VandenBroeck and Keller, Equation (23)

Percentage Accurate: 77.4% → 99.7%

Time: 24.6s

Alternatives: 31

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -42000000000000:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 100000000:\\ \;\;\;\;F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -42000000000000.0)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 100000000.0)
       (- (* 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 <= -42000000000000.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 100000000.0) {
		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 <= -42000000000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 100000000.0)
		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, -42000000000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 100000000.0], 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 < -4.2e13

   1. Initial program 57.4%

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

   3. Taylor expanded in F around -inf 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. add-sqr-sqrt 45.0%

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

      3. sqrt-unprod 51.1%

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

      4. frac-times 51.2%

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

      5. metadata-eval 51.2%

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

      6. metadata-eval 51.2%

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

      7. frac-times 51.1%

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

      8. sqrt-unprod 20.3%

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

      9. add-sqr-sqrt 48.4%

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

      10. metadata-eval 48.4%

          \[\leadsto \frac{\color{blue}{{F}^{0}}}{\sin B} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      11. metadata-eval 48.4%

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

      12. pow-prod-up 48.4%

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

      13. pow1 48.4%

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

      14. inv-pow 48.4%

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

      15. associate-*r/ 48.4%

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

      16. add-sqr-sqrt 20.3%

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

      17. fma-def 20.3%

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

   5. Applied egg-rr 99.8%

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

   if -4.2e13 < F < 1e8

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   if 1e8 < F

   1. Initial program 67.3%

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

   3. Simplified 79.7%

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

   5. Taylor expanded in F around inf 99.7%

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

      1. associate-/r* 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in F around 0 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -42000000000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 100000000:\\ \;\;\;\;F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.6 \cdot 10^{+14}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 130000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \cos B \cdot \frac{x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.6e+14)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 130000000.0)
       (-
        (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5))
        (* (cos B) (/ x (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 <= -1.6e+14) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 130000000.0) {
		tmp = ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (cos(B) * (x / 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 <= (-1.6d+14)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 130000000.0d0) then
        tmp = ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0))) - (cos(b) * (x / 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 <= -1.6e+14) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 130000000.0) {
		tmp = ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (Math.cos(B) * (x / 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 <= -1.6e+14:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 130000000.0:
		tmp = ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (math.cos(B) * (x / 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 <= -1.6e+14)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 130000000.0)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)) - Float64(cos(B) * Float64(x / 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 <= -1.6e+14)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 130000000.0)
		tmp = ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5)) - (cos(B) * (x / 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, -1.6e+14], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 130000000.0], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[B], $MachinePrecision] * N[(x / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.6e14

   1. Initial program 57.4%

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

   3. Taylor expanded in F around -inf 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. add-sqr-sqrt 45.0%

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

      3. sqrt-unprod 51.1%

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

      4. frac-times 51.2%

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

      5. metadata-eval 51.2%

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

      6. metadata-eval 51.2%

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

      7. frac-times 51.1%

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

      8. sqrt-unprod 20.3%

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

      9. add-sqr-sqrt 48.4%

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

      10. metadata-eval 48.4%

          \[\leadsto \frac{\color{blue}{{F}^{0}}}{\sin B} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      11. metadata-eval 48.4%

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

      12. pow-prod-up 48.4%

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

      13. pow1 48.4%

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

      14. inv-pow 48.4%

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

      15. associate-*r/ 48.4%

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

      16. add-sqr-sqrt 20.3%

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

      17. fma-def 20.3%

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

   5. Applied egg-rr 99.8%

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

   if -1.6e14 < F < 1.3e8

   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 x around 0 99.5%

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

      1. associate-*l/ 99.5%

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

      2. *-commutative 99.5%

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

   5. Simplified 99.5%

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

   if 1.3e8 < F

   1. Initial program 67.3%

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

   3. Simplified 79.7%

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

   5. Taylor expanded in F around inf 99.7%

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

      1. associate-/r* 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in F around 0 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.6 \cdot 10^{+14}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 130000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \cos B \cdot \frac{x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -5 \cdot 10^{+14}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 100000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} + x \cdot \frac{-1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -5e+14)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 100000000.0)
       (+
        (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5))
        (* x (/ -1.0 (tan B))))
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -5e+14) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 100000000.0) {
		tmp = ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) + (x * (-1.0 / tan(B)));
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-5d+14)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 100000000.0d0) then
        tmp = ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0))) + (x * ((-1.0d0) / tan(b)))
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -5e+14) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 100000000.0) {
		tmp = ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) + (x * (-1.0 / Math.tan(B)));
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -5e+14:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 100000000.0:
		tmp = ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) + (x * (-1.0 / math.tan(B)))
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -5e+14)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 100000000.0)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)) + Float64(x * Float64(-1.0 / tan(B))));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -5e+14)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 100000000.0)
		tmp = ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5)) + (x * (-1.0 / tan(B)));
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -5e+14], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 100000000.0], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] + N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -5e14

   1. Initial program 57.4%

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

   3. Taylor expanded in F around -inf 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. add-sqr-sqrt 45.0%

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

      3. sqrt-unprod 51.1%

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

      4. frac-times 51.2%

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

      5. metadata-eval 51.2%

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

      6. metadata-eval 51.2%

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

      7. frac-times 51.1%

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

      8. sqrt-unprod 20.3%

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

      9. add-sqr-sqrt 48.4%

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

      10. metadata-eval 48.4%

          \[\leadsto \frac{\color{blue}{{F}^{0}}}{\sin B} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      11. metadata-eval 48.4%

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

      12. pow-prod-up 48.4%

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

      13. pow1 48.4%

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

      14. inv-pow 48.4%

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

      15. associate-*r/ 48.4%

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

      16. add-sqr-sqrt 20.3%

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

      17. fma-def 20.3%

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

   5. Applied egg-rr 99.8%

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

   if -5e14 < F < 1e8

   1. Initial program 99.5%

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

   if 1e8 < F

   1. Initial program 67.3%

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

   3. Simplified 79.7%

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

   5. Taylor expanded in F around inf 99.7%

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

      1. associate-/r* 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in F around 0 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{+14}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 100000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} + x \cdot \frac{-1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.42:\\ \;\;\;\;\frac{-1}{\sin B} - t_1\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;F \cdot \left(t_0 \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 - t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))) (t_1 (/ x (tan B))))
   (if (<= F -1.42)
     (- (/ -1.0 (sin B)) t_1)
     (if (<= F 1.4)
       (- (* 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 <= -1.42) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= 1.4) {
		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 <= (-1.42d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_1
    else if (f <= 1.4d0) 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 <= -1.42) {
		tmp = (-1.0 / Math.sin(B)) - t_1;
	} else if (F <= 1.4) {
		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 <= -1.42:
		tmp = (-1.0 / math.sin(B)) - t_1
	elif F <= 1.4:
		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 <= -1.42)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= 1.4)
		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 <= -1.42)
		tmp = (-1.0 / sin(B)) - t_1;
	elseif (F <= 1.4)
		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, -1.42], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, 1.4], 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 < -1.4199999999999999

   1. Initial program 59.1%

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

   3. Taylor expanded in F around -inf 99.3%

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

      1. +-commutative 99.3%

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

      2. add-sqr-sqrt 44.5%

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

      3. sqrt-unprod 50.6%

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

      4. frac-times 50.6%

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

      5. metadata-eval 50.6%

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

      6. metadata-eval 50.6%

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

      7. frac-times 50.6%

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

      8. sqrt-unprod 20.9%

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

      9. add-sqr-sqrt 49.1%

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

      10. metadata-eval 49.1%

          \[\leadsto \frac{\color{blue}{{F}^{0}}}{\sin B} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      11. metadata-eval 49.1%

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

      12. pow-prod-up 49.1%

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

      13. pow1 49.1%

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

      14. inv-pow 49.1%

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

      15. associate-*r/ 49.1%

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

      16. add-sqr-sqrt 20.9%

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

      17. fma-def 20.9%

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

   5. Applied egg-rr 99.4%

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

   if -1.4199999999999999 < F < 1.3999999999999999

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in F around 0 99.5%

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

   if 1.3999999999999999 < F

   1. Initial program 68.3%

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

   3. Simplified 80.4%

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

   5. Taylor expanded in F around inf 98.9%

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

      1. associate-/r* 98.9%

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

   7. Simplified 98.9%

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

   8. Taylor expanded in F around 0 99.1%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.42:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;F \cdot \left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\right) - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.42:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;\frac{1}{\frac{\sin B}{F \cdot \sqrt{0.5}}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.42)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.4)
       (- (/ 1.0 (/ (sin B) (* F (sqrt 0.5)))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.42) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.4) {
		tmp = (1.0 / (sin(B) / (F * sqrt(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 <= (-1.42d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.4d0) then
        tmp = (1.0d0 / (sin(b) / (f * sqrt(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 <= -1.42) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.4) {
		tmp = (1.0 / (Math.sin(B) / (F * Math.sqrt(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 <= -1.42:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.4:
		tmp = (1.0 / (math.sin(B) / (F * math.sqrt(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 <= -1.42)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.4)
		tmp = Float64(Float64(1.0 / Float64(sin(B) / Float64(F * sqrt(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 <= -1.42)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.4)
		tmp = (1.0 / (sin(B) / (F * sqrt(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, -1.42], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.4], N[(N[(1.0 / N[(N[Sin[B], $MachinePrecision] / N[(F * N[Sqrt[0.5], $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 < -1.4199999999999999

   1. Initial program 59.1%

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

   3. Taylor expanded in F around -inf 99.3%

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

      1. +-commutative 99.3%

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

      2. add-sqr-sqrt 44.5%

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

      3. sqrt-unprod 50.6%

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

      4. frac-times 50.6%

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

      5. metadata-eval 50.6%

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

      6. metadata-eval 50.6%

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

      7. frac-times 50.6%

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

      8. sqrt-unprod 20.9%

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

      9. add-sqr-sqrt 49.1%

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

      10. metadata-eval 49.1%

          \[\leadsto \frac{\color{blue}{{F}^{0}}}{\sin B} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      11. metadata-eval 49.1%

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

      12. pow-prod-up 49.1%

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

      13. pow1 49.1%

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

      14. inv-pow 49.1%

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

      15. associate-*r/ 49.1%

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

      16. add-sqr-sqrt 20.9%

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

      17. fma-def 20.9%

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

   5. Applied egg-rr 99.4%

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

   if -1.4199999999999999 < F < 1.3999999999999999

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in F around 0 99.5%

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

   6. Taylor expanded in x around 0 99.3%

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

      1. associate-*r/ 37.7%

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

      2. clear-num 37.7%

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

   8. Applied egg-rr 99.3%

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

   if 1.3999999999999999 < F

   1. Initial program 68.3%

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

   3. Simplified 80.4%

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

   5. Taylor expanded in F around inf 98.9%

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

      1. associate-/r* 98.9%

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

   7. Simplified 98.9%

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

   8. Taylor expanded in F around 0 99.1%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.42:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;\frac{1}{\frac{\sin B}{F \cdot \sqrt{0.5}}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.42:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.42)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.4)
       (- (* F (/ (sqrt 0.5) (sin B))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.42) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.4) {
		tmp = (F * (sqrt(0.5) / sin(B))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.42d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.4d0) then
        tmp = (f * (sqrt(0.5d0) / sin(b))) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.42) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.4) {
		tmp = (F * (Math.sqrt(0.5) / Math.sin(B))) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.42:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.4:
		tmp = (F * (math.sqrt(0.5) / math.sin(B))) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.42)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.4)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / sin(B))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.42)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.4)
		tmp = (F * (sqrt(0.5) / sin(B))) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.42], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.4], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.4199999999999999

   1. Initial program 59.1%

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

   3. Taylor expanded in F around -inf 99.3%

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

      1. +-commutative 99.3%

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

      2. add-sqr-sqrt 44.5%

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

      3. sqrt-unprod 50.6%

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

      4. frac-times 50.6%

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

      5. metadata-eval 50.6%

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

      6. metadata-eval 50.6%

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

      7. frac-times 50.6%

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

      8. sqrt-unprod 20.9%

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

      9. add-sqr-sqrt 49.1%

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

      10. metadata-eval 49.1%

          \[\leadsto \frac{\color{blue}{{F}^{0}}}{\sin B} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      11. metadata-eval 49.1%

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

      12. pow-prod-up 49.1%

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

      13. pow1 49.1%

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

      14. inv-pow 49.1%

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

      15. associate-*r/ 49.1%

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

      16. add-sqr-sqrt 20.9%

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

      17. fma-def 20.9%

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

   5. Applied egg-rr 99.4%

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

   if -1.4199999999999999 < F < 1.3999999999999999

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in F around 0 99.5%

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

   6. Taylor expanded in x around 0 99.3%

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

   if 1.3999999999999999 < F

   1. Initial program 68.3%

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

   3. Simplified 80.4%

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

   5. Taylor expanded in F around inf 98.9%

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

      1. associate-/r* 98.9%

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

   7. Simplified 98.9%

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

   8. Taylor expanded in F around 0 99.1%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.42:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := x \cdot \frac{-1}{\tan B} + \frac{1}{\frac{B}{F} \cdot \left(F + 0.5 \cdot \frac{2 + x \cdot 2}{F}\right)}\\ t_2 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.019:\\ \;\;\;\;\frac{-1}{\sin B} - t_2\\ \mathbf{elif}\;F \leq -9.5 \cdot 10^{-104}:\\ \;\;\;\;F \cdot \left(t_0 \cdot \sqrt{0.5}\right)\\ \mathbf{elif}\;F \leq 3 \cdot 10^{-78}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 1.75 \cdot 10^{-14}:\\ \;\;\;\;\frac{F}{\frac{\sin B}{\sqrt{0.5}}}\\ \mathbf{elif}\;F \leq 8000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 - t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B)))
        (t_1
         (+
          (* x (/ -1.0 (tan B)))
          (/ 1.0 (* (/ B F) (+ F (* 0.5 (/ (+ 2.0 (* x 2.0)) F)))))))
        (t_2 (/ x (tan B))))
   (if (<= F -0.019)
     (- (/ -1.0 (sin B)) t_2)
     (if (<= F -9.5e-104)
       (* F (* t_0 (sqrt 0.5)))
       (if (<= F 3e-78)
         t_1
         (if (<= F 1.75e-14)
           (/ F (/ (sin B) (sqrt 0.5)))
           (if (<= F 8000.0) t_1 (- t_0 t_2))))))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / sin(B);
	double t_1 = (x * (-1.0 / tan(B))) + (1.0 / ((B / F) * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))));
	double t_2 = x / tan(B);
	double tmp;
	if (F <= -0.019) {
		tmp = (-1.0 / sin(B)) - t_2;
	} else if (F <= -9.5e-104) {
		tmp = F * (t_0 * sqrt(0.5));
	} else if (F <= 3e-78) {
		tmp = t_1;
	} else if (F <= 1.75e-14) {
		tmp = F / (sin(B) / sqrt(0.5));
	} else if (F <= 8000.0) {
		tmp = t_1;
	} else {
		tmp = t_0 - t_2;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 / sin(b)
    t_1 = (x * ((-1.0d0) / tan(b))) + (1.0d0 / ((b / f) * (f + (0.5d0 * ((2.0d0 + (x * 2.0d0)) / f)))))
    t_2 = x / tan(b)
    if (f <= (-0.019d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_2
    else if (f <= (-9.5d-104)) then
        tmp = f * (t_0 * sqrt(0.5d0))
    else if (f <= 3d-78) then
        tmp = t_1
    else if (f <= 1.75d-14) then
        tmp = f / (sin(b) / sqrt(0.5d0))
    else if (f <= 8000.0d0) then
        tmp = t_1
    else
        tmp = t_0 - t_2
    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 * (-1.0 / Math.tan(B))) + (1.0 / ((B / F) * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))));
	double t_2 = x / Math.tan(B);
	double tmp;
	if (F <= -0.019) {
		tmp = (-1.0 / Math.sin(B)) - t_2;
	} else if (F <= -9.5e-104) {
		tmp = F * (t_0 * Math.sqrt(0.5));
	} else if (F <= 3e-78) {
		tmp = t_1;
	} else if (F <= 1.75e-14) {
		tmp = F / (Math.sin(B) / Math.sqrt(0.5));
	} else if (F <= 8000.0) {
		tmp = t_1;
	} else {
		tmp = t_0 - t_2;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = 1.0 / math.sin(B)
	t_1 = (x * (-1.0 / math.tan(B))) + (1.0 / ((B / F) * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))))
	t_2 = x / math.tan(B)
	tmp = 0
	if F <= -0.019:
		tmp = (-1.0 / math.sin(B)) - t_2
	elif F <= -9.5e-104:
		tmp = F * (t_0 * math.sqrt(0.5))
	elif F <= 3e-78:
		tmp = t_1
	elif F <= 1.75e-14:
		tmp = F / (math.sin(B) / math.sqrt(0.5))
	elif F <= 8000.0:
		tmp = t_1
	else:
		tmp = t_0 - t_2
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / sin(B))
	t_1 = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / Float64(Float64(B / F) * Float64(F + Float64(0.5 * Float64(Float64(2.0 + Float64(x * 2.0)) / F))))))
	t_2 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -0.019)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_2);
	elseif (F <= -9.5e-104)
		tmp = Float64(F * Float64(t_0 * sqrt(0.5)));
	elseif (F <= 3e-78)
		tmp = t_1;
	elseif (F <= 1.75e-14)
		tmp = Float64(F / Float64(sin(B) / sqrt(0.5)));
	elseif (F <= 8000.0)
		tmp = t_1;
	else
		tmp = Float64(t_0 - t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = 1.0 / sin(B);
	t_1 = (x * (-1.0 / tan(B))) + (1.0 / ((B / F) * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))));
	t_2 = x / tan(B);
	tmp = 0.0;
	if (F <= -0.019)
		tmp = (-1.0 / sin(B)) - t_2;
	elseif (F <= -9.5e-104)
		tmp = F * (t_0 * sqrt(0.5));
	elseif (F <= 3e-78)
		tmp = t_1;
	elseif (F <= 1.75e-14)
		tmp = F / (sin(B) / sqrt(0.5));
	elseif (F <= 8000.0)
		tmp = t_1;
	else
		tmp = t_0 - t_2;
	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[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[(B / F), $MachinePrecision] * N[(F + N[(0.5 * N[(N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.019], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision], If[LessEqual[F, -9.5e-104], N[(F * N[(t$95$0 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3e-78], t$95$1, If[LessEqual[F, 1.75e-14], N[(F / N[(N[Sin[B], $MachinePrecision] / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 8000.0], t$95$1, N[(t$95$0 - t$95$2), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -0.0189999999999999995

   1. Initial program 59.6%

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

   3. Taylor expanded in F around -inf 99.3%

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

      1. +-commutative 99.3%

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

      2. add-sqr-sqrt 45.2%

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

      3. sqrt-unprod 51.2%

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

      4. frac-times 51.2%

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

      5. metadata-eval 51.2%

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

      6. metadata-eval 51.2%

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

      7. frac-times 51.2%

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

      8. sqrt-unprod 20.6%

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

      9. add-sqr-sqrt 49.8%

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

      10. metadata-eval 49.8%

          \[\leadsto \frac{\color{blue}{{F}^{0}}}{\sin B} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      11. metadata-eval 49.8%

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

      12. pow-prod-up 49.8%

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

      13. pow1 49.8%

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

      14. inv-pow 49.8%

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

      15. associate-*r/ 49.8%

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

      16. add-sqr-sqrt 20.6%

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

      17. fma-def 20.6%

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

   5. Applied egg-rr 99.4%

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

   if -0.0189999999999999995 < F < -9.5000000000000002e-104

   1. Initial program 99.1%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in F around 0 98.5%

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

   6. Taylor expanded in x around 0 70.7%

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

      1. *-lft-identity 70.7%

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

      2. times-frac 70.8%

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

      3. /-rgt-identity 70.8%

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

   8. Simplified 70.8%

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

      1. div-inv 70.8%

         \[\leadsto F \cdot \color{blue}{\left(\sqrt{0.5} \cdot \frac{1}{\sin B}\right)} \]

   10. Applied egg-rr 70.8%

       \[\leadsto F \cdot \color{blue}{\left(\sqrt{0.5} \cdot \frac{1}{\sin B}\right)} \]

   if -9.5000000000000002e-104 < F < 2.99999999999999988e-78 or 1.7500000000000001e-14 < F < 8e3

   1. Initial program 99.7%

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

      1. associate-*l/ 99.6%

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

      2. +-commutative 99.6%

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

      3. *-commutative 99.6%

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

      4. fma-udef 99.6%

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

      5. fma-def 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. clear-num 99.6%

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

      9. fma-def 99.6%

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

      10. fma-udef 99.6%

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

      11. *-commutative 99.6%

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

      12. fma-def 99.6%

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

      13. fma-def 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in B around 0 87.5%

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

   6. Taylor expanded in F around inf 76.2%

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

   if 2.99999999999999988e-78 < F < 1.7500000000000001e-14

   1. Initial program 99.0%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in F around 0 99.5%

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

   6. Taylor expanded in x around 0 62.7%

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

      1. associate-/l* 62.9%

         \[\leadsto \color{blue}{\frac{F}{\frac{\sin B}{\sqrt{0.5}}}} \]

   8. Simplified 62.9%

      \[\leadsto \color{blue}{\frac{F}{\frac{\sin B}{\sqrt{0.5}}}} \]

   if 8e3 < F

   1. Initial program 67.3%

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

   3. Simplified 79.7%

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

   5. Taylor expanded in F around inf 99.7%

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

      1. associate-/r* 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in F around 0 99.9%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.019:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -9.5 \cdot 10^{-104}:\\ \;\;\;\;F \cdot \left(\frac{1}{\sin B} \cdot \sqrt{0.5}\right)\\ \mathbf{elif}\;F \leq 3 \cdot 10^{-78}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\frac{B}{F} \cdot \left(F + 0.5 \cdot \frac{2 + x \cdot 2}{F}\right)}\\ \mathbf{elif}\;F \leq 1.75 \cdot 10^{-14}:\\ \;\;\;\;\frac{F}{\frac{\sin B}{\sqrt{0.5}}}\\ \mathbf{elif}\;F \leq 8000:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\frac{B}{F} \cdot \left(F + 0.5 \cdot \frac{2 + x \cdot 2}{F}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := x \cdot \frac{-1}{\tan B} + \frac{1}{\frac{B}{F} \cdot \left(F + 0.5 \cdot \frac{2 + x \cdot 2}{F}\right)}\\ t_2 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1}{\sin B} - t_2\\ \mathbf{elif}\;F \leq -9.2 \cdot 10^{-104}:\\ \;\;\;\;F \cdot \left(t_0 \cdot \sqrt{0.5}\right)\\ \mathbf{elif}\;F \leq 1.55 \cdot 10^{-79}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 5.3 \cdot 10^{-17}:\\ \;\;\;\;\frac{1}{\frac{\sin B}{F \cdot \sqrt{0.5}}}\\ \mathbf{elif}\;F \leq 54000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 - t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B)))
        (t_1
         (+
          (* x (/ -1.0 (tan B)))
          (/ 1.0 (* (/ B F) (+ F (* 0.5 (/ (+ 2.0 (* x 2.0)) F)))))))
        (t_2 (/ x (tan B))))
   (if (<= F -3.5e-5)
     (- (/ -1.0 (sin B)) t_2)
     (if (<= F -9.2e-104)
       (* F (* t_0 (sqrt 0.5)))
       (if (<= F 1.55e-79)
         t_1
         (if (<= F 5.3e-17)
           (/ 1.0 (/ (sin B) (* F (sqrt 0.5))))
           (if (<= F 54000.0) t_1 (- t_0 t_2))))))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / sin(B);
	double t_1 = (x * (-1.0 / tan(B))) + (1.0 / ((B / F) * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))));
	double t_2 = x / tan(B);
	double tmp;
	if (F <= -3.5e-5) {
		tmp = (-1.0 / sin(B)) - t_2;
	} else if (F <= -9.2e-104) {
		tmp = F * (t_0 * sqrt(0.5));
	} else if (F <= 1.55e-79) {
		tmp = t_1;
	} else if (F <= 5.3e-17) {
		tmp = 1.0 / (sin(B) / (F * sqrt(0.5)));
	} else if (F <= 54000.0) {
		tmp = t_1;
	} else {
		tmp = t_0 - t_2;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 / sin(b)
    t_1 = (x * ((-1.0d0) / tan(b))) + (1.0d0 / ((b / f) * (f + (0.5d0 * ((2.0d0 + (x * 2.0d0)) / f)))))
    t_2 = x / tan(b)
    if (f <= (-3.5d-5)) then
        tmp = ((-1.0d0) / sin(b)) - t_2
    else if (f <= (-9.2d-104)) then
        tmp = f * (t_0 * sqrt(0.5d0))
    else if (f <= 1.55d-79) then
        tmp = t_1
    else if (f <= 5.3d-17) then
        tmp = 1.0d0 / (sin(b) / (f * sqrt(0.5d0)))
    else if (f <= 54000.0d0) then
        tmp = t_1
    else
        tmp = t_0 - t_2
    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 * (-1.0 / Math.tan(B))) + (1.0 / ((B / F) * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))));
	double t_2 = x / Math.tan(B);
	double tmp;
	if (F <= -3.5e-5) {
		tmp = (-1.0 / Math.sin(B)) - t_2;
	} else if (F <= -9.2e-104) {
		tmp = F * (t_0 * Math.sqrt(0.5));
	} else if (F <= 1.55e-79) {
		tmp = t_1;
	} else if (F <= 5.3e-17) {
		tmp = 1.0 / (Math.sin(B) / (F * Math.sqrt(0.5)));
	} else if (F <= 54000.0) {
		tmp = t_1;
	} else {
		tmp = t_0 - t_2;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = 1.0 / math.sin(B)
	t_1 = (x * (-1.0 / math.tan(B))) + (1.0 / ((B / F) * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))))
	t_2 = x / math.tan(B)
	tmp = 0
	if F <= -3.5e-5:
		tmp = (-1.0 / math.sin(B)) - t_2
	elif F <= -9.2e-104:
		tmp = F * (t_0 * math.sqrt(0.5))
	elif F <= 1.55e-79:
		tmp = t_1
	elif F <= 5.3e-17:
		tmp = 1.0 / (math.sin(B) / (F * math.sqrt(0.5)))
	elif F <= 54000.0:
		tmp = t_1
	else:
		tmp = t_0 - t_2
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / sin(B))
	t_1 = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / Float64(Float64(B / F) * Float64(F + Float64(0.5 * Float64(Float64(2.0 + Float64(x * 2.0)) / F))))))
	t_2 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -3.5e-5)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_2);
	elseif (F <= -9.2e-104)
		tmp = Float64(F * Float64(t_0 * sqrt(0.5)));
	elseif (F <= 1.55e-79)
		tmp = t_1;
	elseif (F <= 5.3e-17)
		tmp = Float64(1.0 / Float64(sin(B) / Float64(F * sqrt(0.5))));
	elseif (F <= 54000.0)
		tmp = t_1;
	else
		tmp = Float64(t_0 - t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = 1.0 / sin(B);
	t_1 = (x * (-1.0 / tan(B))) + (1.0 / ((B / F) * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))));
	t_2 = x / tan(B);
	tmp = 0.0;
	if (F <= -3.5e-5)
		tmp = (-1.0 / sin(B)) - t_2;
	elseif (F <= -9.2e-104)
		tmp = F * (t_0 * sqrt(0.5));
	elseif (F <= 1.55e-79)
		tmp = t_1;
	elseif (F <= 5.3e-17)
		tmp = 1.0 / (sin(B) / (F * sqrt(0.5)));
	elseif (F <= 54000.0)
		tmp = t_1;
	else
		tmp = t_0 - t_2;
	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[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[(B / F), $MachinePrecision] * N[(F + N[(0.5 * N[(N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -3.5e-5], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision], If[LessEqual[F, -9.2e-104], N[(F * N[(t$95$0 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.55e-79], t$95$1, If[LessEqual[F, 5.3e-17], N[(1.0 / N[(N[Sin[B], $MachinePrecision] / N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 54000.0], t$95$1, N[(t$95$0 - t$95$2), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -3.4999999999999997e-5

   1. Initial program 59.6%

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

   3. Taylor expanded in F around -inf 99.3%

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

      1. +-commutative 99.3%

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

      2. add-sqr-sqrt 45.2%

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

      3. sqrt-unprod 51.2%

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

      4. frac-times 51.2%

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

      5. metadata-eval 51.2%

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

      6. metadata-eval 51.2%

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

      7. frac-times 51.2%

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

      8. sqrt-unprod 20.6%

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

      9. add-sqr-sqrt 49.8%

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

      10. metadata-eval 49.8%

          \[\leadsto \frac{\color{blue}{{F}^{0}}}{\sin B} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      11. metadata-eval 49.8%

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

      12. pow-prod-up 49.8%

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

      13. pow1 49.8%

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

      14. inv-pow 49.8%

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

      15. associate-*r/ 49.8%

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

      16. add-sqr-sqrt 20.6%

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

      17. fma-def 20.6%

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

   5. Applied egg-rr 99.4%

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

   if -3.4999999999999997e-5 < F < -9.1999999999999998e-104

   1. Initial program 99.1%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in F around 0 98.5%

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

   6. Taylor expanded in x around 0 70.7%

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

      1. *-lft-identity 70.7%

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

      2. times-frac 70.8%

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

      3. /-rgt-identity 70.8%

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

   8. Simplified 70.8%

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

      1. div-inv 70.8%

         \[\leadsto F \cdot \color{blue}{\left(\sqrt{0.5} \cdot \frac{1}{\sin B}\right)} \]

   10. Applied egg-rr 70.8%

       \[\leadsto F \cdot \color{blue}{\left(\sqrt{0.5} \cdot \frac{1}{\sin B}\right)} \]

   if -9.1999999999999998e-104 < F < 1.55e-79 or 5.2999999999999998e-17 < F < 54000

   1. Initial program 99.7%

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

      1. associate-*l/ 99.6%

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

      2. +-commutative 99.6%

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

      3. *-commutative 99.6%

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

      4. fma-udef 99.6%

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

      5. fma-def 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. clear-num 99.6%

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

      9. fma-def 99.6%

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

      10. fma-udef 99.6%

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

      11. *-commutative 99.6%

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

      12. fma-def 99.6%

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

      13. fma-def 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in B around 0 87.5%

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

   6. Taylor expanded in F around inf 76.2%

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

   if 1.55e-79 < F < 5.2999999999999998e-17

   1. Initial program 99.0%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in F around 0 99.5%

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

   6. Taylor expanded in x around 0 62.7%

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

      1. *-lft-identity 62.7%

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

      2. times-frac 62.8%

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

      3. /-rgt-identity 62.8%

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

   8. Simplified 62.8%

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

      1. associate-*r/ 62.7%

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

      2. clear-num 62.9%

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

   10. Applied egg-rr 62.9%

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

   if 54000 < F

   1. Initial program 67.3%

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

   3. Simplified 79.7%

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

   5. Taylor expanded in F around inf 99.7%

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

      1. associate-/r* 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in F around 0 99.9%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -9.2 \cdot 10^{-104}:\\ \;\;\;\;F \cdot \left(\frac{1}{\sin B} \cdot \sqrt{0.5}\right)\\ \mathbf{elif}\;F \leq 1.55 \cdot 10^{-79}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\frac{B}{F} \cdot \left(F + 0.5 \cdot \frac{2 + x \cdot 2}{F}\right)}\\ \mathbf{elif}\;F \leq 5.3 \cdot 10^{-17}:\\ \;\;\;\;\frac{1}{\frac{\sin B}{F \cdot \sqrt{0.5}}}\\ \mathbf{elif}\;F \leq 54000:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\frac{B}{F} \cdot \left(F + 0.5 \cdot \frac{2 + x \cdot 2}{F}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -8100000:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq -2.2 \cdot 10^{-126}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.65 \cdot 10^{-13}:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -8100000.0)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F -2.2e-126)
       (- (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)) (/ x B))
       (if (<= F 1.65e-13)
         (- (* (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 <= -8100000.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= -2.2e-126) {
		tmp = ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B);
	} else if (F <= 1.65e-13) {
		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 <= (-8100000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= (-2.2d-126)) then
        tmp = ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0))) - (x / b)
    else if (f <= 1.65d-13) 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 <= -8100000.0) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= -2.2e-126) {
		tmp = ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B);
	} else if (F <= 1.65e-13) {
		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 <= -8100000.0:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= -2.2e-126:
		tmp = ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B)
	elif F <= 1.65e-13:
		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 <= -8100000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= -2.2e-126)
		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 <= 1.65e-13)
		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 <= -8100000.0)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= -2.2e-126)
		tmp = ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5)) - (x / B);
	elseif (F <= 1.65e-13)
		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, -8100000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -2.2e-126], 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, 1.65e-13], 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 < -8.1e6

   1. Initial program 58.0%

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

   3. Taylor expanded in F around -inf 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. add-sqr-sqrt 44.4%

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

      3. sqrt-unprod 51.8%

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

      4. frac-times 51.8%

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

      5. metadata-eval 51.8%

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

      6. metadata-eval 51.8%

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

      7. frac-times 51.8%

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

      8. sqrt-unprod 21.4%

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

      9. add-sqr-sqrt 49.1%

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

      10. metadata-eval 49.1%

          \[\leadsto \frac{\color{blue}{{F}^{0}}}{\sin B} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      11. metadata-eval 49.1%

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

      12. pow-prod-up 49.1%

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

      13. pow1 49.1%

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

      14. inv-pow 49.1%

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

      15. associate-*r/ 49.1%

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

      16. add-sqr-sqrt 21.4%

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

      17. fma-def 21.4%

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

   5. Applied egg-rr 99.8%

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

   if -8.1e6 < F < -2.20000000000000014e-126

   1. Initial program 99.2%

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

   3. Taylor expanded in B around 0 85.0%

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

   if -2.20000000000000014e-126 < F < 1.65e-13

   1. Initial program 99.6%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in F around 0 99.7%

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

   6. Taylor expanded in B around 0 84.7%

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

   if 1.65e-13 < F

   1. Initial program 69.3%

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

   3. Simplified 81.0%

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

   5. Taylor expanded in F around inf 97.5%

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

      1. associate-/r* 97.5%

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

   7. Simplified 97.5%

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

   8. Taylor expanded in F around 0 97.7%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -8100000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -2.2 \cdot 10^{-126}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.65 \cdot 10^{-13}:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := F \cdot \frac{\sqrt{0.5}}{\sin B}\\ t_1 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -1.3 \cdot 10^{-15}:\\ \;\;\;\;t_1 + \frac{-1}{B}\\ \mathbf{elif}\;F \leq -2.8 \cdot 10^{-104}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 7.8 \cdot 10^{-78}:\\ \;\;\;\;t_1 + \frac{1}{\frac{B}{F} \cdot \left(F + 0.5 \cdot \frac{2 + x \cdot 2}{F}\right)}\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 2.55 \cdot 10^{+66}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* F (/ (sqrt 0.5) (sin B)))) (t_1 (* x (/ -1.0 (tan B)))))
   (if (<= F -1.3e-15)
     (+ t_1 (/ -1.0 B))
     (if (<= F -2.8e-104)
       t_0
       (if (<= F 7.8e-78)
         (+ t_1 (/ 1.0 (* (/ B F) (+ F (* 0.5 (/ (+ 2.0 (* x 2.0)) F))))))
         (if (<= F 1.4)
           t_0
           (if (<= F 2.55e+66)
             (/ 1.0 (sin B))
             (- (/ 1.0 B) (/ x (tan B))))))))))

C

double code(double F, double B, double x) {
	double t_0 = F * (sqrt(0.5) / sin(B));
	double t_1 = x * (-1.0 / tan(B));
	double tmp;
	if (F <= -1.3e-15) {
		tmp = t_1 + (-1.0 / B);
	} else if (F <= -2.8e-104) {
		tmp = t_0;
	} else if (F <= 7.8e-78) {
		tmp = t_1 + (1.0 / ((B / F) * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))));
	} else if (F <= 1.4) {
		tmp = t_0;
	} else if (F <= 2.55e+66) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = (1.0 / B) - (x / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = f * (sqrt(0.5d0) / sin(b))
    t_1 = x * ((-1.0d0) / tan(b))
    if (f <= (-1.3d-15)) then
        tmp = t_1 + ((-1.0d0) / b)
    else if (f <= (-2.8d-104)) then
        tmp = t_0
    else if (f <= 7.8d-78) then
        tmp = t_1 + (1.0d0 / ((b / f) * (f + (0.5d0 * ((2.0d0 + (x * 2.0d0)) / f)))))
    else if (f <= 1.4d0) then
        tmp = t_0
    else if (f <= 2.55d+66) then
        tmp = 1.0d0 / sin(b)
    else
        tmp = (1.0d0 / b) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = F * (Math.sqrt(0.5) / Math.sin(B));
	double t_1 = x * (-1.0 / Math.tan(B));
	double tmp;
	if (F <= -1.3e-15) {
		tmp = t_1 + (-1.0 / B);
	} else if (F <= -2.8e-104) {
		tmp = t_0;
	} else if (F <= 7.8e-78) {
		tmp = t_1 + (1.0 / ((B / F) * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))));
	} else if (F <= 1.4) {
		tmp = t_0;
	} else if (F <= 2.55e+66) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = (1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = F * (math.sqrt(0.5) / math.sin(B))
	t_1 = x * (-1.0 / math.tan(B))
	tmp = 0
	if F <= -1.3e-15:
		tmp = t_1 + (-1.0 / B)
	elif F <= -2.8e-104:
		tmp = t_0
	elif F <= 7.8e-78:
		tmp = t_1 + (1.0 / ((B / F) * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))))
	elif F <= 1.4:
		tmp = t_0
	elif F <= 2.55e+66:
		tmp = 1.0 / math.sin(B)
	else:
		tmp = (1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(F * Float64(sqrt(0.5) / sin(B)))
	t_1 = Float64(x * Float64(-1.0 / tan(B)))
	tmp = 0.0
	if (F <= -1.3e-15)
		tmp = Float64(t_1 + Float64(-1.0 / B));
	elseif (F <= -2.8e-104)
		tmp = t_0;
	elseif (F <= 7.8e-78)
		tmp = Float64(t_1 + Float64(1.0 / Float64(Float64(B / F) * Float64(F + Float64(0.5 * Float64(Float64(2.0 + Float64(x * 2.0)) / F))))));
	elseif (F <= 1.4)
		tmp = t_0;
	elseif (F <= 2.55e+66)
		tmp = Float64(1.0 / sin(B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = F * (sqrt(0.5) / sin(B));
	t_1 = x * (-1.0 / tan(B));
	tmp = 0.0;
	if (F <= -1.3e-15)
		tmp = t_1 + (-1.0 / B);
	elseif (F <= -2.8e-104)
		tmp = t_0;
	elseif (F <= 7.8e-78)
		tmp = t_1 + (1.0 / ((B / F) * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))));
	elseif (F <= 1.4)
		tmp = t_0;
	elseif (F <= 2.55e+66)
		tmp = 1.0 / sin(B);
	else
		tmp = (1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.3e-15], N[(t$95$1 + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -2.8e-104], t$95$0, If[LessEqual[F, 7.8e-78], N[(t$95$1 + N[(1.0 / N[(N[(B / F), $MachinePrecision] * N[(F + N[(0.5 * N[(N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.4], t$95$0, If[LessEqual[F, 2.55e+66], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -1.30000000000000002e-15

   1. Initial program 61.1%

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

   3. Taylor expanded in F around -inf 95.9%

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

   4. Taylor expanded in B around 0 79.2%

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

   if -1.30000000000000002e-15 < F < -2.8e-104 or 7.8000000000000004e-78 < F < 1.3999999999999999

   1. Initial program 99.1%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in F around 0 99.3%

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

   6. Taylor expanded in x around 0 69.2%

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

      1. *-lft-identity 69.2%

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

      2. times-frac 69.3%

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

      3. /-rgt-identity 69.3%

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

   8. Simplified 69.3%

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

   if -2.8e-104 < F < 7.8000000000000004e-78

   1. Initial program 99.7%

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

      1. associate-*l/ 99.6%

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

      2. +-commutative 99.6%

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

      3. *-commutative 99.6%

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

      4. fma-udef 99.6%

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

      5. fma-def 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. clear-num 99.6%

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

      9. fma-def 99.6%

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

      10. fma-udef 99.6%

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

      11. *-commutative 99.6%

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

      12. fma-def 99.6%

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

      13. fma-def 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in B around 0 88.0%

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

   6. Taylor expanded in F around inf 76.4%

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

   if 1.3999999999999999 < F < 2.55000000000000004e66

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in F around inf 95.8%

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

      1. associate-/r* 95.9%

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

   7. Simplified 95.9%

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

   8. Taylor expanded in F around 0 96.2%

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

   9. Taylor expanded in x around 0 74.5%

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

   if 2.55000000000000004e66 < F

   1. Initial program 59.7%

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

   3. Simplified 75.0%

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

   5. Taylor expanded in F around inf 99.7%

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

      1. associate-/r* 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in B around 0 81.4%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.3 \cdot 10^{-15}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq -2.8 \cdot 10^{-104}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 7.8 \cdot 10^{-78}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\frac{B}{F} \cdot \left(F + 0.5 \cdot \frac{2 + x \cdot 2}{F}\right)}\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 2.55 \cdot 10^{+66}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 70.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -3.2 \cdot 10^{-16}:\\ \;\;\;\;t_0 + \frac{-1}{B}\\ \mathbf{elif}\;F \leq -2.1 \cdot 10^{-104}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 3.2 \cdot 10^{-80}:\\ \;\;\;\;t_0 + \frac{1}{\frac{B}{F} \cdot \left(F + 0.5 \cdot \frac{2 + x \cdot 2}{F}\right)}\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;\frac{F}{\frac{\sin B}{\sqrt{0.5}}}\\ \mathbf{elif}\;F \leq 3 \cdot 10^{+58}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B)))))
   (if (<= F -3.2e-16)
     (+ t_0 (/ -1.0 B))
     (if (<= F -2.1e-104)
       (* F (/ (sqrt 0.5) (sin B)))
       (if (<= F 3.2e-80)
         (+ t_0 (/ 1.0 (* (/ B F) (+ F (* 0.5 (/ (+ 2.0 (* x 2.0)) F))))))
         (if (<= F 1.4)
           (/ F (/ (sin B) (sqrt 0.5)))
           (if (<= F 3e+58) (/ 1.0 (sin B)) (- (/ 1.0 B) (/ x (tan B))))))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double tmp;
	if (F <= -3.2e-16) {
		tmp = t_0 + (-1.0 / B);
	} else if (F <= -2.1e-104) {
		tmp = F * (sqrt(0.5) / sin(B));
	} else if (F <= 3.2e-80) {
		tmp = t_0 + (1.0 / ((B / F) * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))));
	} else if (F <= 1.4) {
		tmp = F / (sin(B) / sqrt(0.5));
	} else if (F <= 3e+58) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = (1.0 / B) - (x / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * ((-1.0d0) / tan(b))
    if (f <= (-3.2d-16)) then
        tmp = t_0 + ((-1.0d0) / b)
    else if (f <= (-2.1d-104)) then
        tmp = f * (sqrt(0.5d0) / sin(b))
    else if (f <= 3.2d-80) then
        tmp = t_0 + (1.0d0 / ((b / f) * (f + (0.5d0 * ((2.0d0 + (x * 2.0d0)) / f)))))
    else if (f <= 1.4d0) then
        tmp = f / (sin(b) / sqrt(0.5d0))
    else if (f <= 3d+58) then
        tmp = 1.0d0 / sin(b)
    else
        tmp = (1.0d0 / b) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / Math.tan(B));
	double tmp;
	if (F <= -3.2e-16) {
		tmp = t_0 + (-1.0 / B);
	} else if (F <= -2.1e-104) {
		tmp = F * (Math.sqrt(0.5) / Math.sin(B));
	} else if (F <= 3.2e-80) {
		tmp = t_0 + (1.0 / ((B / F) * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))));
	} else if (F <= 1.4) {
		tmp = F / (Math.sin(B) / Math.sqrt(0.5));
	} else if (F <= 3e+58) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = (1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * (-1.0 / math.tan(B))
	tmp = 0
	if F <= -3.2e-16:
		tmp = t_0 + (-1.0 / B)
	elif F <= -2.1e-104:
		tmp = F * (math.sqrt(0.5) / math.sin(B))
	elif F <= 3.2e-80:
		tmp = t_0 + (1.0 / ((B / F) * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))))
	elif F <= 1.4:
		tmp = F / (math.sin(B) / math.sqrt(0.5))
	elif F <= 3e+58:
		tmp = 1.0 / math.sin(B)
	else:
		tmp = (1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	tmp = 0.0
	if (F <= -3.2e-16)
		tmp = Float64(t_0 + Float64(-1.0 / B));
	elseif (F <= -2.1e-104)
		tmp = Float64(F * Float64(sqrt(0.5) / sin(B)));
	elseif (F <= 3.2e-80)
		tmp = Float64(t_0 + Float64(1.0 / Float64(Float64(B / F) * Float64(F + Float64(0.5 * Float64(Float64(2.0 + Float64(x * 2.0)) / F))))));
	elseif (F <= 1.4)
		tmp = Float64(F / Float64(sin(B) / sqrt(0.5)));
	elseif (F <= 3e+58)
		tmp = Float64(1.0 / sin(B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x * (-1.0 / tan(B));
	tmp = 0.0;
	if (F <= -3.2e-16)
		tmp = t_0 + (-1.0 / B);
	elseif (F <= -2.1e-104)
		tmp = F * (sqrt(0.5) / sin(B));
	elseif (F <= 3.2e-80)
		tmp = t_0 + (1.0 / ((B / F) * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))));
	elseif (F <= 1.4)
		tmp = F / (sin(B) / sqrt(0.5));
	elseif (F <= 3e+58)
		tmp = 1.0 / sin(B);
	else
		tmp = (1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -3.2e-16], N[(t$95$0 + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -2.1e-104], N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.2e-80], N[(t$95$0 + N[(1.0 / N[(N[(B / F), $MachinePrecision] * N[(F + N[(0.5 * N[(N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.4], N[(F / N[(N[Sin[B], $MachinePrecision] / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3e+58], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if F < -3.20000000000000023e-16

   1. Initial program 61.1%

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

   3. Taylor expanded in F around -inf 95.9%

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

   4. Taylor expanded in B around 0 79.2%

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

   if -3.20000000000000023e-16 < F < -2.09999999999999999e-104

   1. Initial program 99.1%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in F around 0 99.2%

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

   6. Taylor expanded in x around 0 77.2%

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

      1. *-lft-identity 77.2%

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

      2. times-frac 77.3%

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

      3. /-rgt-identity 77.3%

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

   8. Simplified 77.3%

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

   if -2.09999999999999999e-104 < F < 3.1999999999999999e-80

   1. Initial program 99.7%

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

      1. associate-*l/ 99.6%

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

      2. +-commutative 99.6%

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

      3. *-commutative 99.6%

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

      4. fma-udef 99.6%

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

      5. fma-def 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. clear-num 99.6%

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

      9. fma-def 99.6%

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

      10. fma-udef 99.6%

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

      11. *-commutative 99.6%

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

      12. fma-def 99.6%

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

      13. fma-def 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in B around 0 88.0%

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

   6. Taylor expanded in F around inf 76.4%

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

   if 3.1999999999999999e-80 < F < 1.3999999999999999

   1. Initial program 99.0%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in F around 0 99.4%

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

   6. Taylor expanded in x around 0 61.2%

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

      1. associate-/l* 61.4%

         \[\leadsto \color{blue}{\frac{F}{\frac{\sin B}{\sqrt{0.5}}}} \]

   8. Simplified 61.4%

      \[\leadsto \color{blue}{\frac{F}{\frac{\sin B}{\sqrt{0.5}}}} \]

   if 1.3999999999999999 < F < 3.0000000000000002e58

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in F around inf 95.8%

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

      1. associate-/r* 95.9%

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

   7. Simplified 95.9%

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

   8. Taylor expanded in F around 0 96.2%

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

   9. Taylor expanded in x around 0 74.5%

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

   if 3.0000000000000002e58 < F

   1. Initial program 59.7%

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

   3. Simplified 75.0%

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

   5. Taylor expanded in F around inf 99.7%

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

      1. associate-/r* 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in B around 0 81.4%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.2 \cdot 10^{-16}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq -2.1 \cdot 10^{-104}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 3.2 \cdot 10^{-80}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\frac{B}{F} \cdot \left(F + 0.5 \cdot \frac{2 + x \cdot 2}{F}\right)}\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;\frac{F}{\frac{\sin B}{\sqrt{0.5}}}\\ \mathbf{elif}\;F \leq 3 \cdot 10^{+58}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 70.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -8.6 \cdot 10^{-20}:\\ \;\;\;\;t_1 + \frac{-1}{B}\\ \mathbf{elif}\;F \leq -2.1 \cdot 10^{-104}:\\ \;\;\;\;F \cdot \left(t_0 \cdot \sqrt{0.5}\right)\\ \mathbf{elif}\;F \leq 1.48 \cdot 10^{-78}:\\ \;\;\;\;t_1 + \frac{1}{\frac{B}{F} \cdot \left(F + 0.5 \cdot \frac{2 + x \cdot 2}{F}\right)}\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;\frac{F}{\frac{\sin B}{\sqrt{0.5}}}\\ \mathbf{elif}\;F \leq 9.4 \cdot 10^{+58}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))) (t_1 (* x (/ -1.0 (tan B)))))
   (if (<= F -8.6e-20)
     (+ t_1 (/ -1.0 B))
     (if (<= F -2.1e-104)
       (* F (* t_0 (sqrt 0.5)))
       (if (<= F 1.48e-78)
         (+ t_1 (/ 1.0 (* (/ B F) (+ F (* 0.5 (/ (+ 2.0 (* x 2.0)) F))))))
         (if (<= F 1.4)
           (/ F (/ (sin B) (sqrt 0.5)))
           (if (<= F 9.4e+58) t_0 (- (/ 1.0 B) (/ x (tan B))))))))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / sin(B);
	double t_1 = x * (-1.0 / tan(B));
	double tmp;
	if (F <= -8.6e-20) {
		tmp = t_1 + (-1.0 / B);
	} else if (F <= -2.1e-104) {
		tmp = F * (t_0 * sqrt(0.5));
	} else if (F <= 1.48e-78) {
		tmp = t_1 + (1.0 / ((B / F) * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))));
	} else if (F <= 1.4) {
		tmp = F / (sin(B) / sqrt(0.5));
	} else if (F <= 9.4e+58) {
		tmp = t_0;
	} else {
		tmp = (1.0 / B) - (x / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 / sin(b)
    t_1 = x * ((-1.0d0) / tan(b))
    if (f <= (-8.6d-20)) then
        tmp = t_1 + ((-1.0d0) / b)
    else if (f <= (-2.1d-104)) then
        tmp = f * (t_0 * sqrt(0.5d0))
    else if (f <= 1.48d-78) then
        tmp = t_1 + (1.0d0 / ((b / f) * (f + (0.5d0 * ((2.0d0 + (x * 2.0d0)) / f)))))
    else if (f <= 1.4d0) then
        tmp = f / (sin(b) / sqrt(0.5d0))
    else if (f <= 9.4d+58) then
        tmp = t_0
    else
        tmp = (1.0d0 / b) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = 1.0 / Math.sin(B);
	double t_1 = x * (-1.0 / Math.tan(B));
	double tmp;
	if (F <= -8.6e-20) {
		tmp = t_1 + (-1.0 / B);
	} else if (F <= -2.1e-104) {
		tmp = F * (t_0 * Math.sqrt(0.5));
	} else if (F <= 1.48e-78) {
		tmp = t_1 + (1.0 / ((B / F) * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))));
	} else if (F <= 1.4) {
		tmp = F / (Math.sin(B) / Math.sqrt(0.5));
	} else if (F <= 9.4e+58) {
		tmp = t_0;
	} else {
		tmp = (1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = 1.0 / math.sin(B)
	t_1 = x * (-1.0 / math.tan(B))
	tmp = 0
	if F <= -8.6e-20:
		tmp = t_1 + (-1.0 / B)
	elif F <= -2.1e-104:
		tmp = F * (t_0 * math.sqrt(0.5))
	elif F <= 1.48e-78:
		tmp = t_1 + (1.0 / ((B / F) * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))))
	elif F <= 1.4:
		tmp = F / (math.sin(B) / math.sqrt(0.5))
	elif F <= 9.4e+58:
		tmp = t_0
	else:
		tmp = (1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / sin(B))
	t_1 = Float64(x * Float64(-1.0 / tan(B)))
	tmp = 0.0
	if (F <= -8.6e-20)
		tmp = Float64(t_1 + Float64(-1.0 / B));
	elseif (F <= -2.1e-104)
		tmp = Float64(F * Float64(t_0 * sqrt(0.5)));
	elseif (F <= 1.48e-78)
		tmp = Float64(t_1 + Float64(1.0 / Float64(Float64(B / F) * Float64(F + Float64(0.5 * Float64(Float64(2.0 + Float64(x * 2.0)) / F))))));
	elseif (F <= 1.4)
		tmp = Float64(F / Float64(sin(B) / sqrt(0.5)));
	elseif (F <= 9.4e+58)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = 1.0 / sin(B);
	t_1 = x * (-1.0 / tan(B));
	tmp = 0.0;
	if (F <= -8.6e-20)
		tmp = t_1 + (-1.0 / B);
	elseif (F <= -2.1e-104)
		tmp = F * (t_0 * sqrt(0.5));
	elseif (F <= 1.48e-78)
		tmp = t_1 + (1.0 / ((B / F) * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))));
	elseif (F <= 1.4)
		tmp = F / (sin(B) / sqrt(0.5));
	elseif (F <= 9.4e+58)
		tmp = t_0;
	else
		tmp = (1.0 / B) - (x / tan(B));
	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[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -8.6e-20], N[(t$95$1 + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -2.1e-104], N[(F * N[(t$95$0 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.48e-78], N[(t$95$1 + N[(1.0 / N[(N[(B / F), $MachinePrecision] * N[(F + N[(0.5 * N[(N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.4], N[(F / N[(N[Sin[B], $MachinePrecision] / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 9.4e+58], t$95$0, N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if F < -8.60000000000000022e-20

   1. Initial program 61.1%

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

   3. Taylor expanded in F around -inf 95.9%

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

   4. Taylor expanded in B around 0 79.2%

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

   if -8.60000000000000022e-20 < F < -2.09999999999999999e-104

   1. Initial program 99.1%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in F around 0 99.2%

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

   6. Taylor expanded in x around 0 77.2%

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

      1. *-lft-identity 77.2%

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

      2. times-frac 77.3%

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

      3. /-rgt-identity 77.3%

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

   8. Simplified 77.3%

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

      1. div-inv 77.3%

         \[\leadsto F \cdot \color{blue}{\left(\sqrt{0.5} \cdot \frac{1}{\sin B}\right)} \]

   10. Applied egg-rr 77.3%

       \[\leadsto F \cdot \color{blue}{\left(\sqrt{0.5} \cdot \frac{1}{\sin B}\right)} \]

   if -2.09999999999999999e-104 < F < 1.48000000000000007e-78

   1. Initial program 99.7%

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

      1. associate-*l/ 99.6%

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

      2. +-commutative 99.6%

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

      3. *-commutative 99.6%

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

      4. fma-udef 99.6%

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

      5. fma-def 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. clear-num 99.6%

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

      9. fma-def 99.6%

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

      10. fma-udef 99.6%

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

      11. *-commutative 99.6%

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

      12. fma-def 99.6%

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

      13. fma-def 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in B around 0 88.0%

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

   6. Taylor expanded in F around inf 76.4%

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

   if 1.48000000000000007e-78 < F < 1.3999999999999999

   1. Initial program 99.0%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in F around 0 99.4%

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

   6. Taylor expanded in x around 0 61.2%

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

      1. associate-/l* 61.4%

         \[\leadsto \color{blue}{\frac{F}{\frac{\sin B}{\sqrt{0.5}}}} \]

   8. Simplified 61.4%

      \[\leadsto \color{blue}{\frac{F}{\frac{\sin B}{\sqrt{0.5}}}} \]

   if 1.3999999999999999 < F < 9.39999999999999944e58

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in F around inf 95.8%

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

      1. associate-/r* 95.9%

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

   7. Simplified 95.9%

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

   8. Taylor expanded in F around 0 96.2%

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

   9. Taylor expanded in x around 0 74.5%

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

   if 9.39999999999999944e58 < F

   1. Initial program 59.7%

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

   3. Simplified 75.0%

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

   5. Taylor expanded in F around inf 99.7%

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

      1. associate-/r* 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in B around 0 81.4%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -8.6 \cdot 10^{-20}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq -2.1 \cdot 10^{-104}:\\ \;\;\;\;F \cdot \left(\frac{1}{\sin B} \cdot \sqrt{0.5}\right)\\ \mathbf{elif}\;F \leq 1.48 \cdot 10^{-78}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\frac{B}{F} \cdot \left(F + 0.5 \cdot \frac{2 + x \cdot 2}{F}\right)}\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;\frac{F}{\frac{\sin B}{\sqrt{0.5}}}\\ \mathbf{elif}\;F \leq 9.4 \cdot 10^{+58}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 78.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.00029:\\ \;\;\;\;\frac{-1}{\sin B} - t_1\\ \mathbf{elif}\;F \leq -5.2 \cdot 10^{-106}:\\ \;\;\;\;F \cdot \left(t_0 \cdot \sqrt{0.5}\right)\\ \mathbf{elif}\;F \leq 6.1 \cdot 10^{-80}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\frac{B}{F} \cdot \left(F + 0.5 \cdot \frac{2 + x \cdot 2}{F}\right)}\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;\frac{F}{\frac{\sin B}{\sqrt{0.5}}}\\ \mathbf{elif}\;F \leq 2.5 \cdot 10^{+58}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))) (t_1 (/ x (tan B))))
   (if (<= F -0.00029)
     (- (/ -1.0 (sin B)) t_1)
     (if (<= F -5.2e-106)
       (* F (* t_0 (sqrt 0.5)))
       (if (<= F 6.1e-80)
         (+
          (* x (/ -1.0 (tan B)))
          (/ 1.0 (* (/ B F) (+ F (* 0.5 (/ (+ 2.0 (* x 2.0)) F))))))
         (if (<= F 1.4)
           (/ F (/ (sin B) (sqrt 0.5)))
           (if (<= F 2.5e+58) t_0 (- (/ 1.0 B) 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 <= -0.00029) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= -5.2e-106) {
		tmp = F * (t_0 * sqrt(0.5));
	} else if (F <= 6.1e-80) {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / ((B / F) * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))));
	} else if (F <= 1.4) {
		tmp = F / (sin(B) / sqrt(0.5));
	} else if (F <= 2.5e+58) {
		tmp = t_0;
	} else {
		tmp = (1.0 / 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 = 1.0d0 / sin(b)
    t_1 = x / tan(b)
    if (f <= (-0.00029d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_1
    else if (f <= (-5.2d-106)) then
        tmp = f * (t_0 * sqrt(0.5d0))
    else if (f <= 6.1d-80) then
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / ((b / f) * (f + (0.5d0 * ((2.0d0 + (x * 2.0d0)) / f)))))
    else if (f <= 1.4d0) then
        tmp = f / (sin(b) / sqrt(0.5d0))
    else if (f <= 2.5d+58) then
        tmp = t_0
    else
        tmp = (1.0d0 / b) - 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 <= -0.00029) {
		tmp = (-1.0 / Math.sin(B)) - t_1;
	} else if (F <= -5.2e-106) {
		tmp = F * (t_0 * Math.sqrt(0.5));
	} else if (F <= 6.1e-80) {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / ((B / F) * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))));
	} else if (F <= 1.4) {
		tmp = F / (Math.sin(B) / Math.sqrt(0.5));
	} else if (F <= 2.5e+58) {
		tmp = t_0;
	} else {
		tmp = (1.0 / B) - 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 <= -0.00029:
		tmp = (-1.0 / math.sin(B)) - t_1
	elif F <= -5.2e-106:
		tmp = F * (t_0 * math.sqrt(0.5))
	elif F <= 6.1e-80:
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / ((B / F) * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))))
	elif F <= 1.4:
		tmp = F / (math.sin(B) / math.sqrt(0.5))
	elif F <= 2.5e+58:
		tmp = t_0
	else:
		tmp = (1.0 / B) - 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 <= -0.00029)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= -5.2e-106)
		tmp = Float64(F * Float64(t_0 * sqrt(0.5)));
	elseif (F <= 6.1e-80)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / Float64(Float64(B / F) * Float64(F + Float64(0.5 * Float64(Float64(2.0 + Float64(x * 2.0)) / F))))));
	elseif (F <= 1.4)
		tmp = Float64(F / Float64(sin(B) / sqrt(0.5)));
	elseif (F <= 2.5e+58)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / B) - 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 <= -0.00029)
		tmp = (-1.0 / sin(B)) - t_1;
	elseif (F <= -5.2e-106)
		tmp = F * (t_0 * sqrt(0.5));
	elseif (F <= 6.1e-80)
		tmp = (x * (-1.0 / tan(B))) + (1.0 / ((B / F) * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))));
	elseif (F <= 1.4)
		tmp = F / (sin(B) / sqrt(0.5));
	elseif (F <= 2.5e+58)
		tmp = t_0;
	else
		tmp = (1.0 / B) - 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, -0.00029], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, -5.2e-106], N[(F * N[(t$95$0 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 6.1e-80], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[(B / F), $MachinePrecision] * N[(F + N[(0.5 * N[(N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.4], N[(F / N[(N[Sin[B], $MachinePrecision] / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.5e+58], t$95$0, N[(N[(1.0 / B), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if F < -2.9e-4

   1. Initial program 59.6%

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

   3. Taylor expanded in F around -inf 99.3%

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

      1. +-commutative 99.3%

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

      2. add-sqr-sqrt 45.2%

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

      3. sqrt-unprod 51.2%

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

      4. frac-times 51.2%

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

      5. metadata-eval 51.2%

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

      6. metadata-eval 51.2%

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

      7. frac-times 51.2%

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

      8. sqrt-unprod 20.6%

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

      9. add-sqr-sqrt 49.8%

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

      10. metadata-eval 49.8%

          \[\leadsto \frac{\color{blue}{{F}^{0}}}{\sin B} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      11. metadata-eval 49.8%

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

      12. pow-prod-up 49.8%

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

      13. pow1 49.8%

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

      14. inv-pow 49.8%

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

      15. associate-*r/ 49.8%

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

      16. add-sqr-sqrt 20.6%

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

      17. fma-def 20.6%

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

   5. Applied egg-rr 99.4%

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

   if -2.9e-4 < F < -5.2000000000000001e-106

   1. Initial program 99.1%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in F around 0 98.5%

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

   6. Taylor expanded in x around 0 70.7%

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

      1. *-lft-identity 70.7%

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

      2. times-frac 70.8%

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

      3. /-rgt-identity 70.8%

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

   8. Simplified 70.8%

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

      1. div-inv 70.8%

         \[\leadsto F \cdot \color{blue}{\left(\sqrt{0.5} \cdot \frac{1}{\sin B}\right)} \]

   10. Applied egg-rr 70.8%

       \[\leadsto F \cdot \color{blue}{\left(\sqrt{0.5} \cdot \frac{1}{\sin B}\right)} \]

   if -5.2000000000000001e-106 < F < 6.1000000000000002e-80

   1. Initial program 99.7%

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

      1. associate-*l/ 99.6%

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

      2. +-commutative 99.6%

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

      3. *-commutative 99.6%

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

      4. fma-udef 99.6%

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

      5. fma-def 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. clear-num 99.6%

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

      9. fma-def 99.6%

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

      10. fma-udef 99.6%

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

      11. *-commutative 99.6%

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

      12. fma-def 99.6%

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

      13. fma-def 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in B around 0 88.0%

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

   6. Taylor expanded in F around inf 76.4%

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

   if 6.1000000000000002e-80 < F < 1.3999999999999999

   1. Initial program 99.0%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in F around 0 99.4%

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

   6. Taylor expanded in x around 0 61.2%

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

      1. associate-/l* 61.4%

         \[\leadsto \color{blue}{\frac{F}{\frac{\sin B}{\sqrt{0.5}}}} \]

   8. Simplified 61.4%

      \[\leadsto \color{blue}{\frac{F}{\frac{\sin B}{\sqrt{0.5}}}} \]

   if 1.3999999999999999 < F < 2.49999999999999993e58

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in F around inf 95.8%

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

      1. associate-/r* 95.9%

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

   7. Simplified 95.9%

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

   8. Taylor expanded in F around 0 96.2%

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

   9. Taylor expanded in x around 0 74.5%

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

   if 2.49999999999999993e58 < F

   1. Initial program 59.7%

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

   3. Simplified 75.0%

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

   5. Taylor expanded in F around inf 99.7%

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

      1. associate-/r* 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in B around 0 81.4%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.00029:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -5.2 \cdot 10^{-106}:\\ \;\;\;\;F \cdot \left(\frac{1}{\sin B} \cdot \sqrt{0.5}\right)\\ \mathbf{elif}\;F \leq 6.1 \cdot 10^{-80}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\frac{B}{F} \cdot \left(F + 0.5 \cdot \frac{2 + x \cdot 2}{F}\right)}\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;\frac{F}{\frac{\sin B}{\sqrt{0.5}}}\\ \mathbf{elif}\;F \leq 2.5 \cdot 10^{+58}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 91.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.5:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 1.65 \cdot 10^{-13}:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -0.5)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.65e-13)
       (- (* (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 <= -0.5) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.65e-13) {
		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 <= (-0.5d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.65d-13) 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 <= -0.5) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.65e-13) {
		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 <= -0.5:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.65e-13:
		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 <= -0.5)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.65e-13)
		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 <= -0.5)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.65e-13)
		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, -0.5], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.65e-13], 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 < -0.5

   1. Initial program 59.1%

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

   3. Taylor expanded in F around -inf 99.3%

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

      1. +-commutative 99.3%

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

      2. add-sqr-sqrt 44.5%

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

      3. sqrt-unprod 50.6%

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

      4. frac-times 50.6%

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

      5. metadata-eval 50.6%

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

      6. metadata-eval 50.6%

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

      7. frac-times 50.6%

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

      8. sqrt-unprod 20.9%

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

      9. add-sqr-sqrt 49.1%

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

      10. metadata-eval 49.1%

          \[\leadsto \frac{\color{blue}{{F}^{0}}}{\sin B} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      11. metadata-eval 49.1%

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

      12. pow-prod-up 49.1%

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

      13. pow1 49.1%

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

      14. inv-pow 49.1%

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

      15. associate-*r/ 49.1%

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

      16. add-sqr-sqrt 20.9%

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

      17. fma-def 20.9%

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

   5. Applied egg-rr 99.4%

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

   if -0.5 < F < 1.65e-13

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in F around 0 99.5%

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

   6. Taylor expanded in B around 0 81.1%

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

   if 1.65e-13 < F

   1. Initial program 69.3%

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

   3. Simplified 81.0%

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

   5. Taylor expanded in F around inf 97.5%

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

      1. associate-/r* 97.5%

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

   7. Simplified 97.5%

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

   8. Taylor expanded in F around 0 97.7%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.5:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.65 \cdot 10^{-13}:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 91.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.47:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 1.65 \cdot 10^{-13}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -0.47)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.65e-13)
       (- (* F (/ (sqrt 0.5) B)) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -0.47) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.65e-13) {
		tmp = (F * (sqrt(0.5) / B)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-0.47d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.65d-13) then
        tmp = (f * (sqrt(0.5d0) / b)) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -0.47) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.65e-13) {
		tmp = (F * (Math.sqrt(0.5) / B)) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -0.47:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.65e-13:
		tmp = (F * (math.sqrt(0.5) / B)) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -0.47)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.65e-13)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / B)) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -0.47)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.65e-13)
		tmp = (F * (sqrt(0.5) / B)) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.47], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.65e-13], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.46999999999999997

   1. Initial program 59.1%

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

   3. Taylor expanded in F around -inf 99.3%

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

      1. +-commutative 99.3%

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

      2. add-sqr-sqrt 44.5%

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

      3. sqrt-unprod 50.6%

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

      4. frac-times 50.6%

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

      5. metadata-eval 50.6%

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

      6. metadata-eval 50.6%

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

      7. frac-times 50.6%

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

      8. sqrt-unprod 20.9%

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

      9. add-sqr-sqrt 49.1%

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

      10. metadata-eval 49.1%

          \[\leadsto \frac{\color{blue}{{F}^{0}}}{\sin B} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      11. metadata-eval 49.1%

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

      12. pow-prod-up 49.1%

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

      13. pow1 49.1%

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

      14. inv-pow 49.1%

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

      15. associate-*r/ 49.1%

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

      16. add-sqr-sqrt 20.9%

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

      17. fma-def 20.9%

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

   5. Applied egg-rr 99.4%

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

   if -0.46999999999999997 < F < 1.65e-13

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in F around 0 99.5%

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

   6. Taylor expanded in x around 0 99.3%

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

   7. Taylor expanded in B around 0 81.1%

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

   if 1.65e-13 < F

   1. Initial program 69.3%

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

   3. Simplified 81.0%

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

   5. Taylor expanded in F around inf 97.5%

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

      1. associate-/r* 97.5%

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

   7. Simplified 97.5%

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

   8. Taylor expanded in F around 0 97.7%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.47:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.65 \cdot 10^{-13}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 70.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -2.75 \cdot 10^{-18}:\\ \;\;\;\;t_0 + \frac{-1}{B}\\ \mathbf{elif}\;F \leq -5.2 \cdot 10^{-135}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{elif}\;F \leq 50000000:\\ \;\;\;\;t_0 + \frac{1}{\frac{B}{F} \cdot \left(F + 0.5 \cdot \frac{2 + x \cdot 2}{F}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B)))))
   (if (<= F -2.75e-18)
     (+ t_0 (/ -1.0 B))
     (if (<= F -5.2e-135)
       (/ (- (* F (sqrt 0.5)) x) B)
       (if (<= F 50000000.0)
         (+ t_0 (/ 1.0 (* (/ B F) (+ F (* 0.5 (/ (+ 2.0 (* x 2.0)) F))))))
         (- (/ 1.0 B) (/ x (tan B))))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double tmp;
	if (F <= -2.75e-18) {
		tmp = t_0 + (-1.0 / B);
	} else if (F <= -5.2e-135) {
		tmp = ((F * sqrt(0.5)) - x) / B;
	} else if (F <= 50000000.0) {
		tmp = t_0 + (1.0 / ((B / F) * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))));
	} else {
		tmp = (1.0 / B) - (x / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * ((-1.0d0) / tan(b))
    if (f <= (-2.75d-18)) then
        tmp = t_0 + ((-1.0d0) / b)
    else if (f <= (-5.2d-135)) then
        tmp = ((f * sqrt(0.5d0)) - x) / b
    else if (f <= 50000000.0d0) then
        tmp = t_0 + (1.0d0 / ((b / f) * (f + (0.5d0 * ((2.0d0 + (x * 2.0d0)) / f)))))
    else
        tmp = (1.0d0 / b) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / Math.tan(B));
	double tmp;
	if (F <= -2.75e-18) {
		tmp = t_0 + (-1.0 / B);
	} else if (F <= -5.2e-135) {
		tmp = ((F * Math.sqrt(0.5)) - x) / B;
	} else if (F <= 50000000.0) {
		tmp = t_0 + (1.0 / ((B / F) * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))));
	} else {
		tmp = (1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * (-1.0 / math.tan(B))
	tmp = 0
	if F <= -2.75e-18:
		tmp = t_0 + (-1.0 / B)
	elif F <= -5.2e-135:
		tmp = ((F * math.sqrt(0.5)) - x) / B
	elif F <= 50000000.0:
		tmp = t_0 + (1.0 / ((B / F) * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))))
	else:
		tmp = (1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	tmp = 0.0
	if (F <= -2.75e-18)
		tmp = Float64(t_0 + Float64(-1.0 / B));
	elseif (F <= -5.2e-135)
		tmp = Float64(Float64(Float64(F * sqrt(0.5)) - x) / B);
	elseif (F <= 50000000.0)
		tmp = Float64(t_0 + Float64(1.0 / Float64(Float64(B / F) * Float64(F + Float64(0.5 * Float64(Float64(2.0 + Float64(x * 2.0)) / F))))));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x * (-1.0 / tan(B));
	tmp = 0.0;
	if (F <= -2.75e-18)
		tmp = t_0 + (-1.0 / B);
	elseif (F <= -5.2e-135)
		tmp = ((F * sqrt(0.5)) - x) / B;
	elseif (F <= 50000000.0)
		tmp = t_0 + (1.0 / ((B / F) * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))));
	else
		tmp = (1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.75e-18], N[(t$95$0 + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -5.2e-135], N[(N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 50000000.0], N[(t$95$0 + N[(1.0 / N[(N[(B / F), $MachinePrecision] * N[(F + N[(0.5 * N[(N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.75e-18

   1. Initial program 61.1%

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

   3. Taylor expanded in F around -inf 95.9%

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

   4. Taylor expanded in B around 0 79.2%

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

   if -2.75e-18 < F < -5.20000000000000008e-135

   1. Initial program 99.2%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in F around 0 99.2%

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

   6. Taylor expanded in B around 0 56.6%

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

   7. Taylor expanded in x around 0 56.6%

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

   if -5.20000000000000008e-135 < F < 5e7

   1. Initial program 99.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. 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-udef 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-def 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. clear-num 99.6%

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

      9. fma-def 99.6%

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

      10. fma-udef 99.6%

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

      11. *-commutative 99.6%

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

      12. fma-def 99.6%

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

      13. fma-def 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in B around 0 84.2%

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

   6. Taylor expanded in F around inf 70.4%

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

   if 5e7 < F

   1. Initial program 67.3%

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

   3. Simplified 79.7%

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

   5. Taylor expanded in F around inf 99.7%

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

      1. associate-/r* 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in B around 0 76.8%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.75 \cdot 10^{-18}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq -5.2 \cdot 10^{-135}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{elif}\;F \leq 50000000:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\frac{B}{F} \cdot \left(F + 0.5 \cdot \frac{2 + x \cdot 2}{F}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 70.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -1.2 \cdot 10^{-15}:\\ \;\;\;\;t_0 + \frac{-1}{B}\\ \mathbf{elif}\;F \leq -1.25 \cdot 10^{-130}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{elif}\;F \leq 2.95 \cdot 10^{-55}:\\ \;\;\;\;t_0 + \frac{1}{\frac{B}{F} \cdot \left(\frac{-1 - x}{F} - F\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B)))))
   (if (<= F -1.2e-15)
     (+ t_0 (/ -1.0 B))
     (if (<= F -1.25e-130)
       (/ (- (* F (sqrt 0.5)) x) B)
       (if (<= F 2.95e-55)
         (+ t_0 (/ 1.0 (* (/ B F) (- (/ (- -1.0 x) F) F))))
         (- (/ 1.0 B) (/ x (tan B))))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double tmp;
	if (F <= -1.2e-15) {
		tmp = t_0 + (-1.0 / B);
	} else if (F <= -1.25e-130) {
		tmp = ((F * sqrt(0.5)) - x) / B;
	} else if (F <= 2.95e-55) {
		tmp = t_0 + (1.0 / ((B / F) * (((-1.0 - x) / F) - F)));
	} else {
		tmp = (1.0 / B) - (x / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * ((-1.0d0) / tan(b))
    if (f <= (-1.2d-15)) then
        tmp = t_0 + ((-1.0d0) / b)
    else if (f <= (-1.25d-130)) then
        tmp = ((f * sqrt(0.5d0)) - x) / b
    else if (f <= 2.95d-55) then
        tmp = t_0 + (1.0d0 / ((b / f) * ((((-1.0d0) - x) / f) - f)))
    else
        tmp = (1.0d0 / b) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / Math.tan(B));
	double tmp;
	if (F <= -1.2e-15) {
		tmp = t_0 + (-1.0 / B);
	} else if (F <= -1.25e-130) {
		tmp = ((F * Math.sqrt(0.5)) - x) / B;
	} else if (F <= 2.95e-55) {
		tmp = t_0 + (1.0 / ((B / F) * (((-1.0 - x) / F) - F)));
	} else {
		tmp = (1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * (-1.0 / math.tan(B))
	tmp = 0
	if F <= -1.2e-15:
		tmp = t_0 + (-1.0 / B)
	elif F <= -1.25e-130:
		tmp = ((F * math.sqrt(0.5)) - x) / B
	elif F <= 2.95e-55:
		tmp = t_0 + (1.0 / ((B / F) * (((-1.0 - x) / F) - F)))
	else:
		tmp = (1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	tmp = 0.0
	if (F <= -1.2e-15)
		tmp = Float64(t_0 + Float64(-1.0 / B));
	elseif (F <= -1.25e-130)
		tmp = Float64(Float64(Float64(F * sqrt(0.5)) - x) / B);
	elseif (F <= 2.95e-55)
		tmp = Float64(t_0 + Float64(1.0 / Float64(Float64(B / F) * Float64(Float64(Float64(-1.0 - x) / F) - F))));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x * (-1.0 / tan(B));
	tmp = 0.0;
	if (F <= -1.2e-15)
		tmp = t_0 + (-1.0 / B);
	elseif (F <= -1.25e-130)
		tmp = ((F * sqrt(0.5)) - x) / B;
	elseif (F <= 2.95e-55)
		tmp = t_0 + (1.0 / ((B / F) * (((-1.0 - x) / F) - F)));
	else
		tmp = (1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.2e-15], N[(t$95$0 + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.25e-130], N[(N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 2.95e-55], N[(t$95$0 + N[(1.0 / N[(N[(B / F), $MachinePrecision] * N[(N[(N[(-1.0 - x), $MachinePrecision] / F), $MachinePrecision] - F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.19999999999999997e-15

   1. Initial program 61.1%

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

   3. Taylor expanded in F around -inf 95.9%

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

   4. Taylor expanded in B around 0 79.2%

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

   if -1.19999999999999997e-15 < F < -1.2499999999999999e-130

   1. Initial program 99.2%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in F around 0 99.2%

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

   6. Taylor expanded in B around 0 56.6%

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

   7. Taylor expanded in x around 0 56.6%

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

   if -1.2499999999999999e-130 < F < 2.9499999999999999e-55

   1. Initial program 99.6%

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

      1. associate-*l/ 99.6%

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

      2. +-commutative 99.6%

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

      3. *-commutative 99.6%

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

      4. fma-udef 99.6%

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

      5. fma-def 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. clear-num 99.6%

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

      9. fma-def 99.6%

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

      10. fma-udef 99.6%

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

      11. *-commutative 99.6%

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

      12. fma-def 99.6%

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

      13. fma-def 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in B around 0 87.4%

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

   6. Taylor expanded in F around -inf 74.2%

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

      1. neg-mul-1 74.2%

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

      2. +-commutative 74.2%

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

      3. unsub-neg 74.2%

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

      4. associate-*r/ 74.2%

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

      5. distribute-lft-in 74.2%

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

      6. metadata-eval 74.2%

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

      7. associate-*r* 74.2%

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

      8. metadata-eval 74.2%

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

      9. neg-mul-1 74.2%

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

   8. Simplified 74.2%

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

   if 2.9499999999999999e-55 < F

   1. Initial program 73.1%

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

   3. Simplified 83.3%

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

   5. Taylor expanded in F around inf 88.8%

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

      1. associate-/r* 88.8%

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

   7. Simplified 88.8%

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

   8. Taylor expanded in B around 0 70.4%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.2 \cdot 10^{-15}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq -1.25 \cdot 10^{-130}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{elif}\;F \leq 2.95 \cdot 10^{-55}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\frac{B}{F} \cdot \left(\frac{-1 - x}{F} - F\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 43.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1 - x}{B}\\ \mathbf{if}\;F \leq -1.68 \cdot 10^{+174}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -1.42 \cdot 10^{+63}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -0.0095:\\ \;\;\;\;t_0 + B \cdot \left(x \cdot 0.3333333333333333 - 0.16666666666666666\right)\\ \mathbf{elif}\;F \leq -7 \cdot 10^{-110}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B}\\ \mathbf{elif}\;F \leq 1.38 \cdot 10^{-33}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 3.1 \cdot 10^{+81}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333 + 0.16666666666666666\right) + \frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- -1.0 x) B)))
   (if (<= F -1.68e+174)
     t_0
     (if (<= F -1.42e+63)
       (/ -1.0 (sin B))
       (if (<= F -0.0095)
         (+ t_0 (* B (- (* x 0.3333333333333333) 0.16666666666666666)))
         (if (<= F -7e-110)
           (* F (/ (sqrt 0.5) B))
           (if (<= F 1.38e-33)
             (/ (- x) B)
             (if (<= F 3.1e+81)
               (/ 1.0 (sin B))
               (+
                (* B (+ (* x 0.3333333333333333) 0.16666666666666666))
                (/ (- 1.0 x) B))))))))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 - x) / B;
	double tmp;
	if (F <= -1.68e+174) {
		tmp = t_0;
	} else if (F <= -1.42e+63) {
		tmp = -1.0 / sin(B);
	} else if (F <= -0.0095) {
		tmp = t_0 + (B * ((x * 0.3333333333333333) - 0.16666666666666666));
	} else if (F <= -7e-110) {
		tmp = F * (sqrt(0.5) / B);
	} else if (F <= 1.38e-33) {
		tmp = -x / B;
	} else if (F <= 3.1e+81) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = (B * ((x * 0.3333333333333333) + 0.16666666666666666)) + ((1.0 - x) / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-1.0d0) - x) / b
    if (f <= (-1.68d+174)) then
        tmp = t_0
    else if (f <= (-1.42d+63)) then
        tmp = (-1.0d0) / sin(b)
    else if (f <= (-0.0095d0)) then
        tmp = t_0 + (b * ((x * 0.3333333333333333d0) - 0.16666666666666666d0))
    else if (f <= (-7d-110)) then
        tmp = f * (sqrt(0.5d0) / b)
    else if (f <= 1.38d-33) then
        tmp = -x / b
    else if (f <= 3.1d+81) then
        tmp = 1.0d0 / sin(b)
    else
        tmp = (b * ((x * 0.3333333333333333d0) + 0.16666666666666666d0)) + ((1.0d0 - x) / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (-1.0 - x) / B;
	double tmp;
	if (F <= -1.68e+174) {
		tmp = t_0;
	} else if (F <= -1.42e+63) {
		tmp = -1.0 / Math.sin(B);
	} else if (F <= -0.0095) {
		tmp = t_0 + (B * ((x * 0.3333333333333333) - 0.16666666666666666));
	} else if (F <= -7e-110) {
		tmp = F * (Math.sqrt(0.5) / B);
	} else if (F <= 1.38e-33) {
		tmp = -x / B;
	} else if (F <= 3.1e+81) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = (B * ((x * 0.3333333333333333) + 0.16666666666666666)) + ((1.0 - x) / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (-1.0 - x) / B
	tmp = 0
	if F <= -1.68e+174:
		tmp = t_0
	elif F <= -1.42e+63:
		tmp = -1.0 / math.sin(B)
	elif F <= -0.0095:
		tmp = t_0 + (B * ((x * 0.3333333333333333) - 0.16666666666666666))
	elif F <= -7e-110:
		tmp = F * (math.sqrt(0.5) / B)
	elif F <= 1.38e-33:
		tmp = -x / B
	elif F <= 3.1e+81:
		tmp = 1.0 / math.sin(B)
	else:
		tmp = (B * ((x * 0.3333333333333333) + 0.16666666666666666)) + ((1.0 - x) / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 - x) / B)
	tmp = 0.0
	if (F <= -1.68e+174)
		tmp = t_0;
	elseif (F <= -1.42e+63)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= -0.0095)
		tmp = Float64(t_0 + Float64(B * Float64(Float64(x * 0.3333333333333333) - 0.16666666666666666)));
	elseif (F <= -7e-110)
		tmp = Float64(F * Float64(sqrt(0.5) / B));
	elseif (F <= 1.38e-33)
		tmp = Float64(Float64(-x) / B);
	elseif (F <= 3.1e+81)
		tmp = Float64(1.0 / sin(B));
	else
		tmp = Float64(Float64(B * Float64(Float64(x * 0.3333333333333333) + 0.16666666666666666)) + Float64(Float64(1.0 - x) / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (-1.0 - x) / B;
	tmp = 0.0;
	if (F <= -1.68e+174)
		tmp = t_0;
	elseif (F <= -1.42e+63)
		tmp = -1.0 / sin(B);
	elseif (F <= -0.0095)
		tmp = t_0 + (B * ((x * 0.3333333333333333) - 0.16666666666666666));
	elseif (F <= -7e-110)
		tmp = F * (sqrt(0.5) / B);
	elseif (F <= 1.38e-33)
		tmp = -x / B;
	elseif (F <= 3.1e+81)
		tmp = 1.0 / sin(B);
	else
		tmp = (B * ((x * 0.3333333333333333) + 0.16666666666666666)) + ((1.0 - x) / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision]}, If[LessEqual[F, -1.68e+174], t$95$0, If[LessEqual[F, -1.42e+63], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -0.0095], N[(t$95$0 + N[(B * N[(N[(x * 0.3333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -7e-110], N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.38e-33], N[((-x) / B), $MachinePrecision], If[LessEqual[F, 3.1e+81], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(B * N[(N[(x * 0.3333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if F < -1.68e174

   1. Initial program 26.6%

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

   3. Taylor expanded in F around -inf 99.8%

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

   4. Taylor expanded in B around 0 65.0%

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

      1. associate-*r/ 65.0%

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

      2. distribute-lft-in 65.0%

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

      3. metadata-eval 65.0%

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

      4. neg-mul-1 65.0%

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

   6. Simplified 65.0%

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

   if -1.68e174 < F < -1.4200000000000001e63

   1. Initial program 78.3%

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

   3. Taylor expanded in F around -inf 99.7%

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

   4. Taylor expanded in x around 0 57.2%

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

   if -1.4200000000000001e63 < F < -0.00949999999999999976

   1. Initial program 94.0%

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

   3. Taylor expanded in F around -inf 97.6%

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

   4. Taylor expanded in B around 0 60.9%

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

   if -0.00949999999999999976 < F < -6.99999999999999947e-110

   1. Initial program 99.1%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in F around 0 98.5%

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

   6. Taylor expanded in B around 0 52.9%

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

   7. Taylor expanded in x around 0 42.9%

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

      1. associate-/l* 42.7%

         \[\leadsto \color{blue}{\frac{F}{\frac{B}{\sqrt{0.5}}}} \]

   9. Simplified 42.7%

      \[\leadsto \color{blue}{\frac{F}{\frac{B}{\sqrt{0.5}}}} \]
   10. Step-by-step derivation

       1. clear-num 42.8%

          \[\leadsto \color{blue}{\frac{1}{\frac{\frac{B}{\sqrt{0.5}}}{F}}} \]

       2. associate-/r/ 42.8%

          \[\leadsto \color{blue}{\frac{1}{\frac{B}{\sqrt{0.5}}} \cdot F} \]

       3. clear-num 42.7%

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

   11. Applied egg-rr 42.7%

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

   if -6.99999999999999947e-110 < F < 1.38e-33

   1. Initial program 99.6%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in F around inf 29.0%

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

      1. associate-/r* 29.0%

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

   7. Simplified 29.0%

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

   8. Taylor expanded in B around 0 16.7%

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

   9. Taylor expanded in x around inf 39.4%

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

       1. neg-mul-1 39.4%

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

       2. distribute-neg-frac 39.4%

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

   11. Simplified 39.4%

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

   if 1.38e-33 < F < 3.1e81

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in F around inf 73.9%

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

      1. associate-/r* 74.0%

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

   7. Simplified 74.0%

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

   8. Taylor expanded in F around 0 74.2%

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

   9. Taylor expanded in x around 0 49.2%

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

   if 3.1e81 < F

   1. Initial program 57.9%

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

   3. Simplified 73.9%

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

   5. Taylor expanded in F around inf 99.7%

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

      1. associate-/r* 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in B around 0 63.5%

      \[\leadsto \color{blue}{\left(B \cdot \left(0.16666666666666666 + 0.3333333333333333 \cdot x\right) + \frac{1}{B}\right) - \frac{x}{B}} \]
   9. Step-by-step derivation

      1. associate--l+ 63.5%

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

      2. +-commutative 63.5%

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

      3. *-commutative 63.5%

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

      4. div-sub 63.5%

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

   10. Simplified 63.5%

       \[\leadsto \color{blue}{B \cdot \left(x \cdot 0.3333333333333333 + 0.16666666666666666\right) + \frac{1 - x}{B}} \]
3. Recombined 7 regimes into one program.

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.68 \cdot 10^{+174}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq -1.42 \cdot 10^{+63}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -0.0095:\\ \;\;\;\;\frac{-1 - x}{B} + B \cdot \left(x \cdot 0.3333333333333333 - 0.16666666666666666\right)\\ \mathbf{elif}\;F \leq -7 \cdot 10^{-110}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B}\\ \mathbf{elif}\;F \leq 1.38 \cdot 10^{-33}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 3.1 \cdot 10^{+81}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333 + 0.16666666666666666\right) + \frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 43.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1 - x}{B}\\ \mathbf{if}\;F \leq -1.15 \cdot 10^{+175}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -4.6 \cdot 10^{+63}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -7.8 \cdot 10^{-5}:\\ \;\;\;\;t_0 + B \cdot \left(x \cdot 0.3333333333333333 - 0.16666666666666666\right)\\ \mathbf{elif}\;F \leq -1.55 \cdot 10^{-114}:\\ \;\;\;\;\frac{F}{\frac{B}{\sqrt{0.5}}}\\ \mathbf{elif}\;F \leq 3.05 \cdot 10^{-33}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 1.2 \cdot 10^{+80}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333 + 0.16666666666666666\right) + \frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- -1.0 x) B)))
   (if (<= F -1.15e+175)
     t_0
     (if (<= F -4.6e+63)
       (/ -1.0 (sin B))
       (if (<= F -7.8e-5)
         (+ t_0 (* B (- (* x 0.3333333333333333) 0.16666666666666666)))
         (if (<= F -1.55e-114)
           (/ F (/ B (sqrt 0.5)))
           (if (<= F 3.05e-33)
             (/ (- x) B)
             (if (<= F 1.2e+80)
               (/ 1.0 (sin B))
               (+
                (* B (+ (* x 0.3333333333333333) 0.16666666666666666))
                (/ (- 1.0 x) B))))))))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 - x) / B;
	double tmp;
	if (F <= -1.15e+175) {
		tmp = t_0;
	} else if (F <= -4.6e+63) {
		tmp = -1.0 / sin(B);
	} else if (F <= -7.8e-5) {
		tmp = t_0 + (B * ((x * 0.3333333333333333) - 0.16666666666666666));
	} else if (F <= -1.55e-114) {
		tmp = F / (B / sqrt(0.5));
	} else if (F <= 3.05e-33) {
		tmp = -x / B;
	} else if (F <= 1.2e+80) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = (B * ((x * 0.3333333333333333) + 0.16666666666666666)) + ((1.0 - x) / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-1.0d0) - x) / b
    if (f <= (-1.15d+175)) then
        tmp = t_0
    else if (f <= (-4.6d+63)) then
        tmp = (-1.0d0) / sin(b)
    else if (f <= (-7.8d-5)) then
        tmp = t_0 + (b * ((x * 0.3333333333333333d0) - 0.16666666666666666d0))
    else if (f <= (-1.55d-114)) then
        tmp = f / (b / sqrt(0.5d0))
    else if (f <= 3.05d-33) then
        tmp = -x / b
    else if (f <= 1.2d+80) then
        tmp = 1.0d0 / sin(b)
    else
        tmp = (b * ((x * 0.3333333333333333d0) + 0.16666666666666666d0)) + ((1.0d0 - x) / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (-1.0 - x) / B;
	double tmp;
	if (F <= -1.15e+175) {
		tmp = t_0;
	} else if (F <= -4.6e+63) {
		tmp = -1.0 / Math.sin(B);
	} else if (F <= -7.8e-5) {
		tmp = t_0 + (B * ((x * 0.3333333333333333) - 0.16666666666666666));
	} else if (F <= -1.55e-114) {
		tmp = F / (B / Math.sqrt(0.5));
	} else if (F <= 3.05e-33) {
		tmp = -x / B;
	} else if (F <= 1.2e+80) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = (B * ((x * 0.3333333333333333) + 0.16666666666666666)) + ((1.0 - x) / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (-1.0 - x) / B
	tmp = 0
	if F <= -1.15e+175:
		tmp = t_0
	elif F <= -4.6e+63:
		tmp = -1.0 / math.sin(B)
	elif F <= -7.8e-5:
		tmp = t_0 + (B * ((x * 0.3333333333333333) - 0.16666666666666666))
	elif F <= -1.55e-114:
		tmp = F / (B / math.sqrt(0.5))
	elif F <= 3.05e-33:
		tmp = -x / B
	elif F <= 1.2e+80:
		tmp = 1.0 / math.sin(B)
	else:
		tmp = (B * ((x * 0.3333333333333333) + 0.16666666666666666)) + ((1.0 - x) / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 - x) / B)
	tmp = 0.0
	if (F <= -1.15e+175)
		tmp = t_0;
	elseif (F <= -4.6e+63)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= -7.8e-5)
		tmp = Float64(t_0 + Float64(B * Float64(Float64(x * 0.3333333333333333) - 0.16666666666666666)));
	elseif (F <= -1.55e-114)
		tmp = Float64(F / Float64(B / sqrt(0.5)));
	elseif (F <= 3.05e-33)
		tmp = Float64(Float64(-x) / B);
	elseif (F <= 1.2e+80)
		tmp = Float64(1.0 / sin(B));
	else
		tmp = Float64(Float64(B * Float64(Float64(x * 0.3333333333333333) + 0.16666666666666666)) + Float64(Float64(1.0 - x) / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (-1.0 - x) / B;
	tmp = 0.0;
	if (F <= -1.15e+175)
		tmp = t_0;
	elseif (F <= -4.6e+63)
		tmp = -1.0 / sin(B);
	elseif (F <= -7.8e-5)
		tmp = t_0 + (B * ((x * 0.3333333333333333) - 0.16666666666666666));
	elseif (F <= -1.55e-114)
		tmp = F / (B / sqrt(0.5));
	elseif (F <= 3.05e-33)
		tmp = -x / B;
	elseif (F <= 1.2e+80)
		tmp = 1.0 / sin(B);
	else
		tmp = (B * ((x * 0.3333333333333333) + 0.16666666666666666)) + ((1.0 - x) / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision]}, If[LessEqual[F, -1.15e+175], t$95$0, If[LessEqual[F, -4.6e+63], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -7.8e-5], N[(t$95$0 + N[(B * N[(N[(x * 0.3333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.55e-114], N[(F / N[(B / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.05e-33], N[((-x) / B), $MachinePrecision], If[LessEqual[F, 1.2e+80], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(B * N[(N[(x * 0.3333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if F < -1.15e175

   1. Initial program 26.6%

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

   3. Taylor expanded in F around -inf 99.8%

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

   4. Taylor expanded in B around 0 65.0%

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

      1. associate-*r/ 65.0%

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

      2. distribute-lft-in 65.0%

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

      3. metadata-eval 65.0%

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

      4. neg-mul-1 65.0%

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

   6. Simplified 65.0%

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

   if -1.15e175 < F < -4.59999999999999986e63

   1. Initial program 78.3%

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

   3. Taylor expanded in F around -inf 99.7%

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

   4. Taylor expanded in x around 0 57.2%

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

   if -4.59999999999999986e63 < F < -7.7999999999999999e-5

   1. Initial program 94.0%

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

   3. Taylor expanded in F around -inf 97.6%

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

   4. Taylor expanded in B around 0 60.9%

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

   if -7.7999999999999999e-5 < F < -1.55e-114

   1. Initial program 99.1%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in F around 0 98.5%

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

   6. Taylor expanded in B around 0 52.9%

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

   7. Taylor expanded in x around 0 42.9%

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

      1. associate-/l* 42.7%

         \[\leadsto \color{blue}{\frac{F}{\frac{B}{\sqrt{0.5}}}} \]

   9. Simplified 42.7%

      \[\leadsto \color{blue}{\frac{F}{\frac{B}{\sqrt{0.5}}}} \]

   if -1.55e-114 < F < 3.0500000000000001e-33

   1. Initial program 99.6%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in F around inf 29.0%

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

      1. associate-/r* 29.0%

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

   7. Simplified 29.0%

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

   8. Taylor expanded in B around 0 16.7%

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

   9. Taylor expanded in x around inf 39.4%

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

       1. neg-mul-1 39.4%

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

       2. distribute-neg-frac 39.4%

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

   11. Simplified 39.4%

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

   if 3.0500000000000001e-33 < F < 1.1999999999999999e80

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in F around inf 73.9%

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

      1. associate-/r* 74.0%

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

   7. Simplified 74.0%

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

   8. Taylor expanded in F around 0 74.2%

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

   9. Taylor expanded in x around 0 49.2%

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

   if 1.1999999999999999e80 < F

   1. Initial program 57.9%

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

   3. Simplified 73.9%

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

   5. Taylor expanded in F around inf 99.7%

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

      1. associate-/r* 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in B around 0 63.5%

      \[\leadsto \color{blue}{\left(B \cdot \left(0.16666666666666666 + 0.3333333333333333 \cdot x\right) + \frac{1}{B}\right) - \frac{x}{B}} \]
   9. Step-by-step derivation

      1. associate--l+ 63.5%

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

      2. +-commutative 63.5%

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

      3. *-commutative 63.5%

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

      4. div-sub 63.5%

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

   10. Simplified 63.5%

       \[\leadsto \color{blue}{B \cdot \left(x \cdot 0.3333333333333333 + 0.16666666666666666\right) + \frac{1 - x}{B}} \]
3. Recombined 7 regimes into one program.

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.15 \cdot 10^{+175}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq -4.6 \cdot 10^{+63}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -7.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1 - x}{B} + B \cdot \left(x \cdot 0.3333333333333333 - 0.16666666666666666\right)\\ \mathbf{elif}\;F \leq -1.55 \cdot 10^{-114}:\\ \;\;\;\;\frac{F}{\frac{B}{\sqrt{0.5}}}\\ \mathbf{elif}\;F \leq 3.05 \cdot 10^{-33}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 1.2 \cdot 10^{+80}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333 + 0.16666666666666666\right) + \frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 43.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1 - x}{B}\\ \mathbf{if}\;F \leq -1.1 \cdot 10^{+174}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -2.9 \cdot 10^{+62}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -0.0051:\\ \;\;\;\;t_0 + B \cdot \left(x \cdot 0.3333333333333333 - 0.16666666666666666\right)\\ \mathbf{elif}\;F \leq -8 \cdot 10^{-107}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{B}\\ \mathbf{elif}\;F \leq 3.05 \cdot 10^{-33}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 7.6 \cdot 10^{+79}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333 + 0.16666666666666666\right) + \frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- -1.0 x) B)))
   (if (<= F -1.1e+174)
     t_0
     (if (<= F -2.9e+62)
       (/ -1.0 (sin B))
       (if (<= F -0.0051)
         (+ t_0 (* B (- (* x 0.3333333333333333) 0.16666666666666666)))
         (if (<= F -8e-107)
           (/ (* F (sqrt 0.5)) B)
           (if (<= F 3.05e-33)
             (/ (- x) B)
             (if (<= F 7.6e+79)
               (/ 1.0 (sin B))
               (+
                (* B (+ (* x 0.3333333333333333) 0.16666666666666666))
                (/ (- 1.0 x) B))))))))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 - x) / B;
	double tmp;
	if (F <= -1.1e+174) {
		tmp = t_0;
	} else if (F <= -2.9e+62) {
		tmp = -1.0 / sin(B);
	} else if (F <= -0.0051) {
		tmp = t_0 + (B * ((x * 0.3333333333333333) - 0.16666666666666666));
	} else if (F <= -8e-107) {
		tmp = (F * sqrt(0.5)) / B;
	} else if (F <= 3.05e-33) {
		tmp = -x / B;
	} else if (F <= 7.6e+79) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = (B * ((x * 0.3333333333333333) + 0.16666666666666666)) + ((1.0 - x) / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-1.0d0) - x) / b
    if (f <= (-1.1d+174)) then
        tmp = t_0
    else if (f <= (-2.9d+62)) then
        tmp = (-1.0d0) / sin(b)
    else if (f <= (-0.0051d0)) then
        tmp = t_0 + (b * ((x * 0.3333333333333333d0) - 0.16666666666666666d0))
    else if (f <= (-8d-107)) then
        tmp = (f * sqrt(0.5d0)) / b
    else if (f <= 3.05d-33) then
        tmp = -x / b
    else if (f <= 7.6d+79) then
        tmp = 1.0d0 / sin(b)
    else
        tmp = (b * ((x * 0.3333333333333333d0) + 0.16666666666666666d0)) + ((1.0d0 - x) / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (-1.0 - x) / B;
	double tmp;
	if (F <= -1.1e+174) {
		tmp = t_0;
	} else if (F <= -2.9e+62) {
		tmp = -1.0 / Math.sin(B);
	} else if (F <= -0.0051) {
		tmp = t_0 + (B * ((x * 0.3333333333333333) - 0.16666666666666666));
	} else if (F <= -8e-107) {
		tmp = (F * Math.sqrt(0.5)) / B;
	} else if (F <= 3.05e-33) {
		tmp = -x / B;
	} else if (F <= 7.6e+79) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = (B * ((x * 0.3333333333333333) + 0.16666666666666666)) + ((1.0 - x) / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (-1.0 - x) / B
	tmp = 0
	if F <= -1.1e+174:
		tmp = t_0
	elif F <= -2.9e+62:
		tmp = -1.0 / math.sin(B)
	elif F <= -0.0051:
		tmp = t_0 + (B * ((x * 0.3333333333333333) - 0.16666666666666666))
	elif F <= -8e-107:
		tmp = (F * math.sqrt(0.5)) / B
	elif F <= 3.05e-33:
		tmp = -x / B
	elif F <= 7.6e+79:
		tmp = 1.0 / math.sin(B)
	else:
		tmp = (B * ((x * 0.3333333333333333) + 0.16666666666666666)) + ((1.0 - x) / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 - x) / B)
	tmp = 0.0
	if (F <= -1.1e+174)
		tmp = t_0;
	elseif (F <= -2.9e+62)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= -0.0051)
		tmp = Float64(t_0 + Float64(B * Float64(Float64(x * 0.3333333333333333) - 0.16666666666666666)));
	elseif (F <= -8e-107)
		tmp = Float64(Float64(F * sqrt(0.5)) / B);
	elseif (F <= 3.05e-33)
		tmp = Float64(Float64(-x) / B);
	elseif (F <= 7.6e+79)
		tmp = Float64(1.0 / sin(B));
	else
		tmp = Float64(Float64(B * Float64(Float64(x * 0.3333333333333333) + 0.16666666666666666)) + Float64(Float64(1.0 - x) / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (-1.0 - x) / B;
	tmp = 0.0;
	if (F <= -1.1e+174)
		tmp = t_0;
	elseif (F <= -2.9e+62)
		tmp = -1.0 / sin(B);
	elseif (F <= -0.0051)
		tmp = t_0 + (B * ((x * 0.3333333333333333) - 0.16666666666666666));
	elseif (F <= -8e-107)
		tmp = (F * sqrt(0.5)) / B;
	elseif (F <= 3.05e-33)
		tmp = -x / B;
	elseif (F <= 7.6e+79)
		tmp = 1.0 / sin(B);
	else
		tmp = (B * ((x * 0.3333333333333333) + 0.16666666666666666)) + ((1.0 - x) / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision]}, If[LessEqual[F, -1.1e+174], t$95$0, If[LessEqual[F, -2.9e+62], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -0.0051], N[(t$95$0 + N[(B * N[(N[(x * 0.3333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -8e-107], N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 3.05e-33], N[((-x) / B), $MachinePrecision], If[LessEqual[F, 7.6e+79], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(B * N[(N[(x * 0.3333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if F < -1.1000000000000001e174

   1. Initial program 26.6%

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

   3. Taylor expanded in F around -inf 99.8%

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

   4. Taylor expanded in B around 0 65.0%

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

      1. associate-*r/ 65.0%

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

      2. distribute-lft-in 65.0%

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

      3. metadata-eval 65.0%

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

      4. neg-mul-1 65.0%

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

   6. Simplified 65.0%

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

   if -1.1000000000000001e174 < F < -2.89999999999999984e62

   1. Initial program 78.3%

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

   3. Taylor expanded in F around -inf 99.7%

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

   4. Taylor expanded in x around 0 57.2%

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

   if -2.89999999999999984e62 < F < -0.0051000000000000004

   1. Initial program 94.0%

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

   3. Taylor expanded in F around -inf 97.6%

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

   4. Taylor expanded in B around 0 60.9%

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

   if -0.0051000000000000004 < F < -8e-107

   1. Initial program 99.1%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in F around 0 98.5%

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

   6. Taylor expanded in B around 0 52.9%

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

   7. Taylor expanded in x around 0 42.9%

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

   if -8e-107 < F < 3.0500000000000001e-33

   1. Initial program 99.6%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in F around inf 29.0%

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

      1. associate-/r* 29.0%

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

   7. Simplified 29.0%

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

   8. Taylor expanded in B around 0 16.7%

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

   9. Taylor expanded in x around inf 39.4%

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

       1. neg-mul-1 39.4%

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

       2. distribute-neg-frac 39.4%

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

   11. Simplified 39.4%

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

   if 3.0500000000000001e-33 < F < 7.6000000000000005e79

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in F around inf 73.9%

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

      1. associate-/r* 74.0%

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

   7. Simplified 74.0%

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

   8. Taylor expanded in F around 0 74.2%

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

   9. Taylor expanded in x around 0 49.2%

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

   if 7.6000000000000005e79 < F

   1. Initial program 57.9%

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

   3. Simplified 73.9%

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

   5. Taylor expanded in F around inf 99.7%

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

      1. associate-/r* 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in B around 0 63.5%

      \[\leadsto \color{blue}{\left(B \cdot \left(0.16666666666666666 + 0.3333333333333333 \cdot x\right) + \frac{1}{B}\right) - \frac{x}{B}} \]
   9. Step-by-step derivation

      1. associate--l+ 63.5%

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

      2. +-commutative 63.5%

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

      3. *-commutative 63.5%

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

      4. div-sub 63.5%

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

   10. Simplified 63.5%

       \[\leadsto \color{blue}{B \cdot \left(x \cdot 0.3333333333333333 + 0.16666666666666666\right) + \frac{1 - x}{B}} \]
3. Recombined 7 regimes into one program.

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.1 \cdot 10^{+174}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq -2.9 \cdot 10^{+62}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -0.0051:\\ \;\;\;\;\frac{-1 - x}{B} + B \cdot \left(x \cdot 0.3333333333333333 - 0.16666666666666666\right)\\ \mathbf{elif}\;F \leq -8 \cdot 10^{-107}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{B}\\ \mathbf{elif}\;F \leq 3.05 \cdot 10^{-33}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 7.6 \cdot 10^{+79}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333 + 0.16666666666666666\right) + \frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 56.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F \cdot \sqrt{0.5} - x}{B}\\ t_1 := \frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;x \leq -1.4 \cdot 10^{-51}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2.5 \cdot 10^{-111}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{-148}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-186}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-86}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- (* F (sqrt 0.5)) x) B)) (t_1 (- (/ 1.0 B) (/ x (tan B)))))
   (if (<= x -1.4e-51)
     t_1
     (if (<= x -2.5e-111)
       t_0
       (if (<= x -8.5e-148)
         t_1
         (if (<= x 4.8e-186) (/ -1.0 (sin B)) (if (<= x 2.8e-86) t_0 t_1)))))))

C

double code(double F, double B, double x) {
	double t_0 = ((F * sqrt(0.5)) - x) / B;
	double t_1 = (1.0 / B) - (x / tan(B));
	double tmp;
	if (x <= -1.4e-51) {
		tmp = t_1;
	} else if (x <= -2.5e-111) {
		tmp = t_0;
	} else if (x <= -8.5e-148) {
		tmp = t_1;
	} else if (x <= 4.8e-186) {
		tmp = -1.0 / sin(B);
	} else if (x <= 2.8e-86) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((f * sqrt(0.5d0)) - x) / b
    t_1 = (1.0d0 / b) - (x / tan(b))
    if (x <= (-1.4d-51)) then
        tmp = t_1
    else if (x <= (-2.5d-111)) then
        tmp = t_0
    else if (x <= (-8.5d-148)) then
        tmp = t_1
    else if (x <= 4.8d-186) then
        tmp = (-1.0d0) / sin(b)
    else if (x <= 2.8d-86) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = ((F * Math.sqrt(0.5)) - x) / B;
	double t_1 = (1.0 / B) - (x / Math.tan(B));
	double tmp;
	if (x <= -1.4e-51) {
		tmp = t_1;
	} else if (x <= -2.5e-111) {
		tmp = t_0;
	} else if (x <= -8.5e-148) {
		tmp = t_1;
	} else if (x <= 4.8e-186) {
		tmp = -1.0 / Math.sin(B);
	} else if (x <= 2.8e-86) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = ((F * math.sqrt(0.5)) - x) / B
	t_1 = (1.0 / B) - (x / math.tan(B))
	tmp = 0
	if x <= -1.4e-51:
		tmp = t_1
	elif x <= -2.5e-111:
		tmp = t_0
	elif x <= -8.5e-148:
		tmp = t_1
	elif x <= 4.8e-186:
		tmp = -1.0 / math.sin(B)
	elif x <= 2.8e-86:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(Float64(F * sqrt(0.5)) - x) / B)
	t_1 = Float64(Float64(1.0 / B) - Float64(x / tan(B)))
	tmp = 0.0
	if (x <= -1.4e-51)
		tmp = t_1;
	elseif (x <= -2.5e-111)
		tmp = t_0;
	elseif (x <= -8.5e-148)
		tmp = t_1;
	elseif (x <= 4.8e-186)
		tmp = Float64(-1.0 / sin(B));
	elseif (x <= 2.8e-86)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = ((F * sqrt(0.5)) - x) / B;
	t_1 = (1.0 / B) - (x / tan(B));
	tmp = 0.0;
	if (x <= -1.4e-51)
		tmp = t_1;
	elseif (x <= -2.5e-111)
		tmp = t_0;
	elseif (x <= -8.5e-148)
		tmp = t_1;
	elseif (x <= 4.8e-186)
		tmp = -1.0 / sin(B);
	elseif (x <= 2.8e-86)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.4e-51], t$95$1, If[LessEqual[x, -2.5e-111], t$95$0, If[LessEqual[x, -8.5e-148], t$95$1, If[LessEqual[x, 4.8e-186], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.8e-86], t$95$0, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.4e-51 or -2.5000000000000001e-111 < x < -8.49999999999999989e-148 or 2.80000000000000009e-86 < x

   1. Initial program 82.5%

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

   3. Simplified 94.6%

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

   5. Taylor expanded in F around inf 71.1%

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

      1. associate-/r* 71.1%

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

   7. Simplified 71.1%

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

   8. Taylor expanded in B around 0 78.7%

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

   if -1.4e-51 < x < -2.5000000000000001e-111 or 4.80000000000000006e-186 < x < 2.80000000000000009e-86

   1. Initial program 81.5%

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

   3. Simplified 84.2%

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

   5. Taylor expanded in F around 0 64.7%

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

   6. Taylor expanded in B around 0 53.6%

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

   7. Taylor expanded in x around 0 53.6%

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

   if -8.49999999999999989e-148 < x < 4.80000000000000006e-186

   1. Initial program 74.1%

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

   3. Taylor expanded in F around -inf 35.3%

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

   4. Taylor expanded in x around 0 35.3%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-51}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;x \leq -2.5 \cdot 10^{-111}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{-148}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-186}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-86}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 43.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1 - x}{B}\\ \mathbf{if}\;F \leq -5.1 \cdot 10^{+175}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -3.05 \cdot 10^{+62}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -4 \cdot 10^{-60}:\\ \;\;\;\;t_0 + B \cdot \left(x \cdot 0.3333333333333333 - 0.16666666666666666\right)\\ \mathbf{elif}\;F \leq 1.2 \cdot 10^{-34}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 4.1 \cdot 10^{+79}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333 + 0.16666666666666666\right) + \frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- -1.0 x) B)))
   (if (<= F -5.1e+175)
     t_0
     (if (<= F -3.05e+62)
       (/ -1.0 (sin B))
       (if (<= F -4e-60)
         (+ t_0 (* B (- (* x 0.3333333333333333) 0.16666666666666666)))
         (if (<= F 1.2e-34)
           (/ (- x) B)
           (if (<= F 4.1e+79)
             (/ 1.0 (sin B))
             (+
              (* B (+ (* x 0.3333333333333333) 0.16666666666666666))
              (/ (- 1.0 x) B)))))))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 - x) / B;
	double tmp;
	if (F <= -5.1e+175) {
		tmp = t_0;
	} else if (F <= -3.05e+62) {
		tmp = -1.0 / sin(B);
	} else if (F <= -4e-60) {
		tmp = t_0 + (B * ((x * 0.3333333333333333) - 0.16666666666666666));
	} else if (F <= 1.2e-34) {
		tmp = -x / B;
	} else if (F <= 4.1e+79) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = (B * ((x * 0.3333333333333333) + 0.16666666666666666)) + ((1.0 - x) / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-1.0d0) - x) / b
    if (f <= (-5.1d+175)) then
        tmp = t_0
    else if (f <= (-3.05d+62)) then
        tmp = (-1.0d0) / sin(b)
    else if (f <= (-4d-60)) then
        tmp = t_0 + (b * ((x * 0.3333333333333333d0) - 0.16666666666666666d0))
    else if (f <= 1.2d-34) then
        tmp = -x / b
    else if (f <= 4.1d+79) then
        tmp = 1.0d0 / sin(b)
    else
        tmp = (b * ((x * 0.3333333333333333d0) + 0.16666666666666666d0)) + ((1.0d0 - x) / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (-1.0 - x) / B;
	double tmp;
	if (F <= -5.1e+175) {
		tmp = t_0;
	} else if (F <= -3.05e+62) {
		tmp = -1.0 / Math.sin(B);
	} else if (F <= -4e-60) {
		tmp = t_0 + (B * ((x * 0.3333333333333333) - 0.16666666666666666));
	} else if (F <= 1.2e-34) {
		tmp = -x / B;
	} else if (F <= 4.1e+79) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = (B * ((x * 0.3333333333333333) + 0.16666666666666666)) + ((1.0 - x) / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (-1.0 - x) / B
	tmp = 0
	if F <= -5.1e+175:
		tmp = t_0
	elif F <= -3.05e+62:
		tmp = -1.0 / math.sin(B)
	elif F <= -4e-60:
		tmp = t_0 + (B * ((x * 0.3333333333333333) - 0.16666666666666666))
	elif F <= 1.2e-34:
		tmp = -x / B
	elif F <= 4.1e+79:
		tmp = 1.0 / math.sin(B)
	else:
		tmp = (B * ((x * 0.3333333333333333) + 0.16666666666666666)) + ((1.0 - x) / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 - x) / B)
	tmp = 0.0
	if (F <= -5.1e+175)
		tmp = t_0;
	elseif (F <= -3.05e+62)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= -4e-60)
		tmp = Float64(t_0 + Float64(B * Float64(Float64(x * 0.3333333333333333) - 0.16666666666666666)));
	elseif (F <= 1.2e-34)
		tmp = Float64(Float64(-x) / B);
	elseif (F <= 4.1e+79)
		tmp = Float64(1.0 / sin(B));
	else
		tmp = Float64(Float64(B * Float64(Float64(x * 0.3333333333333333) + 0.16666666666666666)) + Float64(Float64(1.0 - x) / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (-1.0 - x) / B;
	tmp = 0.0;
	if (F <= -5.1e+175)
		tmp = t_0;
	elseif (F <= -3.05e+62)
		tmp = -1.0 / sin(B);
	elseif (F <= -4e-60)
		tmp = t_0 + (B * ((x * 0.3333333333333333) - 0.16666666666666666));
	elseif (F <= 1.2e-34)
		tmp = -x / B;
	elseif (F <= 4.1e+79)
		tmp = 1.0 / sin(B);
	else
		tmp = (B * ((x * 0.3333333333333333) + 0.16666666666666666)) + ((1.0 - x) / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision]}, If[LessEqual[F, -5.1e+175], t$95$0, If[LessEqual[F, -3.05e+62], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -4e-60], N[(t$95$0 + N[(B * N[(N[(x * 0.3333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.2e-34], N[((-x) / B), $MachinePrecision], If[LessEqual[F, 4.1e+79], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(B * N[(N[(x * 0.3333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if F < -5.10000000000000007e175

   1. Initial program 26.6%

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

   3. Taylor expanded in F around -inf 99.8%

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

   4. Taylor expanded in B around 0 65.0%

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

      1. associate-*r/ 65.0%

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

      2. distribute-lft-in 65.0%

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

      3. metadata-eval 65.0%

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

      4. neg-mul-1 65.0%

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

   6. Simplified 65.0%

      \[\leadsto \color{blue}{\frac{-1 + \left(-x\right)}{B}} \]

   if -5.10000000000000007e175 < F < -3.0499999999999998e62

   1. Initial program 78.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf 99.7%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   4. Taylor expanded in x around 0 57.2%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   if -3.0499999999999998e62 < F < -3.9999999999999999e-60

   1. Initial program 96.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf 58.7%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   4. Taylor expanded in B around 0 36.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B} + B \cdot \left(0.3333333333333333 \cdot x - 0.16666666666666666\right)} \]

   if -3.9999999999999999e-60 < F < 1.19999999999999996e-34

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in F around inf 27.8%

      \[\leadsto F \cdot \color{blue}{\frac{1}{F \cdot \sin B}} - \frac{x}{\tan B} \]
   6. Step-by-step derivation

      1. associate-/r* 27.8%

         \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   7. Simplified 27.8%

      \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   8. Taylor expanded in B around 0 16.6%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

   9. Taylor expanded in x around inf 38.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   10. Step-by-step derivation

       1. neg-mul-1 38.0%

          \[\leadsto \color{blue}{-\frac{x}{B}} \]

       2. distribute-neg-frac 38.0%

          \[\leadsto \color{blue}{\frac{-x}{B}} \]

   11. Simplified 38.0%

       \[\leadsto \color{blue}{\frac{-x}{B}} \]

   if 1.19999999999999996e-34 < F < 4.1e79

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in F around inf 73.9%

      \[\leadsto F \cdot \color{blue}{\frac{1}{F \cdot \sin B}} - \frac{x}{\tan B} \]
   6. Step-by-step derivation

      1. associate-/r* 74.0%

         \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   7. Simplified 74.0%

      \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   8. Taylor expanded in F around 0 74.2%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} - \frac{x}{\tan B} \]

   9. Taylor expanded in x around 0 49.2%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]

   if 4.1e79 < F

   1. Initial program 57.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Simplified 73.9%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in F around inf 99.7%

      \[\leadsto F \cdot \color{blue}{\frac{1}{F \cdot \sin B}} - \frac{x}{\tan B} \]
   6. Step-by-step derivation

      1. associate-/r* 99.7%

         \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   7. Simplified 99.7%

      \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   8. Taylor expanded in B around 0 63.5%

      \[\leadsto \color{blue}{\left(B \cdot \left(0.16666666666666666 + 0.3333333333333333 \cdot x\right) + \frac{1}{B}\right) - \frac{x}{B}} \]
   9. Step-by-step derivation

      1. associate--l+ 63.5%

         \[\leadsto \color{blue}{B \cdot \left(0.16666666666666666 + 0.3333333333333333 \cdot x\right) + \left(\frac{1}{B} - \frac{x}{B}\right)} \]

      2. +-commutative 63.5%

         \[\leadsto B \cdot \color{blue}{\left(0.3333333333333333 \cdot x + 0.16666666666666666\right)} + \left(\frac{1}{B} - \frac{x}{B}\right) \]

      3. *-commutative 63.5%

         \[\leadsto B \cdot \left(\color{blue}{x \cdot 0.3333333333333333} + 0.16666666666666666\right) + \left(\frac{1}{B} - \frac{x}{B}\right) \]

      4. div-sub 63.5%

         \[\leadsto B \cdot \left(x \cdot 0.3333333333333333 + 0.16666666666666666\right) + \color{blue}{\frac{1 - x}{B}} \]

   10. Simplified 63.5%

       \[\leadsto \color{blue}{B \cdot \left(x \cdot 0.3333333333333333 + 0.16666666666666666\right) + \frac{1 - x}{B}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5.1 \cdot 10^{+175}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq -3.05 \cdot 10^{+62}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -4 \cdot 10^{-60}:\\ \;\;\;\;\frac{-1 - x}{B} + B \cdot \left(x \cdot 0.3333333333333333 - 0.16666666666666666\right)\\ \mathbf{elif}\;F \leq 1.2 \cdot 10^{-34}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 4.1 \cdot 10^{+79}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333 + 0.16666666666666666\right) + \frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 62.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -6.4 \cdot 10^{-16}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq -4.5 \cdot 10^{-135} \lor \neg \left(F \leq -2.1 \cdot 10^{-223}\right) \land F \leq 3.5 \cdot 10^{-29}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -6.4e-16)
   (+ (* x (/ -1.0 (tan B))) (/ -1.0 B))
   (if (or (<= F -4.5e-135) (and (not (<= F -2.1e-223)) (<= F 3.5e-29)))
     (/ (- (* F (sqrt 0.5)) x) B)
     (- (/ 1.0 B) (/ x (tan B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -6.4e-16) {
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / B);
	} else if ((F <= -4.5e-135) || (!(F <= -2.1e-223) && (F <= 3.5e-29))) {
		tmp = ((F * sqrt(0.5)) - x) / B;
	} else {
		tmp = (1.0 / B) - (x / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-6.4d-16)) then
        tmp = (x * ((-1.0d0) / tan(b))) + ((-1.0d0) / b)
    else if ((f <= (-4.5d-135)) .or. (.not. (f <= (-2.1d-223))) .and. (f <= 3.5d-29)) then
        tmp = ((f * sqrt(0.5d0)) - x) / b
    else
        tmp = (1.0d0 / b) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -6.4e-16) {
		tmp = (x * (-1.0 / Math.tan(B))) + (-1.0 / B);
	} else if ((F <= -4.5e-135) || (!(F <= -2.1e-223) && (F <= 3.5e-29))) {
		tmp = ((F * Math.sqrt(0.5)) - x) / B;
	} else {
		tmp = (1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -6.4e-16:
		tmp = (x * (-1.0 / math.tan(B))) + (-1.0 / B)
	elif (F <= -4.5e-135) or (not (F <= -2.1e-223) and (F <= 3.5e-29)):
		tmp = ((F * math.sqrt(0.5)) - x) / B
	else:
		tmp = (1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -6.4e-16)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(-1.0 / B));
	elseif ((F <= -4.5e-135) || (!(F <= -2.1e-223) && (F <= 3.5e-29)))
		tmp = Float64(Float64(Float64(F * sqrt(0.5)) - x) / B);
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -6.4e-16)
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / B);
	elseif ((F <= -4.5e-135) || (~((F <= -2.1e-223)) && (F <= 3.5e-29)))
		tmp = ((F * sqrt(0.5)) - x) / B;
	else
		tmp = (1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -6.4e-16], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[F, -4.5e-135], And[N[Not[LessEqual[F, -2.1e-223]], $MachinePrecision], LessEqual[F, 3.5e-29]]], N[(N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -6.40000000000000046e-16

   1. Initial program 61.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf 95.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   4. Taylor expanded in B around 0 79.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{B}} \]

   if -6.40000000000000046e-16 < F < -4.49999999999999987e-135 or -2.09999999999999982e-223 < F < 3.4999999999999997e-29

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in F around 0 99.6%

      \[\leadsto F \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 + 2 \cdot x}}\right)} - \frac{x}{\tan B} \]

   6. Taylor expanded in B around 0 57.9%

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + 2 \cdot x}} - x}{B}} \]

   7. Taylor expanded in x around 0 57.9%

      \[\leadsto \frac{\color{blue}{F \cdot \sqrt{0.5}} - x}{B} \]

   if -4.49999999999999987e-135 < F < -2.09999999999999982e-223 or 3.4999999999999997e-29 < F

   1. Initial program 75.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Simplified 85.0%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in F around inf 85.0%

      \[\leadsto F \cdot \color{blue}{\frac{1}{F \cdot \sin B}} - \frac{x}{\tan B} \]
   6. Step-by-step derivation

      1. associate-/r* 85.0%

         \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   7. Simplified 85.0%

      \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   8. Taylor expanded in B around 0 72.5%

      \[\leadsto \color{blue}{\frac{1}{B}} - \frac{x}{\tan B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6.4 \cdot 10^{-16}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq -4.5 \cdot 10^{-135} \lor \neg \left(F \leq -2.1 \cdot 10^{-223}\right) \land F \leq 3.5 \cdot 10^{-29}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 57.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.9 \cdot 10^{-148} \lor \neg \left(x \leq 5.7 \cdot 10^{-151}\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (or (<= x -5.9e-148) (not (<= x 5.7e-151)))
   (- (/ 1.0 B) (/ x (tan B)))
   (/ -1.0 (sin B))))

C

double code(double F, double B, double x) {
	double tmp;
	if ((x <= -5.9e-148) || !(x <= 5.7e-151)) {
		tmp = (1.0 / B) - (x / tan(B));
	} else {
		tmp = -1.0 / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-5.9d-148)) .or. (.not. (x <= 5.7d-151))) then
        tmp = (1.0d0 / b) - (x / tan(b))
    else
        tmp = (-1.0d0) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if ((x <= -5.9e-148) || !(x <= 5.7e-151)) {
		tmp = (1.0 / B) - (x / Math.tan(B));
	} else {
		tmp = -1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if (x <= -5.9e-148) or not (x <= 5.7e-151):
		tmp = (1.0 / B) - (x / math.tan(B))
	else:
		tmp = -1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if ((x <= -5.9e-148) || !(x <= 5.7e-151))
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	else
		tmp = Float64(-1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if ((x <= -5.9e-148) || ~((x <= 5.7e-151)))
		tmp = (1.0 / B) - (x / tan(B));
	else
		tmp = -1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[Or[LessEqual[x, -5.9e-148], N[Not[LessEqual[x, 5.7e-151]], $MachinePrecision]], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.90000000000000016e-148 or 5.69999999999999989e-151 < x

   1. Initial program 82.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Simplified 93.0%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in F around inf 63.9%

      \[\leadsto F \cdot \color{blue}{\frac{1}{F \cdot \sin B}} - \frac{x}{\tan B} \]
   6. Step-by-step derivation

      1. associate-/r* 63.9%

         \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   7. Simplified 63.9%

      \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   8. Taylor expanded in B around 0 70.6%

      \[\leadsto \color{blue}{\frac{1}{B}} - \frac{x}{\tan B} \]

   if -5.90000000000000016e-148 < x < 5.69999999999999989e-151

   1. Initial program 75.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf 33.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   4. Taylor expanded in x around 0 33.1%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.9 \cdot 10^{-148} \lor \neg \left(x \leq 5.7 \cdot 10^{-151}\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 43.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1 - x}{B}\\ \mathbf{if}\;F \leq -1.68 \cdot 10^{+174}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -6 \cdot 10^{+63}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -1.95 \cdot 10^{-60}:\\ \;\;\;\;t_0 + B \cdot \left(x \cdot 0.3333333333333333 - 0.16666666666666666\right)\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{-55}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333 + 0.16666666666666666\right) + \frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- -1.0 x) B)))
   (if (<= F -1.68e+174)
     t_0
     (if (<= F -6e+63)
       (/ -1.0 (sin B))
       (if (<= F -1.95e-60)
         (+ t_0 (* B (- (* x 0.3333333333333333) 0.16666666666666666)))
         (if (<= F 2.3e-55)
           (/ (- x) B)
           (+
            (* B (+ (* x 0.3333333333333333) 0.16666666666666666))
            (/ (- 1.0 x) B))))))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 - x) / B;
	double tmp;
	if (F <= -1.68e+174) {
		tmp = t_0;
	} else if (F <= -6e+63) {
		tmp = -1.0 / sin(B);
	} else if (F <= -1.95e-60) {
		tmp = t_0 + (B * ((x * 0.3333333333333333) - 0.16666666666666666));
	} else if (F <= 2.3e-55) {
		tmp = -x / B;
	} else {
		tmp = (B * ((x * 0.3333333333333333) + 0.16666666666666666)) + ((1.0 - x) / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-1.0d0) - x) / b
    if (f <= (-1.68d+174)) then
        tmp = t_0
    else if (f <= (-6d+63)) then
        tmp = (-1.0d0) / sin(b)
    else if (f <= (-1.95d-60)) then
        tmp = t_0 + (b * ((x * 0.3333333333333333d0) - 0.16666666666666666d0))
    else if (f <= 2.3d-55) then
        tmp = -x / b
    else
        tmp = (b * ((x * 0.3333333333333333d0) + 0.16666666666666666d0)) + ((1.0d0 - x) / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (-1.0 - x) / B;
	double tmp;
	if (F <= -1.68e+174) {
		tmp = t_0;
	} else if (F <= -6e+63) {
		tmp = -1.0 / Math.sin(B);
	} else if (F <= -1.95e-60) {
		tmp = t_0 + (B * ((x * 0.3333333333333333) - 0.16666666666666666));
	} else if (F <= 2.3e-55) {
		tmp = -x / B;
	} else {
		tmp = (B * ((x * 0.3333333333333333) + 0.16666666666666666)) + ((1.0 - x) / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (-1.0 - x) / B
	tmp = 0
	if F <= -1.68e+174:
		tmp = t_0
	elif F <= -6e+63:
		tmp = -1.0 / math.sin(B)
	elif F <= -1.95e-60:
		tmp = t_0 + (B * ((x * 0.3333333333333333) - 0.16666666666666666))
	elif F <= 2.3e-55:
		tmp = -x / B
	else:
		tmp = (B * ((x * 0.3333333333333333) + 0.16666666666666666)) + ((1.0 - x) / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 - x) / B)
	tmp = 0.0
	if (F <= -1.68e+174)
		tmp = t_0;
	elseif (F <= -6e+63)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= -1.95e-60)
		tmp = Float64(t_0 + Float64(B * Float64(Float64(x * 0.3333333333333333) - 0.16666666666666666)));
	elseif (F <= 2.3e-55)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(B * Float64(Float64(x * 0.3333333333333333) + 0.16666666666666666)) + Float64(Float64(1.0 - x) / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (-1.0 - x) / B;
	tmp = 0.0;
	if (F <= -1.68e+174)
		tmp = t_0;
	elseif (F <= -6e+63)
		tmp = -1.0 / sin(B);
	elseif (F <= -1.95e-60)
		tmp = t_0 + (B * ((x * 0.3333333333333333) - 0.16666666666666666));
	elseif (F <= 2.3e-55)
		tmp = -x / B;
	else
		tmp = (B * ((x * 0.3333333333333333) + 0.16666666666666666)) + ((1.0 - x) / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision]}, If[LessEqual[F, -1.68e+174], t$95$0, If[LessEqual[F, -6e+63], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.95e-60], N[(t$95$0 + N[(B * N[(N[(x * 0.3333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.3e-55], N[((-x) / B), $MachinePrecision], N[(N[(B * N[(N[(x * 0.3333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -1.68e174

   1. Initial program 26.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf 99.8%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   4. Taylor expanded in B around 0 65.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   5. Step-by-step derivation

      1. associate-*r/ 65.0%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

      2. distribute-lft-in 65.0%

         \[\leadsto \frac{\color{blue}{-1 \cdot 1 + -1 \cdot x}}{B} \]

      3. metadata-eval 65.0%

         \[\leadsto \frac{\color{blue}{-1} + -1 \cdot x}{B} \]

      4. neg-mul-1 65.0%

         \[\leadsto \frac{-1 + \color{blue}{\left(-x\right)}}{B} \]

   6. Simplified 65.0%

      \[\leadsto \color{blue}{\frac{-1 + \left(-x\right)}{B}} \]

   if -1.68e174 < F < -5.99999999999999998e63

   1. Initial program 78.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf 99.7%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   4. Taylor expanded in x around 0 57.2%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   if -5.99999999999999998e63 < F < -1.9500000000000001e-60

   1. Initial program 96.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf 58.7%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   4. Taylor expanded in B around 0 36.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B} + B \cdot \left(0.3333333333333333 \cdot x - 0.16666666666666666\right)} \]

   if -1.9500000000000001e-60 < F < 2.30000000000000011e-55

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in F around inf 27.4%

      \[\leadsto F \cdot \color{blue}{\frac{1}{F \cdot \sin B}} - \frac{x}{\tan B} \]
   6. Step-by-step derivation

      1. associate-/r* 27.4%

         \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   7. Simplified 27.4%

      \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   8. Taylor expanded in B around 0 15.9%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

   9. Taylor expanded in x around inf 38.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   10. Step-by-step derivation

       1. neg-mul-1 38.1%

          \[\leadsto \color{blue}{-\frac{x}{B}} \]

       2. distribute-neg-frac 38.1%

          \[\leadsto \color{blue}{\frac{-x}{B}} \]

   11. Simplified 38.1%

       \[\leadsto \color{blue}{\frac{-x}{B}} \]

   if 2.30000000000000011e-55 < F

   1. Initial program 73.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Simplified 83.3%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in F around inf 88.8%

      \[\leadsto F \cdot \color{blue}{\frac{1}{F \cdot \sin B}} - \frac{x}{\tan B} \]
   6. Step-by-step derivation

      1. associate-/r* 88.8%

         \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   7. Simplified 88.8%

      \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   8. Taylor expanded in B around 0 52.0%

      \[\leadsto \color{blue}{\left(B \cdot \left(0.16666666666666666 + 0.3333333333333333 \cdot x\right) + \frac{1}{B}\right) - \frac{x}{B}} \]
   9. Step-by-step derivation

      1. associate--l+ 52.0%

         \[\leadsto \color{blue}{B \cdot \left(0.16666666666666666 + 0.3333333333333333 \cdot x\right) + \left(\frac{1}{B} - \frac{x}{B}\right)} \]

      2. +-commutative 52.0%

         \[\leadsto B \cdot \color{blue}{\left(0.3333333333333333 \cdot x + 0.16666666666666666\right)} + \left(\frac{1}{B} - \frac{x}{B}\right) \]

      3. *-commutative 52.0%

         \[\leadsto B \cdot \left(\color{blue}{x \cdot 0.3333333333333333} + 0.16666666666666666\right) + \left(\frac{1}{B} - \frac{x}{B}\right) \]

      4. div-sub 52.0%

         \[\leadsto B \cdot \left(x \cdot 0.3333333333333333 + 0.16666666666666666\right) + \color{blue}{\frac{1 - x}{B}} \]

   10. Simplified 52.0%

       \[\leadsto \color{blue}{B \cdot \left(x \cdot 0.3333333333333333 + 0.16666666666666666\right) + \frac{1 - x}{B}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.68 \cdot 10^{+174}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq -6 \cdot 10^{+63}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -1.95 \cdot 10^{-60}:\\ \;\;\;\;\frac{-1 - x}{B} + B \cdot \left(x \cdot 0.3333333333333333 - 0.16666666666666666\right)\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{-55}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333 + 0.16666666666666666\right) + \frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 43.2% accurate, 14.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.5 \cdot 10^{-60}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 4.4 \cdot 10^{-55}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333 + 0.16666666666666666\right) + \frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.5e-60)
   (/ (- -1.0 x) B)
   (if (<= F 4.4e-55)
     (/ (- x) B)
     (+
      (* B (+ (* x 0.3333333333333333) 0.16666666666666666))
      (/ (- 1.0 x) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.5e-60) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 4.4e-55) {
		tmp = -x / B;
	} else {
		tmp = (B * ((x * 0.3333333333333333) + 0.16666666666666666)) + ((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 <= (-3.5d-60)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 4.4d-55) then
        tmp = -x / b
    else
        tmp = (b * ((x * 0.3333333333333333d0) + 0.16666666666666666d0)) + ((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 <= -3.5e-60) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 4.4e-55) {
		tmp = -x / B;
	} else {
		tmp = (B * ((x * 0.3333333333333333) + 0.16666666666666666)) + ((1.0 - x) / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.5e-60:
		tmp = (-1.0 - x) / B
	elif F <= 4.4e-55:
		tmp = -x / B
	else:
		tmp = (B * ((x * 0.3333333333333333) + 0.16666666666666666)) + ((1.0 - x) / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.5e-60)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 4.4e-55)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(B * Float64(Float64(x * 0.3333333333333333) + 0.16666666666666666)) + Float64(Float64(1.0 - x) / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -3.5e-60)
		tmp = (-1.0 - x) / B;
	elseif (F <= 4.4e-55)
		tmp = -x / B;
	else
		tmp = (B * ((x * 0.3333333333333333) + 0.16666666666666666)) + ((1.0 - x) / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.5e-60], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 4.4e-55], N[((-x) / B), $MachinePrecision], N[(N[(B * N[(N[(x * 0.3333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.49999999999999976e-60

   1. Initial program 65.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf 86.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   4. Taylor expanded in B around 0 48.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   5. Step-by-step derivation

      1. associate-*r/ 48.1%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

      2. distribute-lft-in 48.1%

         \[\leadsto \frac{\color{blue}{-1 \cdot 1 + -1 \cdot x}}{B} \]

      3. metadata-eval 48.1%

         \[\leadsto \frac{\color{blue}{-1} + -1 \cdot x}{B} \]

      4. neg-mul-1 48.1%

         \[\leadsto \frac{-1 + \color{blue}{\left(-x\right)}}{B} \]

   6. Simplified 48.1%

      \[\leadsto \color{blue}{\frac{-1 + \left(-x\right)}{B}} \]

   if -3.49999999999999976e-60 < F < 4.3999999999999999e-55

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in F around inf 27.4%

      \[\leadsto F \cdot \color{blue}{\frac{1}{F \cdot \sin B}} - \frac{x}{\tan B} \]
   6. Step-by-step derivation

      1. associate-/r* 27.4%

         \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   7. Simplified 27.4%

      \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   8. Taylor expanded in B around 0 15.9%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

   9. Taylor expanded in x around inf 38.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   10. Step-by-step derivation

       1. neg-mul-1 38.1%

          \[\leadsto \color{blue}{-\frac{x}{B}} \]

       2. distribute-neg-frac 38.1%

          \[\leadsto \color{blue}{\frac{-x}{B}} \]

   11. Simplified 38.1%

       \[\leadsto \color{blue}{\frac{-x}{B}} \]

   if 4.3999999999999999e-55 < F

   1. Initial program 73.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Simplified 83.3%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in F around inf 88.8%

      \[\leadsto F \cdot \color{blue}{\frac{1}{F \cdot \sin B}} - \frac{x}{\tan B} \]
   6. Step-by-step derivation

      1. associate-/r* 88.8%

         \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   7. Simplified 88.8%

      \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   8. Taylor expanded in B around 0 52.0%

      \[\leadsto \color{blue}{\left(B \cdot \left(0.16666666666666666 + 0.3333333333333333 \cdot x\right) + \frac{1}{B}\right) - \frac{x}{B}} \]
   9. Step-by-step derivation

      1. associate--l+ 52.0%

         \[\leadsto \color{blue}{B \cdot \left(0.16666666666666666 + 0.3333333333333333 \cdot x\right) + \left(\frac{1}{B} - \frac{x}{B}\right)} \]

      2. +-commutative 52.0%

         \[\leadsto B \cdot \color{blue}{\left(0.3333333333333333 \cdot x + 0.16666666666666666\right)} + \left(\frac{1}{B} - \frac{x}{B}\right) \]

      3. *-commutative 52.0%

         \[\leadsto B \cdot \left(\color{blue}{x \cdot 0.3333333333333333} + 0.16666666666666666\right) + \left(\frac{1}{B} - \frac{x}{B}\right) \]

      4. div-sub 52.0%

         \[\leadsto B \cdot \left(x \cdot 0.3333333333333333 + 0.16666666666666666\right) + \color{blue}{\frac{1 - x}{B}} \]

   10. Simplified 52.0%

       \[\leadsto \color{blue}{B \cdot \left(x \cdot 0.3333333333333333 + 0.16666666666666666\right) + \frac{1 - x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 45.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.5 \cdot 10^{-60}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 4.4 \cdot 10^{-55}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333 + 0.16666666666666666\right) + \frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 43.2% accurate, 14.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.4 \cdot 10^{-60}:\\ \;\;\;\;\frac{-1 - x}{B} + B \cdot \left(x \cdot 0.3333333333333333 - 0.16666666666666666\right)\\ \mathbf{elif}\;F \leq 1.6 \cdot 10^{-55}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333 + 0.16666666666666666\right) + \frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.4e-60)
   (+ (/ (- -1.0 x) B) (* B (- (* x 0.3333333333333333) 0.16666666666666666)))
   (if (<= F 1.6e-55)
     (/ (- x) B)
     (+
      (* B (+ (* x 0.3333333333333333) 0.16666666666666666))
      (/ (- 1.0 x) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.4e-60) {
		tmp = ((-1.0 - x) / B) + (B * ((x * 0.3333333333333333) - 0.16666666666666666));
	} else if (F <= 1.6e-55) {
		tmp = -x / B;
	} else {
		tmp = (B * ((x * 0.3333333333333333) + 0.16666666666666666)) + ((1.0 - x) / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-2.4d-60)) then
        tmp = (((-1.0d0) - x) / b) + (b * ((x * 0.3333333333333333d0) - 0.16666666666666666d0))
    else if (f <= 1.6d-55) then
        tmp = -x / b
    else
        tmp = (b * ((x * 0.3333333333333333d0) + 0.16666666666666666d0)) + ((1.0d0 - x) / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.4e-60) {
		tmp = ((-1.0 - x) / B) + (B * ((x * 0.3333333333333333) - 0.16666666666666666));
	} else if (F <= 1.6e-55) {
		tmp = -x / B;
	} else {
		tmp = (B * ((x * 0.3333333333333333) + 0.16666666666666666)) + ((1.0 - x) / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.4e-60:
		tmp = ((-1.0 - x) / B) + (B * ((x * 0.3333333333333333) - 0.16666666666666666))
	elif F <= 1.6e-55:
		tmp = -x / B
	else:
		tmp = (B * ((x * 0.3333333333333333) + 0.16666666666666666)) + ((1.0 - x) / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.4e-60)
		tmp = Float64(Float64(Float64(-1.0 - x) / B) + Float64(B * Float64(Float64(x * 0.3333333333333333) - 0.16666666666666666)));
	elseif (F <= 1.6e-55)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(B * Float64(Float64(x * 0.3333333333333333) + 0.16666666666666666)) + Float64(Float64(1.0 - x) / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.4e-60)
		tmp = ((-1.0 - x) / B) + (B * ((x * 0.3333333333333333) - 0.16666666666666666));
	elseif (F <= 1.6e-55)
		tmp = -x / B;
	else
		tmp = (B * ((x * 0.3333333333333333) + 0.16666666666666666)) + ((1.0 - x) / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.4e-60], N[(N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision] + N[(B * N[(N[(x * 0.3333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.6e-55], N[((-x) / B), $MachinePrecision], N[(N[(B * N[(N[(x * 0.3333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.40000000000000009e-60

   1. Initial program 65.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf 86.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   4. Taylor expanded in B around 0 48.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B} + B \cdot \left(0.3333333333333333 \cdot x - 0.16666666666666666\right)} \]

   if -2.40000000000000009e-60 < F < 1.6000000000000001e-55

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in F around inf 27.4%

      \[\leadsto F \cdot \color{blue}{\frac{1}{F \cdot \sin B}} - \frac{x}{\tan B} \]
   6. Step-by-step derivation

      1. associate-/r* 27.4%

         \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   7. Simplified 27.4%

      \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   8. Taylor expanded in B around 0 15.9%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

   9. Taylor expanded in x around inf 38.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   10. Step-by-step derivation

       1. neg-mul-1 38.1%

          \[\leadsto \color{blue}{-\frac{x}{B}} \]

       2. distribute-neg-frac 38.1%

          \[\leadsto \color{blue}{\frac{-x}{B}} \]

   11. Simplified 38.1%

       \[\leadsto \color{blue}{\frac{-x}{B}} \]

   if 1.6000000000000001e-55 < F

   1. Initial program 73.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Simplified 83.3%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in F around inf 88.8%

      \[\leadsto F \cdot \color{blue}{\frac{1}{F \cdot \sin B}} - \frac{x}{\tan B} \]
   6. Step-by-step derivation

      1. associate-/r* 88.8%

         \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   7. Simplified 88.8%

      \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   8. Taylor expanded in B around 0 52.0%

      \[\leadsto \color{blue}{\left(B \cdot \left(0.16666666666666666 + 0.3333333333333333 \cdot x\right) + \frac{1}{B}\right) - \frac{x}{B}} \]
   9. Step-by-step derivation

      1. associate--l+ 52.0%

         \[\leadsto \color{blue}{B \cdot \left(0.16666666666666666 + 0.3333333333333333 \cdot x\right) + \left(\frac{1}{B} - \frac{x}{B}\right)} \]

      2. +-commutative 52.0%

         \[\leadsto B \cdot \color{blue}{\left(0.3333333333333333 \cdot x + 0.16666666666666666\right)} + \left(\frac{1}{B} - \frac{x}{B}\right) \]

      3. *-commutative 52.0%

         \[\leadsto B \cdot \left(\color{blue}{x \cdot 0.3333333333333333} + 0.16666666666666666\right) + \left(\frac{1}{B} - \frac{x}{B}\right) \]

      4. div-sub 52.0%

         \[\leadsto B \cdot \left(x \cdot 0.3333333333333333 + 0.16666666666666666\right) + \color{blue}{\frac{1 - x}{B}} \]

   10. Simplified 52.0%

       \[\leadsto \color{blue}{B \cdot \left(x \cdot 0.3333333333333333 + 0.16666666666666666\right) + \frac{1 - x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.4 \cdot 10^{-60}:\\ \;\;\;\;\frac{-1 - x}{B} + B \cdot \left(x \cdot 0.3333333333333333 - 0.16666666666666666\right)\\ \mathbf{elif}\;F \leq 1.6 \cdot 10^{-55}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(x \cdot 0.3333333333333333 + 0.16666666666666666\right) + \frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 28: 43.2% accurate, 21.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -4 \cdot 10^{-60}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.95 \cdot 10^{-55}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4e-60)
   (/ (- -1.0 x) B)
   (if (<= F 2.95e-55) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4e-60) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 2.95e-55) {
		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 <= (-4d-60)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 2.95d-55) 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 <= -4e-60) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 2.95e-55) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -4e-60:
		tmp = (-1.0 - x) / B
	elif F <= 2.95e-55:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -4e-60)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 2.95e-55)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -4e-60)
		tmp = (-1.0 - x) / B;
	elseif (F <= 2.95e-55)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -4e-60], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 2.95e-55], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.9999999999999999e-60

   1. Initial program 65.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf 86.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   4. Taylor expanded in B around 0 48.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   5. Step-by-step derivation

      1. associate-*r/ 48.1%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(1 + x\right)}{B}} \]

      2. distribute-lft-in 48.1%

         \[\leadsto \frac{\color{blue}{-1 \cdot 1 + -1 \cdot x}}{B} \]

      3. metadata-eval 48.1%

         \[\leadsto \frac{\color{blue}{-1} + -1 \cdot x}{B} \]

      4. neg-mul-1 48.1%

         \[\leadsto \frac{-1 + \color{blue}{\left(-x\right)}}{B} \]

   6. Simplified 48.1%

      \[\leadsto \color{blue}{\frac{-1 + \left(-x\right)}{B}} \]

   if -3.9999999999999999e-60 < F < 2.9499999999999999e-55

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in F around inf 27.4%

      \[\leadsto F \cdot \color{blue}{\frac{1}{F \cdot \sin B}} - \frac{x}{\tan B} \]
   6. Step-by-step derivation

      1. associate-/r* 27.4%

         \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   7. Simplified 27.4%

      \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   8. Taylor expanded in B around 0 15.9%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

   9. Taylor expanded in x around inf 38.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   10. Step-by-step derivation

       1. neg-mul-1 38.1%

          \[\leadsto \color{blue}{-\frac{x}{B}} \]

       2. distribute-neg-frac 38.1%

          \[\leadsto \color{blue}{\frac{-x}{B}} \]

   11. Simplified 38.1%

       \[\leadsto \color{blue}{\frac{-x}{B}} \]

   if 2.9499999999999999e-55 < F

   1. Initial program 73.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Simplified 83.3%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in F around inf 88.8%

      \[\leadsto F \cdot \color{blue}{\frac{1}{F \cdot \sin B}} - \frac{x}{\tan B} \]
   6. Step-by-step derivation

      1. associate-/r* 88.8%

         \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   7. Simplified 88.8%

      \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   8. Taylor expanded in B around 0 50.5%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4 \cdot 10^{-60}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.95 \cdot 10^{-55}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 29: 36.0% accurate, 32.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 1.8 \cdot 10^{-55}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F 1.8e-55) (/ (- x) B) (/ (- 1.0 x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= 1.8e-55) {
		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 <= 1.8d-55) 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 <= 1.8e-55) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= 1.8e-55:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= 1.8e-55)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= 1.8e-55)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, 1.8e-55], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 1.8e-55

   1. Initial program 82.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Simplified 90.1%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in F around inf 35.0%

      \[\leadsto F \cdot \color{blue}{\frac{1}{F \cdot \sin B}} - \frac{x}{\tan B} \]
   6. Step-by-step derivation

      1. associate-/r* 35.0%

         \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   7. Simplified 35.0%

      \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   8. Taylor expanded in B around 0 18.8%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

   9. Taylor expanded in x around inf 30.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   10. Step-by-step derivation

       1. neg-mul-1 30.6%

          \[\leadsto \color{blue}{-\frac{x}{B}} \]

       2. distribute-neg-frac 30.6%

          \[\leadsto \color{blue}{\frac{-x}{B}} \]

   11. Simplified 30.6%

       \[\leadsto \color{blue}{\frac{-x}{B}} \]

   if 1.8e-55 < F

   1. Initial program 73.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Simplified 83.3%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in F around inf 88.8%

      \[\leadsto F \cdot \color{blue}{\frac{1}{F \cdot \sin B}} - \frac{x}{\tan B} \]
   6. Step-by-step derivation

      1. associate-/r* 88.8%

         \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   7. Simplified 88.8%

      \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

   8. Taylor expanded in B around 0 50.5%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.8 \cdot 10^{-55}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 30: 29.0% accurate, 81.0× speedup

Math

\[\begin{array}{l} \\ \frac{-x}{B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (/ (- x) B))

C

double code(double F, double B, double x) {
	return -x / 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 = -x / b
end function

Java

public static double code(double F, double B, double x) {
	return -x / B;
}

Python

def code(F, B, x):
	return -x / B

Julia

function code(F, B, x)
	return Float64(Float64(-x) / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = -x / B;
end

Wolfram

code[F_, B_, x_] := N[((-x) / B), $MachinePrecision]

Derivation

1. Initial program 80.2%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

3. Simplified 88.2%

   \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
4. Add Preprocessing

5. Taylor expanded in F around inf 49.9%

   \[\leadsto F \cdot \color{blue}{\frac{1}{F \cdot \sin B}} - \frac{x}{\tan B} \]
6. Step-by-step derivation

   1. associate-/r* 49.9%

      \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

7. Simplified 49.9%

   \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

8. Taylor expanded in B around 0 27.6%

   \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

9. Taylor expanded in x around inf 29.2%

   \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
10. Step-by-step derivation

    1. neg-mul-1 29.2%

       \[\leadsto \color{blue}{-\frac{x}{B}} \]

    2. distribute-neg-frac 29.2%

       \[\leadsto \color{blue}{\frac{-x}{B}} \]

11. Simplified 29.2%

    \[\leadsto \color{blue}{\frac{-x}{B}} \]

12. Final simplification 29.2%

    \[\leadsto \frac{-x}{B} \]
13. Add Preprocessing

Alternative 31: 9.8% accurate, 108.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (/ 1.0 B))

C

double code(double F, double B, double x) {
	return 1.0 / B;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = 1.0d0 / b
end function

Java

public static double code(double F, double B, double x) {
	return 1.0 / B;
}

Python

def code(F, B, x):
	return 1.0 / B

Julia

function code(F, B, x)
	return Float64(1.0 / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = 1.0 / B;
end

Wolfram

code[F_, B_, x_] := N[(1.0 / B), $MachinePrecision]

Derivation

1. Initial program 80.2%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

3. Simplified 88.2%

   \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
4. Add Preprocessing

5. Taylor expanded in F around inf 49.9%

   \[\leadsto F \cdot \color{blue}{\frac{1}{F \cdot \sin B}} - \frac{x}{\tan B} \]
6. Step-by-step derivation

   1. associate-/r* 49.9%

      \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

7. Simplified 49.9%

   \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\sin B}} - \frac{x}{\tan B} \]

8. Taylor expanded in B around 0 27.6%

   \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

9. Taylor expanded in x around 0 9.6%

   \[\leadsto \color{blue}{\frac{1}{B}} \]

10. Final simplification 9.6%

    \[\leadsto \frac{1}{B} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024019 
(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))))))