tmp = t_1;
	elseif (F <= -4.5e-270)
		tmp = Float64(Float64(Float64(F / B) * Float64(-1.0 / F)) - t_0);
	elseif (F <= 1.4e-177)
		tmp = Float64(x / Float64(-B));
	elseif (F <= 2.55e+39)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / B));
	else
		tmp = Float64(Float64(F / Float64(F * sin(B))) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	t_1 = (-1.0 / sin(B)) - (x / B);
	tmp = 0.0;
	if (F <= -3e+96)
		tmp = t_1;
	elseif (F <= -9.5e+59)
		tmp = (-1.0 / B) - t_0;
	elseif (F <= -1.25e-35)
		tmp = t_1;
	elseif (F <= -4.5e-270)
		tmp = ((F / B) * (-1.0 / F)) - t_0;
	elseif (F <= 1.4e-177)
		tmp = x / -B;
	elseif (F <= 2.55e+39)
		tmp = (x * (-1.0 / tan(B))) + (1.0 / B);
	else
		tmp = (F / (F * sin(B))) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -3e+96], t$95$1, If[LessEqual[F, -9.5e+59], N[(N[(-1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -1.25e-35], t$95$1, If[LessEqual[F, -4.5e-270], N[(N[(N[(F / B), $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.4e-177], N[(x / (-B)), $MachinePrecision], If[LessEqual[F, 2.55e+39], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / B), $MachinePrecision]), $MachinePrecision], N[(N[(F / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if F < -3e96 or -9.50000000000000023e59 < F < -1.24999999999999991e-35

   1. Initial program 58.6%

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

   3. Taylor expanded in F around -inf 90.9%

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

   4. Taylor expanded in B around 0 75.7%

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

   if -3e96 < F < -9.50000000000000023e59

   1. Initial program 99.1%

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

   3. Taylor expanded in F around -inf 99.4%

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

      1. +-commutative 99.4%

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

      2. unsub-neg 99.4%

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

      3. un-div-inv 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in B around 0 99.9%

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

   if -1.24999999999999991e-35 < F < -4.49999999999999998e-270

   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 99.5%

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in B around 0 81.4%

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

   6. Taylor expanded in F around -inf 51.5%

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

   if -4.49999999999999998e-270 < F < 1.39999999999999993e-177

   1. Initial program 99.4%

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

   3. Taylor expanded in F around -inf 25.4%

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

   4. Taylor expanded in B around 0 9.7%

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

      1. mul-1-neg 9.7%

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

      2. distribute-neg-frac2 9.7%

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

   6. Simplified 9.7%

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

   7. Taylor expanded in x around inf 51.0%

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

      1. associate-*r/ 51.0%

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

      2. neg-mul-1 51.0%

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

   9. Simplified 51.0%

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

   if 1.39999999999999993e-177 < F < 2.5499999999999999e39

   1. Initial program 99.3%

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

   3. Taylor expanded in F around inf 47.7%

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

   4. Taylor expanded in B around 0 56.4%

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

   if 2.5499999999999999e39 < F

   1. Initial program 52.3%

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

   3. Taylor expanded in F around inf 73.5%

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

   4. Taylor expanded in B around 0 50.1%

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

      1. un-div-inv 50.1%

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

      2. associate-/l/ 76.4%

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

   6. Applied egg-rr 76.4%

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{F}{F \cdot \sin B}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3 \cdot 10^{+96}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -9.5 \cdot 10^{+59}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.25 \cdot 10^{-35}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -4.5 \cdot 10^{-270}:\\ \;\;\;\;\frac{F}{B} \cdot \frac{-1}{F} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{-177}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;F \leq 2.55 \cdot 10^{+39}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{F \cdot \sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 60.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{B} - \frac{x}{\tan B}\\ t_1 := \frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{if}\;F \leq -3 \cdot 10^{+96}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;F \leq -1.25 \cdot 10^{+60}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq -3.7 \cdot 10^{-35}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;F \leq -1.75 \cdot 10^{-234}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 9.4 \cdot 10^{-178}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ -1.0 B) (/ x (tan B)))) (t_1 (- (/ -1.0 (sin B)) (/ x B))))
   (if (<= F -3e+96)
     t_1
     (if (<= F -1.25e+60)
       t_0
       (if (<= F -3.7e-35)
         t_1
         (if (<= F -1.75e-234)
           t_0
           (if (<= F 9.4e-178)
             (/ x (- B))
             (+ (* x (/ -1.0 (tan B))) (/ 1.0 B)))))))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 / B) - (x / tan(B));
	double t_1 = (-1.0 / sin(B)) - (x / B);
	double tmp;
	if (F <= -3e+96) {
		tmp = t_1;
	} else if (F <= -1.25e+60) {
		tmp = t_0;
	} else if (F <= -3.7e-35) {
		tmp = t_1;
	} else if (F <= -1.75e-234) {
		tmp = t_0;
	} else if (F <= 9.4e-178) {
		tmp = x / -B;
	} else {
		tmp = (x * (-1.0 / tan(B))) + (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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((-1.0d0) / b) - (x / tan(b))
    t_1 = ((-1.0d0) / sin(b)) - (x / b)
    if (f <= (-3d+96)) then
        tmp = t_1
    else if (f <= (-1.25d+60)) then
        tmp = t_0
    else if (f <= (-3.7d-35)) then
        tmp = t_1
    else if (f <= (-1.75d-234)) then
        tmp = t_0
    else if (f <= 9.4d-178) then
        tmp = x / -b
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (-1.0 / B) - (x / Math.tan(B));
	double t_1 = (-1.0 / Math.sin(B)) - (x / B);
	double tmp;
	if (F <= -3e+96) {
		tmp = t_1;
	} else if (F <= -1.25e+60) {
		tmp = t_0;
	} else if (F <= -3.7e-35) {
		tmp = t_1;
	} else if (F <= -1.75e-234) {
		tmp = t_0;
	} else if (F <= 9.4e-178) {
		tmp = x / -B;
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (-1.0 / B) - (x / math.tan(B))
	t_1 = (-1.0 / math.sin(B)) - (x / B)
	tmp = 0
	if F <= -3e+96:
		tmp = t_1
	elif F <= -1.25e+60:
		tmp = t_0
	elif F <= -3.7e-35:
		tmp = t_1
	elif F <= -1.75e-234:
		tmp = t_0
	elif F <= 9.4e-178:
		tmp = x / -B
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 / B) - Float64(x / tan(B)))
	t_1 = Float64(Float64(-1.0 / sin(B)) - Float64(x / B))
	tmp = 0.0
	if (F <= -3e+96)
		tmp = t_1;
	elseif (F <= -1.25e+60)
		tmp = t_0;
	elseif (F <= -3.7e-35)
		tmp = t_1;
	elseif (F <= -1.75e-234)
		tmp = t_0;
	elseif (F <= 9.4e-178)
		tmp = Float64(x / Float64(-B));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (-1.0 / B) - (x / tan(B));
	t_1 = (-1.0 / sin(B)) - (x / B);
	tmp = 0.0;
	if (F <= -3e+96)
		tmp = t_1;
	elseif (F <= -1.25e+60)
		tmp = t_0;
	elseif (F <= -3.7e-35)
		tmp = t_1;
	elseif (F <= -1.75e-234)
		tmp = t_0;
	elseif (F <= 9.4e-178)
		tmp = x / -B;
	else
		tmp = (x * (-1.0 / tan(B))) + (1.0 / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -3e+96], t$95$1, If[LessEqual[F, -1.25e+60], t$95$0, If[LessEqual[F, -3.7e-35], t$95$1, If[LessEqual[F, -1.75e-234], t$95$0, If[LessEqual[F, 9.4e-178], N[(x / (-B)), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3e96 or -1.24999999999999994e60 < F < -3.6999999999999999e-35

   1. Initial program 58.6%

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

   3. Taylor expanded in F around -inf 90.9%

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

   4. Taylor expanded in B around 0 75.7%

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

   if -3e96 < F < -1.24999999999999994e60 or -3.6999999999999999e-35 < F < -1.7500000000000001e-234

   1. Initial program 99.3%

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

   3. Taylor expanded in F around -inf 54.4%

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

      1. +-commutative 54.4%

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

      2. unsub-neg 54.4%

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

      3. un-div-inv 54.7%

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

   5. Applied egg-rr 54.7%

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

   6. Taylor expanded in B around 0 62.9%

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

   if -1.7500000000000001e-234 < F < 9.39999999999999999e-178

   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 24.5%

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

   4. Taylor expanded in B around 0 9.0%

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

      1. mul-1-neg 9.0%

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

      2. distribute-neg-frac2 9.0%

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

   6. Simplified 9.0%

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

   7. Taylor expanded in x around inf 48.5%

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

      1. associate-*r/ 48.5%

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

      2. neg-mul-1 48.5%

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

   9. Simplified 48.5%

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

   if 9.39999999999999999e-178 < F

   1. Initial program 69.7%

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

   3. Taylor expanded in F around inf 64.0%

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

   4. Taylor expanded in B around 0 64.9%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3 \cdot 10^{+96}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.25 \cdot 10^{+60}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -3.7 \cdot 10^{-35}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.75 \cdot 10^{-234}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 9.4 \cdot 10^{-178}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 52.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -5 \cdot 10^{-234}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 2 \cdot 10^{-117}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;F \leq 1.5 \cdot 10^{+183}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ -1.0 B) (/ x (tan B)))))
   (if (<= F -5e-234)
     t_0
     (if (<= F 2e-117) (/ x (- B)) (if (<= F 1.5e+183) t_0 (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 / B) - (x / tan(B));
	double tmp;
	if (F <= -5e-234) {
		tmp = t_0;
	} else if (F <= 2e-117) {
		tmp = x / -B;
	} else if (F <= 1.5e+183) {
		tmp = t_0;
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-1.0d0) / b) - (x / tan(b))
    if (f <= (-5d-234)) then
        tmp = t_0
    else if (f <= 2d-117) then
        tmp = x / -b
    else if (f <= 1.5d+183) then
        tmp = t_0
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (-1.0 / B) - (x / Math.tan(B));
	double tmp;
	if (F <= -5e-234) {
		tmp = t_0;
	} else if (F <= 2e-117) {
		tmp = x / -B;
	} else if (F <= 1.5e+183) {
		tmp = t_0;
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (-1.0 / B) - (x / math.tan(B))
	tmp = 0
	if F <= -5e-234:
		tmp = t_0
	elif F <= 2e-117:
		tmp = x / -B
	elif F <= 1.5e+183:
		tmp = t_0
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 / B) - Float64(x / tan(B)))
	tmp = 0.0
	if (F <= -5e-234)
		tmp = t_0;
	elseif (F <= 2e-117)
		tmp = Float64(x / Float64(-B));
	elseif (F <= 1.5e+183)
		tmp = t_0;
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (-1.0 / B) - (x / tan(B));
	tmp = 0.0;
	if (F <= -5e-234)
		tmp = t_0;
	elseif (F <= 2e-117)
		tmp = x / -B;
	elseif (F <= 1.5e+183)
		tmp = t_0;
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -5e-234], t$95$0, If[LessEqual[F, 2e-117], N[(x / (-B)), $MachinePrecision], If[LessEqual[F, 1.5e+183], t$95$0, N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.99999999999999979e-234 or 2.00000000000000006e-117 < F < 1.49999999999999998e183

   1. Initial program 78.6%

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

   3. Taylor expanded in F around -inf 69.9%

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

      1. +-commutative 69.9%

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

      2. unsub-neg 69.9%

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

      3. un-div-inv 69.9%

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

   5. Applied egg-rr 69.9%

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

   6. Taylor expanded in B around 0 59.8%

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

   if -4.99999999999999979e-234 < F < 2.00000000000000006e-117

   1. Initial program 99.4%

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

   3. Taylor expanded in F around -inf 26.6%

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

   4. Taylor expanded in B around 0 9.8%

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

      1. mul-1-neg 9.8%

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

      2. distribute-neg-frac2 9.8%

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

   6. Simplified 9.8%

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

   7. Taylor expanded in x around inf 48.1%

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

      1. associate-*r/ 48.1%

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

      2. neg-mul-1 48.1%

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

   9. Simplified 48.1%

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

   if 1.49999999999999998e183 < F

   1. Initial program 21.1%

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

   3. Taylor expanded in F around inf 61.7%

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

   4. Taylor expanded in B around 0 39.4%

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

   5. Taylor expanded in x around 0 64.5%

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

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{-234}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2 \cdot 10^{-117}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;F \leq 1.5 \cdot 10^{+183}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 56.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;x \leq -0.00026:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-122}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;x \leq 1.42 \cdot 10^{-11}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ -1.0 B) (/ x (tan B)))))
   (if (<= x -0.00026)
     t_0
     (if (<= x -5e-122)
       (/ x (- B))
       (if (<= x 1.42e-11) (- (/ -1.0 (sin B)) (/ x B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 / B) - (x / tan(B));
	double tmp;
	if (x <= -0.00026) {
		tmp = t_0;
	} else if (x <= -5e-122) {
		tmp = x / -B;
	} else if (x <= 1.42e-11) {
		tmp = (-1.0 / sin(B)) - (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) / b) - (x / tan(b))
    if (x <= (-0.00026d0)) then
        tmp = t_0
    else if (x <= (-5d-122)) then
        tmp = x / -b
    else if (x <= 1.42d-11) then
        tmp = ((-1.0d0) / sin(b)) - (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 / B) - (x / Math.tan(B));
	double tmp;
	if (x <= -0.00026) {
		tmp = t_0;
	} else if (x <= -5e-122) {
		tmp = x / -B;
	} else if (x <= 1.42e-11) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (-1.0 / B) - (x / math.tan(B))
	tmp = 0
	if x <= -0.00026:
		tmp = t_0
	elif x <= -5e-122:
		tmp = x / -B
	elif x <= 1.42e-11:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	else:
		tmp = t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 / B) - Float64(x / tan(B)))
	tmp = 0.0
	if (x <= -0.00026)
		tmp = t_0;
	elseif (x <= -5e-122)
		tmp = Float64(x / Float64(-B));
	elseif (x <= 1.42e-11)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (-1.0 / B) - (x / tan(B));
	tmp = 0.0;
	if (x <= -0.00026)
		tmp = t_0;
	elseif (x <= -5e-122)
		tmp = x / -B;
	elseif (x <= 1.42e-11)
		tmp = (-1.0 / sin(B)) - (x / B);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.00026], t$95$0, If[LessEqual[x, -5e-122], N[(x / (-B)), $MachinePrecision], If[LessEqual[x, 1.42e-11], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.59999999999999977e-4 or 1.42e-11 < x

   1. Initial program 85.3%

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

   3. Taylor expanded in F around -inf 94.4%

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

      1. +-commutative 94.4%

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

      2. unsub-neg 94.4%

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

      3. un-div-inv 94.6%

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

   5. Applied egg-rr 94.6%

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

   6. Taylor expanded in B around 0 96.6%

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

   if -2.59999999999999977e-4 < x < -4.9999999999999999e-122

   1. Initial program 80.4%

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

   3. Taylor expanded in F around -inf 21.5%

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

   4. Taylor expanded in B around 0 15.8%

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

      1. mul-1-neg 15.8%

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

      2. distribute-neg-frac2 15.8%

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

   6. Simplified 15.8%

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

   7. Taylor expanded in x around inf 41.2%

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

      1. associate-*r/ 41.2%

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

      2. neg-mul-1 41.2%

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

   9. Simplified 41.2%

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

   if -4.9999999999999999e-122 < x < 1.42e-11

   1. Initial program 65.1%

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

   3. Taylor expanded in F around -inf 37.2%

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

   4. Taylor expanded in B around 0 37.2%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00026:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-122}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;x \leq 1.42 \cdot 10^{-11}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 43.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.42 \cdot 10^{-33}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.2 \cdot 10^{+18}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.42e-33)
   (/ (- -1.0 x) B)
   (if (<= F 1.2e+18) (/ x (- B)) (/ 1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.42e-33) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.2e+18) {
		tmp = x / -B;
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.42d-33)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 1.2d+18) then
        tmp = x / -b
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.42e-33) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.2e+18) {
		tmp = x / -B;
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.42e-33:
		tmp = (-1.0 - x) / B
	elif F <= 1.2e+18:
		tmp = x / -B
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.42e-33)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 1.2e+18)
		tmp = Float64(x / Float64(-B));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.42e-33)
		tmp = (-1.0 - x) / B;
	elseif (F <= 1.2e+18)
		tmp = x / -B;
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.42e-33], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.2e+18], N[(x / (-B)), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.42000000000000007e-33

   1. Initial program 62.9%

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

   3. Taylor expanded in F around -inf 92.8%

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

   4. Taylor expanded in B around 0 47.6%

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

      1. mul-1-neg 47.6%

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

      2. distribute-neg-frac2 47.6%

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

   6. Simplified 47.6%

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

   if -1.42000000000000007e-33 < F < 1.2e18

   1. Initial program 99.3%

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

   3. Taylor expanded in F around -inf 34.7%

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

   4. Taylor expanded in B around 0 16.6%

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

      1. mul-1-neg 16.6%

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

      2. distribute-neg-frac2 16.6%

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

   6. Simplified 16.6%

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

   7. Taylor expanded in x around inf 37.4%

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

      1. associate-*r/ 37.4%

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

      2. neg-mul-1 37.4%

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

   9. Simplified 37.4%

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

   if 1.2e18 < F

   1. Initial program 53.1%

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

   3. Taylor expanded in F around inf 73.9%

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

   4. Taylor expanded in B around 0 49.4%

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

   5. Taylor expanded in x around 0 54.0%

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

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.42 \cdot 10^{-33}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.2 \cdot 10^{+18}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 36.2% accurate, 21.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.7 \cdot 10^{+137}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq 1.9 \cdot 10^{-41}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.7e+137)
   (/ -1.0 B)
   (if (<= F 1.9e-41) (/ x (- B)) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.7e+137) {
		tmp = -1.0 / B;
	} else if (F <= 1.9e-41) {
		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 <= (-1.7d+137)) then
        tmp = (-1.0d0) / b
    else if (f <= 1.9d-41) 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 <= -1.7e+137) {
		tmp = -1.0 / B;
	} else if (F <= 1.9e-41) {
		tmp = x / -B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.7e+137:
		tmp = -1.0 / B
	elif F <= 1.9e-41:
		tmp = x / -B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.7e+137)
		tmp = Float64(-1.0 / B);
	elseif (F <= 1.9e-41)
		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 <= -1.7e+137)
		tmp = -1.0 / B;
	elseif (F <= 1.9e-41)
		tmp = x / -B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.7e+137], N[(-1.0 / B), $MachinePrecision], If[LessEqual[F, 1.9e-41], N[(x / (-B)), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.69999999999999993e137

   1. Initial program 35.7%

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

   3. Taylor expanded in F around -inf 99.8%

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

   4. Taylor expanded in B around 0 44.1%

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

      1. mul-1-neg 44.1%

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

      2. distribute-neg-frac2 44.1%

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

   6. Simplified 44.1%

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

   7. Taylor expanded in x around 0 29.1%

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

   if -1.69999999999999993e137 < F < 1.8999999999999999e-41

   1. Initial program 98.7%

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

   3. Taylor expanded in F around -inf 48.1%

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

   4. Taylor expanded in B around 0 26.4%

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

      1. mul-1-neg 26.4%

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

      2. distribute-neg-frac2 26.4%

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

   6. Simplified 26.4%

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

   7. Taylor expanded in x around inf 36.1%

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

      1. associate-*r/ 36.1%

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

      2. neg-mul-1 36.1%

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

   9. Simplified 36.1%

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

   if 1.8999999999999999e-41 < F

   1. Initial program 60.4%

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

   3. Taylor expanded in F around inf 72.9%

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

   4. Taylor expanded in B around 0 48.1%

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

   5. Taylor expanded in B around 0 46.3%

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

4. Final simplification 37.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.7 \cdot 10^{+137}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq 1.9 \cdot 10^{-41}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 43.2% accurate, 21.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -5.5 \cdot 10^{-34}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 4.6 \cdot 10^{-41}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -5.5e-34)
   (/ (- -1.0 x) B)
   (if (<= F 4.6e-41) (/ x (- B)) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -5.5e-34) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 4.6e-41) {
		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 <= (-5.5d-34)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 4.6d-41) 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 <= -5.5e-34) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 4.6e-41) {
		tmp = x / -B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -5.5e-34:
		tmp = (-1.0 - x) / B
	elif F <= 4.6e-41:
		tmp = x / -B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -5.5e-34)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 4.6e-41)
		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 <= -5.5e-34)
		tmp = (-1.0 - x) / B;
	elseif (F <= 4.6e-41)
		tmp = x / -B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -5.5e-34], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 4.6e-41], N[(x / (-B)), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.50000000000000014e-34

   1. Initial program 62.9%

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

   3. Taylor expanded in F around -inf 92.8%

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

   4. Taylor expanded in B around 0 47.6%

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

      1. mul-1-neg 47.6%

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

      2. distribute-neg-frac2 47.6%

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

   6. Simplified 47.6%

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

   if -5.50000000000000014e-34 < F < 4.6000000000000002e-41

   1. Initial program 99.3%

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

   3. Taylor expanded in F around -inf 32.4%

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

   4. Taylor expanded in B around 0 15.2%

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

      1. mul-1-neg 15.2%

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

      2. distribute-neg-frac2 15.2%

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

   6. Simplified 15.2%

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

   7. Taylor expanded in x around inf 38.2%

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

      1. associate-*r/ 38.2%

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

      2. neg-mul-1 38.2%

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

   9. Simplified 38.2%

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

   if 4.6000000000000002e-41 < F

   1. Initial program 60.4%

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

   3. Taylor expanded in F around inf 72.9%

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

   4. Taylor expanded in B around 0 48.1%

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

   5. Taylor expanded in B around 0 46.3%

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

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5.5 \cdot 10^{-34}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 4.6 \cdot 10^{-41}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 29.3% accurate, 23.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.42 \cdot 10^{+137}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq 2.5 \cdot 10^{+253}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.42e+137) (/ -1.0 B) (if (<= F 2.5e+253) (/ x (- B)) (/ 1.0 B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.42e+137) {
		tmp = -1.0 / B;
	} else if (F <= 2.5e+253) {
		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 (f <= (-1.42d+137)) then
        tmp = (-1.0d0) / b
    else if (f <= 2.5d+253) 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 (F <= -1.42e+137) {
		tmp = -1.0 / B;
	} else if (F <= 2.5e+253) {
		tmp = x / -B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.42e+137:
		tmp = -1.0 / B
	elif F <= 2.5e+253:
		tmp = x / -B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.42e+137)
		tmp = Float64(-1.0 / B);
	elseif (F <= 2.5e+253)
		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 (F <= -1.42e+137)
		tmp = -1.0 / B;
	elseif (F <= 2.5e+253)
		tmp = x / -B;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.42e+137], N[(-1.0 / B), $MachinePrecision], If[LessEqual[F, 2.5e+253], N[(x / (-B)), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.42e137

   1. Initial program 35.7%

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

   3. Taylor expanded in F around -inf 99.8%

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

   4. Taylor expanded in B around 0 44.1%

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

      1. mul-1-neg 44.1%

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

      2. distribute-neg-frac2 44.1%

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

   6. Simplified 44.1%

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

   7. Taylor expanded in x around 0 29.1%

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

   if -1.42e137 < F < 2.4999999999999998e253

   1. Initial program 90.2%

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

   3. Taylor expanded in F around -inf 49.5%

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

   4. Taylor expanded in B around 0 27.3%

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

      1. mul-1-neg 27.3%

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

      2. distribute-neg-frac2 27.3%

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

   6. Simplified 27.3%

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

   7. Taylor expanded in x around inf 34.4%

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

      1. associate-*r/ 34.4%

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

      2. neg-mul-1 34.4%

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

   9. Simplified 34.4%

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

   if 2.4999999999999998e253 < F

   1. Initial program 22.6%

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

   3. Taylor expanded in F around -inf 22.2%

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

   4. Taylor expanded in B around 0 2.4%

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

      1. mul-1-neg 2.4%

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

      2. distribute-neg-frac2 2.4%

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

   6. Simplified 2.4%

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

      1. add-sqr-sqrt 1.2%

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

      2. *-un-lft-identity 1.2%

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

      3. sqrt-unprod 9.9%

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

      4. sqr-neg 9.9%

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

      5. sqrt-unprod 21.7%

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

      6. times-frac 21.7%

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

      7. +-commutative 21.7%

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

   8. Applied egg-rr 21.7%

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

      1. associate-*l/ 21.7%

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

      2. *-lft-identity 21.7%

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

   10. Simplified 21.7%

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

   11. Taylor expanded in x around 0 44.4%

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

4. Final simplification 33.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.42 \cdot 10^{+137}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq 2.5 \cdot 10^{+253}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 16.6% accurate, 40.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.45 \cdot 10^{-281}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.45e-281) (/ -1.0 B) (/ 1.0 B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.45e-281) {
		tmp = -1.0 / 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 (f <= (-2.45d-281)) then
        tmp = (-1.0d0) / 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 (F <= -2.45e-281) {
		tmp = -1.0 / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.45e-281:
		tmp = -1.0 / B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.45e-281)
		tmp = Float64(-1.0 / B);
	else
		tmp = Float64(1.0 / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.45e-281)
		tmp = -1.0 / B;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.45e-281], N[(-1.0 / B), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -2.45e-281

   1. Initial program 74.4%

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

   3. Taylor expanded in F around -inf 75.5%

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

   4. Taylor expanded in B around 0 39.7%

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

      1. mul-1-neg 39.7%

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

      2. distribute-neg-frac2 39.7%

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

   6. Simplified 39.7%

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

   7. Taylor expanded in x around 0 19.8%

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

   if -2.45e-281 < F

   1. Initial program 76.6%

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

   3. Taylor expanded in F around -inf 38.7%

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

   4. Taylor expanded in B around 0 17.7%

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

      1. mul-1-neg 17.7%

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

      2. distribute-neg-frac2 17.7%

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

   6. Simplified 17.7%

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

      1. add-sqr-sqrt 8.5%

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

      2. *-un-lft-identity 8.5%

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

      3. sqrt-unprod 12.6%

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

      4. sqr-neg 12.6%

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

      5. sqrt-unprod 10.5%

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

      6. times-frac 10.4%

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

      7. +-commutative 10.4%

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

   8. Applied egg-rr 10.4%

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

      1. associate-*l/ 10.5%

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

      2. *-lft-identity 10.5%

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

   10. Simplified 10.5%

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

   11. Taylor expanded in x around 0 15.7%

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

4. Final simplification 17.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.45 \cdot 10^{-281}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 10.1% 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 75.5%

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

3. Taylor expanded in F around -inf 58.2%

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

4. Taylor expanded in B around 0 29.4%

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

   1. mul-1-neg 29.4%

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

   2. distribute-neg-frac2 29.4%

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

6. Simplified 29.4%

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

7. Taylor expanded in x around 0 11.9%

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

8. Final simplification 11.9%

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

Reproduce

herbie shell --seed 2024077 
(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))))))