VandenBroeck and Keller, Equation (23)

Percentage Accurate: 77.7% → 99.1%

Time: 21.0s

Alternatives: 21

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: 77.7% 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: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.7 \cdot 10^{+110}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 200000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.7e+110)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 200000000.0)
       (fma
        (/ F (sin B))
        (pow (fma x 2.0 (fma F F 2.0)) -0.5)
        (/ (- x) (tan B)))
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.7e+110) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 200000000.0) {
		tmp = fma((F / sin(B)), pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5), (-x / tan(B)));
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.7e+110)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 200000000.0)
		tmp = fma(Float64(F / sin(B)), (fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5), Float64(Float64(-x) / tan(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.7e+110], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 200000000.0], N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] + N[((-x) / N[Tan[B], $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 < -1.7000000000000001e110

   1. Initial program 59.8%

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

      1. distribute-lft-neg-in 59.8%

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

      2. +-commutative 59.8%

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

      3. cancel-sign-sub-inv 59.8%

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

   3. Simplified 72.6%

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

      1. clear-num 72.6%

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

      2. inv-pow 72.6%

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

      3. fma-def 72.6%

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

      4. fma-udef 72.6%

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

      5. *-commutative 72.6%

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

      6. fma-def 72.6%

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

      7. fma-def 72.6%

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

   5. Applied egg-rr 72.6%

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

      1. unpow-1 72.6%

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

   7. Simplified 72.6%

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

   8. Taylor expanded in F around -inf 99.9%

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

   if -1.7000000000000001e110 < F < 2e8

   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. Step-by-step derivation

      1. distribute-lft-neg-in 99.5%

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

      2. +-commutative 99.5%

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

      3. fma-def 99.5%

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

      4. +-commutative 99.5%

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

      5. *-commutative 99.5%

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

      6. fma-def 99.5%

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

      7. fma-def 99.5%

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

      8. metadata-eval 99.5%

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

      9. metadata-eval 99.5%

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

      10. associate-*r/ 99.7%

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

      11. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   if 2e8 < F

   1. Initial program 61.8%

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

      1. distribute-lft-neg-in 61.8%

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

      2. +-commutative 61.8%

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

      3. cancel-sign-sub-inv 61.8%

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

   3. Simplified 75.4%

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

      1. clear-num 75.4%

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

      2. inv-pow 75.4%

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

      3. fma-def 75.4%

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

      4. fma-udef 75.4%

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

      5. *-commutative 75.4%

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

      6. fma-def 75.4%

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

      7. fma-def 75.4%

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

   5. Applied egg-rr 75.4%

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

      1. unpow-1 75.4%

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

   7. Simplified 75.4%

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

   8. Taylor expanded in F around inf 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.7 \cdot 10^{+110}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 200000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 2: 99.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -104000000:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 115000000:\\ \;\;\;\;F \cdot \frac{1}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -104000000.0)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 115000000.0)
       (- (* F (/ 1.0 (/ (sin B) (pow (fma 2.0 x (fma F F 2.0)) -0.5)))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -104000000.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 115000000.0) {
		tmp = (F * (1.0 / (sin(B) / pow(fma(2.0, x, fma(F, F, 2.0)), -0.5)))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -104000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 115000000.0)
		tmp = Float64(Float64(F * Float64(1.0 / Float64(sin(B) / (fma(2.0, x, fma(F, F, 2.0)) ^ -0.5)))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -104000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 115000000.0], N[(N[(F * N[(1.0 / N[(N[Sin[B], $MachinePrecision] / N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -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 < -1.04e8

   1. Initial program 74.5%

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

      1. distribute-lft-neg-in 74.5%

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

      2. +-commutative 74.5%

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

      3. cancel-sign-sub-inv 74.5%

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

   3. Simplified 82.6%

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

      1. clear-num 82.5%

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

      2. inv-pow 82.5%

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

      3. fma-def 82.5%

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

      4. fma-udef 82.5%

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

      5. *-commutative 82.5%

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

      6. fma-def 82.5%

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

      7. fma-def 82.5%

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

   5. Applied egg-rr 82.5%

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

      1. unpow-1 82.5%

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

   7. Simplified 82.5%

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

   8. Taylor expanded in F around -inf 99.8%

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

   if -1.04e8 < F < 1.15e8

   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. Step-by-step derivation

      1. distribute-lft-neg-in 99.5%

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

      2. +-commutative 99.5%

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

      3. cancel-sign-sub-inv 99.5%

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

   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. Step-by-step derivation

      1. clear-num 99.6%

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

      2. inv-pow 99.6%

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

      3. fma-def 99.6%

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

      4. fma-udef 99.6%

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

      5. *-commutative 99.6%

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

      6. fma-def 99.6%

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

      7. fma-def 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. unpow-1 99.6%

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

   7. Simplified 99.6%

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

   if 1.15e8 < F

   1. Initial program 61.8%

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

      1. distribute-lft-neg-in 61.8%

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

      2. +-commutative 61.8%

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

      3. cancel-sign-sub-inv 61.8%

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

   3. Simplified 75.4%

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

      1. clear-num 75.4%

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

      2. inv-pow 75.4%

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

      3. fma-def 75.4%

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

      4. fma-udef 75.4%

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

      5. *-commutative 75.4%

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

      6. fma-def 75.4%

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

      7. fma-def 75.4%

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

   5. Applied egg-rr 75.4%

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

      1. unpow-1 75.4%

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

   7. Simplified 75.4%

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

   8. Taylor expanded in F around inf 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -104000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 115000000:\\ \;\;\;\;F \cdot \frac{1}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 3: 99.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -7 \cdot 10^{+15}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 200000000:\\ \;\;\;\;F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -7e+15)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 200000000.0)
       (- (* 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 <= -7e+15) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 200000000.0) {
		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 <= -7e+15)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 200000000.0)
		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, -7e+15], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 200000000.0], 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 < -7e15

   1. Initial program 73.7%

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

      1. distribute-lft-neg-in 73.7%

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

      2. +-commutative 73.7%

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

      3. cancel-sign-sub-inv 73.7%

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

   3. Simplified 82.0%

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

      1. clear-num 81.9%

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

      2. inv-pow 81.9%

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

      3. fma-def 81.9%

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

      4. fma-udef 81.9%

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

      5. *-commutative 81.9%

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

      6. fma-def 81.9%

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

      7. fma-def 81.9%

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

   5. Applied egg-rr 81.9%

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

      1. unpow-1 81.9%

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

   7. Simplified 81.9%

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

   8. Taylor expanded in F around -inf 99.8%

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

   if -7e15 < F < 2e8

   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. Step-by-step derivation

      1. distribute-lft-neg-in 99.5%

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

      2. +-commutative 99.5%

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

      3. cancel-sign-sub-inv 99.5%

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

   3. Simplified 99.6%

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

   if 2e8 < F

   1. Initial program 61.8%

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

      1. distribute-lft-neg-in 61.8%

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

      2. +-commutative 61.8%

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

      3. cancel-sign-sub-inv 61.8%

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

   3. Simplified 75.4%

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

      1. clear-num 75.4%

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

      2. inv-pow 75.4%

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

      3. fma-def 75.4%

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

      4. fma-udef 75.4%

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

      5. *-commutative 75.4%

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

      6. fma-def 75.4%

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

      7. fma-def 75.4%

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

   5. Applied egg-rr 75.4%

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

      1. unpow-1 75.4%

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

   7. Simplified 75.4%

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

   8. Taylor expanded in F around inf 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7 \cdot 10^{+15}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 200000000:\\ \;\;\;\;F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 4: 99.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.7 \cdot 10^{+110}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 50:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.7e+110)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 50.0)
       (- (/ (/ F (sin B)) (sqrt (fma 2.0 x (fma F F 2.0)))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.7e+110) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 50.0) {
		tmp = ((F / sin(B)) / sqrt(fma(2.0, x, fma(F, F, 2.0)))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.7e+110)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 50.0)
		tmp = Float64(Float64(Float64(F / sin(B)) / sqrt(fma(2.0, x, fma(F, F, 2.0)))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.7e+110], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 50.0], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.7000000000000001e110

   1. Initial program 59.8%

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

      1. distribute-lft-neg-in 59.8%

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

      2. +-commutative 59.8%

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

      3. cancel-sign-sub-inv 59.8%

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

   3. Simplified 72.6%

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

      1. clear-num 72.6%

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

      2. inv-pow 72.6%

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

      3. fma-def 72.6%

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

      4. fma-udef 72.6%

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

      5. *-commutative 72.6%

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

      6. fma-def 72.6%

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

      7. fma-def 72.6%

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

   5. Applied egg-rr 72.6%

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

      1. unpow-1 72.6%

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

   7. Simplified 72.6%

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

   8. Taylor expanded in F around -inf 99.9%

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

   if -1.7000000000000001e110 < F < 50

   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. Step-by-step derivation

      1. distribute-lft-neg-in 99.5%

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

      2. +-commutative 99.5%

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

      3. cancel-sign-sub-inv 99.5%

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

   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. Step-by-step derivation

      1. clear-num 99.5%

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

      2. inv-pow 99.5%

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

      3. fma-def 99.5%

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

      4. fma-udef 99.5%

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

      5. *-commutative 99.5%

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

      6. fma-def 99.5%

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

      7. fma-def 99.5%

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

   5. Applied egg-rr 99.5%

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

      1. unpow-1 99.5%

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

   7. Simplified 99.5%

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

      1. expm1-log1p-u 81.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(F \cdot \frac{1}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}\right)\right)} - \frac{x}{\tan B} \]

      2. expm1-udef 66.4%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(F \cdot \frac{1}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}\right)} - 1\right)} - \frac{x}{\tan B} \]

      3. un-div-inv 66.4%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      4. div-inv 66.4%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\color{blue}{\sin B \cdot \frac{1}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      5. pow-flip 66.4%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot \color{blue}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(--0.5\right)}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      6. metadata-eval 66.4%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{0.5}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      7. pow1/2 66.4%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot \color{blue}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}\right)} - 1\right) - \frac{x}{\tan B} \]

   9. Applied egg-rr 66.4%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\right)} - 1\right)} - \frac{x}{\tan B} \]
   10. Step-by-step derivation

       1. expm1-def 81.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\right)\right)} - \frac{x}{\tan B} \]

       2. expm1-log1p 99.6%

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

       3. associate-/r* 99.6%

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

   11. Simplified 99.6%

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

   if 50 < F

   1. Initial program 61.8%

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

      1. distribute-lft-neg-in 61.8%

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

      2. +-commutative 61.8%

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

      3. cancel-sign-sub-inv 61.8%

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

   3. Simplified 75.4%

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

      1. clear-num 75.4%

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

      2. inv-pow 75.4%

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

      3. fma-def 75.4%

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

      4. fma-udef 75.4%

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

      5. *-commutative 75.4%

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

      6. fma-def 75.4%

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

      7. fma-def 75.4%

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

   5. Applied egg-rr 75.4%

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

      1. unpow-1 75.4%

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

   7. Simplified 75.4%

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

   8. Taylor expanded in F around inf 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.7 \cdot 10^{+110}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 50:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 5: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -590000:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 4.6 \cdot 10^{+14}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -590000.0)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 4.6e+14)
       (+
        (* x (/ -1.0 (tan B)))
        (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)))
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -590000.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 4.6e+14) {
		tmp = (x * (-1.0 / tan(B))) + ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5));
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-590000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 4.6d+14) then
        tmp = (x * ((-1.0d0) / tan(b))) + ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)))
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -590000.0) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 4.6e+14) {
		tmp = (x * (-1.0 / Math.tan(B))) + ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5));
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -590000.0:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 4.6e+14:
		tmp = (x * (-1.0 / math.tan(B))) + ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5))
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -590000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 4.6e+14)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -590000.0)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 4.6e+14)
		tmp = (x * (-1.0 / tan(B))) + ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5));
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -590000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 4.6e+14], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.9e5

   1. Initial program 74.9%

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

      1. distribute-lft-neg-in 74.9%

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

      2. +-commutative 74.9%

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

      3. cancel-sign-sub-inv 74.9%

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

   3. Simplified 82.8%

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

      1. clear-num 82.7%

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

      2. inv-pow 82.7%

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

      3. fma-def 82.7%

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

      4. fma-udef 82.7%

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

      5. *-commutative 82.7%

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

      6. fma-def 82.7%

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

      7. fma-def 82.7%

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

   5. Applied egg-rr 82.7%

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

      1. unpow-1 82.7%

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

   7. Simplified 82.7%

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

   8. Taylor expanded in F around -inf 99.8%

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

   if -5.9e5 < F < 4.6e14

   1. Initial program 99.5%

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

   if 4.6e14 < F

   1. Initial program 61.8%

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

      1. distribute-lft-neg-in 61.8%

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

      2. +-commutative 61.8%

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

      3. cancel-sign-sub-inv 61.8%

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

   3. Simplified 75.4%

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

      1. clear-num 75.4%

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

      2. inv-pow 75.4%

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

      3. fma-def 75.4%

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

      4. fma-udef 75.4%

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

      5. *-commutative 75.4%

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

      6. fma-def 75.4%

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

      7. fma-def 75.4%

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

   5. Applied egg-rr 75.4%

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

      1. unpow-1 75.4%

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

   7. Simplified 75.4%

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

   8. Taylor expanded in F around inf 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -590000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 4.6 \cdot 10^{+14}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 6: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -600000:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 0.9:\\ \;\;\;\;F \cdot \left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{\sin B \cdot \left(F + \frac{1}{F}\right)} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -600000.0)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 0.9)
       (- (* F (* (/ 1.0 (sin B)) (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))))) t_0)
       (- (/ F (* (sin B) (+ F (/ 1.0 F)))) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -600000.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 0.9) {
		tmp = (F * ((1.0 / sin(B)) * sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_0;
	} else {
		tmp = (F / (sin(B) * (F + (1.0 / F)))) - 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 <= (-600000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 0.9d0) then
        tmp = (f * ((1.0d0 / sin(b)) * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))))) - t_0
    else
        tmp = (f / (sin(b) * (f + (1.0d0 / f)))) - 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 <= -600000.0) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 0.9) {
		tmp = (F * ((1.0 / Math.sin(B)) * Math.sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_0;
	} else {
		tmp = (F / (Math.sin(B) * (F + (1.0 / F)))) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -600000.0:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 0.9:
		tmp = (F * ((1.0 / math.sin(B)) * math.sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_0
	else:
		tmp = (F / (math.sin(B) * (F + (1.0 / F)))) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -600000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 0.9)
		tmp = Float64(Float64(F * Float64(Float64(1.0 / sin(B)) * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0)))))) - t_0);
	else
		tmp = Float64(Float64(F / Float64(sin(B) * Float64(F + Float64(1.0 / F)))) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -600000.0)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 0.9)
		tmp = (F * ((1.0 / sin(B)) * sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_0;
	else
		tmp = (F / (sin(B) * (F + (1.0 / F)))) - 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, -600000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.9], N[(N[(F * N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * N[(F + N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -6e5

   1. Initial program 74.5%

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

      1. distribute-lft-neg-in 74.5%

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

      2. +-commutative 74.5%

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

      3. cancel-sign-sub-inv 74.5%

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

   3. Simplified 82.6%

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

      1. clear-num 82.5%

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

      2. inv-pow 82.5%

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

      3. fma-def 82.5%

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

      4. fma-udef 82.5%

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

      5. *-commutative 82.5%

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

      6. fma-def 82.5%

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

      7. fma-def 82.5%

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

   5. Applied egg-rr 82.5%

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

      1. unpow-1 82.5%

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

   7. Simplified 82.5%

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

   8. Taylor expanded in F around -inf 99.8%

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

   if -6e5 < F < 0.900000000000000022

   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. Step-by-step derivation

      1. distribute-lft-neg-in 99.5%

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

      2. +-commutative 99.5%

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

      3. cancel-sign-sub-inv 99.5%

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

   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. 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 0.900000000000000022 < F

   1. Initial program 62.3%

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

      1. distribute-lft-neg-in 62.3%

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

      2. +-commutative 62.3%

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

      3. cancel-sign-sub-inv 62.3%

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

   3. Simplified 75.7%

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

      1. clear-num 75.7%

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

      2. inv-pow 75.7%

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

      3. fma-def 75.7%

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

      4. fma-udef 75.7%

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

      5. *-commutative 75.7%

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

      6. fma-def 75.7%

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

      7. fma-def 75.7%

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

   5. Applied egg-rr 75.7%

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

      1. unpow-1 75.7%

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

   7. Simplified 75.7%

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

      1. expm1-log1p-u 63.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(F \cdot \frac{1}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}\right)\right)} - \frac{x}{\tan B} \]

      2. expm1-udef 63.1%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(F \cdot \frac{1}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}\right)} - 1\right)} - \frac{x}{\tan B} \]

      3. un-div-inv 63.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      4. div-inv 63.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\color{blue}{\sin B \cdot \frac{1}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      5. pow-flip 63.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot \color{blue}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(--0.5\right)}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      6. metadata-eval 63.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{0.5}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      7. pow1/2 63.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot \color{blue}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}\right)} - 1\right) - \frac{x}{\tan B} \]

   9. Applied egg-rr 63.1%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\right)} - 1\right)} - \frac{x}{\tan B} \]
   10. Step-by-step derivation

       1. expm1-def 63.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\right)\right)} - \frac{x}{\tan B} \]

       2. expm1-log1p 75.8%

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

       3. associate-/r* 62.4%

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

   11. Simplified 62.4%

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

   12. Taylor expanded in F around inf 86.1%

       \[\leadsto \frac{\frac{F}{\sin B}}{\color{blue}{F + 0.5 \cdot \frac{2 + 2 \cdot x}{F}}} - \frac{x}{\tan B} \]

   13. Taylor expanded in x around 0 99.2%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -600000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.9:\\ \;\;\;\;F \cdot \left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\right) - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{\sin B \cdot \left(F + \frac{1}{F}\right)} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 7: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -600000:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 1.3:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{\sqrt{2 + x \cdot 2}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{\sin B \cdot \left(F + \frac{1}{F}\right)} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -600000.0)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.3)
       (- (/ (/ F (sin B)) (sqrt (+ 2.0 (* x 2.0)))) t_0)
       (- (/ F (* (sin B) (+ F (/ 1.0 F)))) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -600000.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.3) {
		tmp = ((F / sin(B)) / sqrt((2.0 + (x * 2.0)))) - t_0;
	} else {
		tmp = (F / (sin(B) * (F + (1.0 / F)))) - 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 <= (-600000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.3d0) then
        tmp = ((f / sin(b)) / sqrt((2.0d0 + (x * 2.0d0)))) - t_0
    else
        tmp = (f / (sin(b) * (f + (1.0d0 / f)))) - 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 <= -600000.0) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.3) {
		tmp = ((F / Math.sin(B)) / Math.sqrt((2.0 + (x * 2.0)))) - t_0;
	} else {
		tmp = (F / (Math.sin(B) * (F + (1.0 / F)))) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -600000.0:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.3:
		tmp = ((F / math.sin(B)) / math.sqrt((2.0 + (x * 2.0)))) - t_0
	else:
		tmp = (F / (math.sin(B) * (F + (1.0 / F)))) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -600000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.3)
		tmp = Float64(Float64(Float64(F / sin(B)) / sqrt(Float64(2.0 + Float64(x * 2.0)))) - t_0);
	else
		tmp = Float64(Float64(F / Float64(sin(B) * Float64(F + Float64(1.0 / F)))) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -600000.0)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.3)
		tmp = ((F / sin(B)) / sqrt((2.0 + (x * 2.0)))) - t_0;
	else
		tmp = (F / (sin(B) * (F + (1.0 / F)))) - 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, -600000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.3], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * N[(F + N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -6e5

   1. Initial program 74.5%

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

      1. distribute-lft-neg-in 74.5%

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

      2. +-commutative 74.5%

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

      3. cancel-sign-sub-inv 74.5%

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

   3. Simplified 82.6%

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

      1. clear-num 82.5%

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

      2. inv-pow 82.5%

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

      3. fma-def 82.5%

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

      4. fma-udef 82.5%

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

      5. *-commutative 82.5%

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

      6. fma-def 82.5%

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

      7. fma-def 82.5%

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

   5. Applied egg-rr 82.5%

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

      1. unpow-1 82.5%

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

   7. Simplified 82.5%

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

   8. Taylor expanded in F around -inf 99.8%

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

   if -6e5 < F < 1.30000000000000004

   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. Step-by-step derivation

      1. distribute-lft-neg-in 99.5%

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

      2. +-commutative 99.5%

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

      3. cancel-sign-sub-inv 99.5%

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

   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. Step-by-step derivation

      1. clear-num 99.6%

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

      2. inv-pow 99.6%

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

      3. fma-def 99.6%

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

      4. fma-udef 99.6%

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

      5. *-commutative 99.6%

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

      6. fma-def 99.6%

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

      7. fma-def 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. unpow-1 99.6%

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

   7. Simplified 99.6%

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

      1. expm1-log1p-u 89.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(F \cdot \frac{1}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}\right)\right)} - \frac{x}{\tan B} \]

      2. expm1-udef 70.7%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(F \cdot \frac{1}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}\right)} - 1\right)} - \frac{x}{\tan B} \]

      3. un-div-inv 70.7%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      4. div-inv 70.7%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\color{blue}{\sin B \cdot \frac{1}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      5. pow-flip 70.7%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot \color{blue}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(--0.5\right)}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      6. metadata-eval 70.7%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{0.5}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      7. pow1/2 70.7%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot \color{blue}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}\right)} - 1\right) - \frac{x}{\tan B} \]

   9. Applied egg-rr 70.7%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\right)} - 1\right)} - \frac{x}{\tan B} \]
   10. Step-by-step derivation

       1. expm1-def 89.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\right)\right)} - \frac{x}{\tan B} \]

       2. expm1-log1p 99.6%

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

       3. associate-/r* 99.6%

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

   11. Simplified 99.6%

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

   12. Taylor expanded in F around 0 99.4%

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

   if 1.30000000000000004 < F

   1. Initial program 62.3%

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

      1. distribute-lft-neg-in 62.3%

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

      2. +-commutative 62.3%

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

      3. cancel-sign-sub-inv 62.3%

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

   3. Simplified 75.7%

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

      1. clear-num 75.7%

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

      2. inv-pow 75.7%

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

      3. fma-def 75.7%

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

      4. fma-udef 75.7%

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

      5. *-commutative 75.7%

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

      6. fma-def 75.7%

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

      7. fma-def 75.7%

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

   5. Applied egg-rr 75.7%

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

      1. unpow-1 75.7%

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

   7. Simplified 75.7%

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

      1. expm1-log1p-u 63.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(F \cdot \frac{1}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}\right)\right)} - \frac{x}{\tan B} \]

      2. expm1-udef 63.1%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(F \cdot \frac{1}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}\right)} - 1\right)} - \frac{x}{\tan B} \]

      3. un-div-inv 63.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      4. div-inv 63.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\color{blue}{\sin B \cdot \frac{1}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      5. pow-flip 63.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot \color{blue}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(--0.5\right)}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      6. metadata-eval 63.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{0.5}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      7. pow1/2 63.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot \color{blue}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}\right)} - 1\right) - \frac{x}{\tan B} \]

   9. Applied egg-rr 63.1%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\right)} - 1\right)} - \frac{x}{\tan B} \]
   10. Step-by-step derivation

       1. expm1-def 63.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\right)\right)} - \frac{x}{\tan B} \]

       2. expm1-log1p 75.8%

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

       3. associate-/r* 62.4%

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

   11. Simplified 62.4%

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

   12. Taylor expanded in F around inf 86.1%

       \[\leadsto \frac{\frac{F}{\sin B}}{\color{blue}{F + 0.5 \cdot \frac{2 + 2 \cdot x}{F}}} - \frac{x}{\tan B} \]

   13. Taylor expanded in x around 0 99.2%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -600000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.3:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{\sqrt{2 + x \cdot 2}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{\sin B \cdot \left(F + \frac{1}{F}\right)} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 8: 91.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.42 \cdot 10^{+20}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq -2.9 \cdot 10^{-121}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.15 \cdot 10^{-27}:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{\sin B \cdot \left(F + \frac{1}{F}\right)} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.42e+20)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F -2.9e-121)
       (- (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)) (/ x B))
       (if (<= F 3.15e-27)
         (- (* (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))) (/ F B)) t_0)
         (- (/ F (* (sin B) (+ F (/ 1.0 F)))) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.42e+20) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= -2.9e-121) {
		tmp = ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B);
	} else if (F <= 3.15e-27) {
		tmp = (sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - t_0;
	} else {
		tmp = (F / (sin(B) * (F + (1.0 / F)))) - 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 <= (-1.42d+20)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= (-2.9d-121)) then
        tmp = ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0))) - (x / b)
    else if (f <= 3.15d-27) then
        tmp = (sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))) * (f / b)) - t_0
    else
        tmp = (f / (sin(b) * (f + (1.0d0 / f)))) - 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 <= -1.42e+20) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= -2.9e-121) {
		tmp = ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B);
	} else if (F <= 3.15e-27) {
		tmp = (Math.sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - t_0;
	} else {
		tmp = (F / (Math.sin(B) * (F + (1.0 / F)))) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.42e+20:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= -2.9e-121:
		tmp = ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B)
	elif F <= 3.15e-27:
		tmp = (math.sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - t_0
	else:
		tmp = (F / (math.sin(B) * (F + (1.0 / F)))) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.42e+20)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= -2.9e-121)
		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 <= 3.15e-27)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0)))) * Float64(F / B)) - t_0);
	else
		tmp = Float64(Float64(F / Float64(sin(B) * Float64(F + Float64(1.0 / F)))) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.42e+20)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= -2.9e-121)
		tmp = ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5)) - (x / B);
	elseif (F <= 3.15e-27)
		tmp = (sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - t_0;
	else
		tmp = (F / (sin(B) * (F + (1.0 / F)))) - 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, -1.42e+20], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -2.9e-121], 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, 3.15e-27], 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[(F / N[(N[Sin[B], $MachinePrecision] * N[(F + N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.42e20

   1. Initial program 73.3%

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

      1. distribute-lft-neg-in 73.3%

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

      2. +-commutative 73.3%

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

      3. cancel-sign-sub-inv 73.3%

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

   3. Simplified 81.7%

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

      1. clear-num 81.6%

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

      2. inv-pow 81.6%

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

      3. fma-def 81.6%

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

      4. fma-udef 81.6%

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

      5. *-commutative 81.6%

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

      6. fma-def 81.6%

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

      7. fma-def 81.6%

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

   5. Applied egg-rr 81.6%

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

      1. unpow-1 81.6%

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

   7. Simplified 81.6%

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

   8. Taylor expanded in F around -inf 99.8%

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

   if -1.42e20 < F < -2.9e-121

   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. Taylor expanded in B around 0 95.2%

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

   if -2.9e-121 < F < 3.15000000000000005e-27

   1. Initial program 99.5%

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

      1. distribute-lft-neg-in 99.5%

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

      2. +-commutative 99.5%

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

      3. cancel-sign-sub-inv 99.5%

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

   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. Taylor expanded in F around 0 99.7%

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

   5. Taylor expanded in B around 0 93.0%

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

   if 3.15000000000000005e-27 < F

   1. Initial program 63.3%

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

      1. distribute-lft-neg-in 63.3%

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

      2. +-commutative 63.3%

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

      3. cancel-sign-sub-inv 63.3%

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

   3. Simplified 76.4%

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

      1. clear-num 76.3%

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

      2. inv-pow 76.3%

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

      3. fma-def 76.3%

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

      4. fma-udef 76.3%

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

      5. *-commutative 76.3%

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

      6. fma-def 76.3%

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

      7. fma-def 76.3%

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

   5. Applied egg-rr 76.3%

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

      1. unpow-1 76.3%

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

   7. Simplified 76.3%

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

      1. expm1-log1p-u 64.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(F \cdot \frac{1}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}\right)\right)} - \frac{x}{\tan B} \]

      2. expm1-udef 63.8%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(F \cdot \frac{1}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}\right)} - 1\right)} - \frac{x}{\tan B} \]

      3. un-div-inv 63.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      4. div-inv 63.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\color{blue}{\sin B \cdot \frac{1}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      5. pow-flip 63.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot \color{blue}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(--0.5\right)}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      6. metadata-eval 63.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{0.5}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      7. pow1/2 63.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot \color{blue}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}\right)} - 1\right) - \frac{x}{\tan B} \]

   9. Applied egg-rr 63.8%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\right)} - 1\right)} - \frac{x}{\tan B} \]
   10. Step-by-step derivation

       1. expm1-def 64.1%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\right)\right)} - \frac{x}{\tan B} \]

       2. expm1-log1p 76.4%

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

       3. associate-/r* 63.4%

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

   11. Simplified 63.4%

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

   12. Taylor expanded in F around inf 85.4%

       \[\leadsto \frac{\frac{F}{\sin B}}{\color{blue}{F + 0.5 \cdot \frac{2 + 2 \cdot x}{F}}} - \frac{x}{\tan B} \]

   13. Taylor expanded in x around 0 98.0%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.42 \cdot 10^{+20}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -2.9 \cdot 10^{-121}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.15 \cdot 10^{-27}:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{\sin B \cdot \left(F + \frac{1}{F}\right)} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 9: 85.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ t_1 := 2 + x \cdot 2\\ \mathbf{if}\;F \leq -0.00055:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq -2.1 \cdot 10^{-120}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{0.5}\\ \mathbf{elif}\;F \leq 9.5 \cdot 10^{-107}:\\ \;\;\;\;\frac{\frac{F}{B}}{F + \frac{0.5 \cdot t_1}{F}} - t_0\\ \mathbf{elif}\;F \leq 2.8 \cdot 10^{-27}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{t_1}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{\sin B \cdot \left(F + \frac{1}{F}\right)} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))) (t_1 (+ 2.0 (* x 2.0))))
   (if (<= F -0.00055)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F -2.1e-120)
       (* (/ F (sin B)) (sqrt 0.5))
       (if (<= F 9.5e-107)
         (- (/ (/ F B) (+ F (/ (* 0.5 t_1) F))) t_0)
         (if (<= F 2.8e-27)
           (/ (- (* F (sqrt (/ 1.0 t_1))) x) B)
           (- (/ F (* (sin B) (+ F (/ 1.0 F)))) t_0)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double t_1 = 2.0 + (x * 2.0);
	double tmp;
	if (F <= -0.00055) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= -2.1e-120) {
		tmp = (F / sin(B)) * sqrt(0.5);
	} else if (F <= 9.5e-107) {
		tmp = ((F / B) / (F + ((0.5 * t_1) / F))) - t_0;
	} else if (F <= 2.8e-27) {
		tmp = ((F * sqrt((1.0 / t_1))) - x) / B;
	} else {
		tmp = (F / (sin(B) * (F + (1.0 / F)))) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x / tan(b)
    t_1 = 2.0d0 + (x * 2.0d0)
    if (f <= (-0.00055d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= (-2.1d-120)) then
        tmp = (f / sin(b)) * sqrt(0.5d0)
    else if (f <= 9.5d-107) then
        tmp = ((f / b) / (f + ((0.5d0 * t_1) / f))) - t_0
    else if (f <= 2.8d-27) then
        tmp = ((f * sqrt((1.0d0 / t_1))) - x) / b
    else
        tmp = (f / (sin(b) * (f + (1.0d0 / f)))) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double t_1 = 2.0 + (x * 2.0);
	double tmp;
	if (F <= -0.00055) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= -2.1e-120) {
		tmp = (F / Math.sin(B)) * Math.sqrt(0.5);
	} else if (F <= 9.5e-107) {
		tmp = ((F / B) / (F + ((0.5 * t_1) / F))) - t_0;
	} else if (F <= 2.8e-27) {
		tmp = ((F * Math.sqrt((1.0 / t_1))) - x) / B;
	} else {
		tmp = (F / (Math.sin(B) * (F + (1.0 / F)))) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	t_1 = 2.0 + (x * 2.0)
	tmp = 0
	if F <= -0.00055:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= -2.1e-120:
		tmp = (F / math.sin(B)) * math.sqrt(0.5)
	elif F <= 9.5e-107:
		tmp = ((F / B) / (F + ((0.5 * t_1) / F))) - t_0
	elif F <= 2.8e-27:
		tmp = ((F * math.sqrt((1.0 / t_1))) - x) / B
	else:
		tmp = (F / (math.sin(B) * (F + (1.0 / F)))) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	t_1 = Float64(2.0 + Float64(x * 2.0))
	tmp = 0.0
	if (F <= -0.00055)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= -2.1e-120)
		tmp = Float64(Float64(F / sin(B)) * sqrt(0.5));
	elseif (F <= 9.5e-107)
		tmp = Float64(Float64(Float64(F / B) / Float64(F + Float64(Float64(0.5 * t_1) / F))) - t_0);
	elseif (F <= 2.8e-27)
		tmp = Float64(Float64(Float64(F * sqrt(Float64(1.0 / t_1))) - x) / B);
	else
		tmp = Float64(Float64(F / Float64(sin(B) * Float64(F + Float64(1.0 / F)))) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	t_1 = 2.0 + (x * 2.0);
	tmp = 0.0;
	if (F <= -0.00055)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= -2.1e-120)
		tmp = (F / sin(B)) * sqrt(0.5);
	elseif (F <= 9.5e-107)
		tmp = ((F / B) / (F + ((0.5 * t_1) / F))) - t_0;
	elseif (F <= 2.8e-27)
		tmp = ((F * sqrt((1.0 / t_1))) - x) / B;
	else
		tmp = (F / (sin(B) * (F + (1.0 / F)))) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.00055], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -2.1e-120], N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 9.5e-107], N[(N[(N[(F / B), $MachinePrecision] / N[(F + N[(N[(0.5 * t$95$1), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 2.8e-27], N[(N[(N[(F * N[Sqrt[N[(1.0 / t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * N[(F + N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -5.50000000000000033e-4

   1. Initial program 74.9%

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

      1. distribute-lft-neg-in 74.9%

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

      2. +-commutative 74.9%

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

      3. cancel-sign-sub-inv 74.9%

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

   3. Simplified 82.8%

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

      1. clear-num 82.7%

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

      2. inv-pow 82.7%

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

      3. fma-def 82.7%

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

      4. fma-udef 82.7%

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

      5. *-commutative 82.7%

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

      6. fma-def 82.7%

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

      7. fma-def 82.7%

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

   5. Applied egg-rr 82.7%

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

      1. unpow-1 82.7%

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

   7. Simplified 82.7%

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

   8. Taylor expanded in F around -inf 99.8%

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

   if -5.50000000000000033e-4 < F < -2.1e-120

   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. Step-by-step derivation

      1. distribute-lft-neg-in 99.6%

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

      2. +-commutative 99.6%

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

      3. cancel-sign-sub-inv 99.6%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in F around 0 99.4%

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

   5. Taylor expanded in x around 0 83.9%

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

      1. div-inv 83.8%

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

      2. *-commutative 83.8%

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

      3. associate-*l* 83.9%

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

      4. div-inv 84.2%

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

   7. Applied egg-rr 84.2%

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

   if -2.1e-120 < F < 9.4999999999999999e-107

   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. Step-by-step derivation

      1. distribute-lft-neg-in 99.5%

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

      2. +-commutative 99.5%

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

      3. cancel-sign-sub-inv 99.5%

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

   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. Step-by-step derivation

      1. clear-num 99.7%

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

      2. inv-pow 99.7%

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

      3. fma-def 99.7%

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

      4. fma-udef 99.7%

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

      5. *-commutative 99.7%

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

      6. fma-def 99.7%

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

      7. fma-def 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. unpow-1 99.7%

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

   7. Simplified 99.7%

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

      1. expm1-log1p-u 90.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(F \cdot \frac{1}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}\right)\right)} - \frac{x}{\tan B} \]

      2. expm1-udef 81.3%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(F \cdot \frac{1}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}\right)} - 1\right)} - \frac{x}{\tan B} \]

      3. un-div-inv 81.3%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      4. div-inv 81.3%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\color{blue}{\sin B \cdot \frac{1}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      5. pow-flip 81.3%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot \color{blue}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(--0.5\right)}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      6. metadata-eval 81.3%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{0.5}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      7. pow1/2 81.3%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot \color{blue}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}\right)} - 1\right) - \frac{x}{\tan B} \]

   9. Applied egg-rr 81.3%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\right)} - 1\right)} - \frac{x}{\tan B} \]
   10. Step-by-step derivation

       1. expm1-def 90.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\right)\right)} - \frac{x}{\tan B} \]

       2. expm1-log1p 99.7%

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

       3. associate-/r* 99.7%

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

   11. Simplified 99.7%

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

   12. Taylor expanded in F around inf 84.5%

       \[\leadsto \frac{\frac{F}{\sin B}}{\color{blue}{F + 0.5 \cdot \frac{2 + 2 \cdot x}{F}}} - \frac{x}{\tan B} \]

   13. Taylor expanded in B around 0 84.5%

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

       1. associate-/r* 84.5%

          \[\leadsto \color{blue}{\frac{\frac{F}{B}}{F + 0.5 \cdot \frac{2 + 2 \cdot x}{F}}} - \frac{x}{\tan B} \]

       2. associate-*r/ 84.5%

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

   15. Simplified 84.5%

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

   if 9.4999999999999999e-107 < F < 2.8e-27

   1. Initial program 99.4%

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

      1. distribute-lft-neg-in 99.4%

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

      2. +-commutative 99.4%

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

      3. cancel-sign-sub-inv 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in F around 0 99.6%

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

   5. Taylor expanded in B around 0 77.3%

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

   if 2.8e-27 < F

   1. Initial program 63.3%

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

      1. distribute-lft-neg-in 63.3%

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

      2. +-commutative 63.3%

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

      3. cancel-sign-sub-inv 63.3%

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

   3. Simplified 76.4%

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

      1. clear-num 76.3%

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

      2. inv-pow 76.3%

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

      3. fma-def 76.3%

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

      4. fma-udef 76.3%

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

      5. *-commutative 76.3%

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

      6. fma-def 76.3%

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

      7. fma-def 76.3%

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

   5. Applied egg-rr 76.3%

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

      1. unpow-1 76.3%

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

   7. Simplified 76.3%

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

      1. expm1-log1p-u 64.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(F \cdot \frac{1}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}\right)\right)} - \frac{x}{\tan B} \]

      2. expm1-udef 63.8%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(F \cdot \frac{1}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}\right)} - 1\right)} - \frac{x}{\tan B} \]

      3. un-div-inv 63.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      4. div-inv 63.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\color{blue}{\sin B \cdot \frac{1}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      5. pow-flip 63.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot \color{blue}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(--0.5\right)}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      6. metadata-eval 63.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{0.5}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      7. pow1/2 63.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot \color{blue}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}\right)} - 1\right) - \frac{x}{\tan B} \]

   9. Applied egg-rr 63.8%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\right)} - 1\right)} - \frac{x}{\tan B} \]
   10. Step-by-step derivation

       1. expm1-def 64.1%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\right)\right)} - \frac{x}{\tan B} \]

       2. expm1-log1p 76.4%

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

       3. associate-/r* 63.4%

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

   11. Simplified 63.4%

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

   12. Taylor expanded in F around inf 85.4%

       \[\leadsto \frac{\frac{F}{\sin B}}{\color{blue}{F + 0.5 \cdot \frac{2 + 2 \cdot x}{F}}} - \frac{x}{\tan B} \]

   13. Taylor expanded in x around 0 98.0%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.00055:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -2.1 \cdot 10^{-120}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{0.5}\\ \mathbf{elif}\;F \leq 9.5 \cdot 10^{-107}:\\ \;\;\;\;\frac{\frac{F}{B}}{F + \frac{0.5 \cdot \left(2 + x \cdot 2\right)}{F}} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.8 \cdot 10^{-27}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{\sin B \cdot \left(F + \frac{1}{F}\right)} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 10: 89.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.0044:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq -2.1 \cdot 10^{-120}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{0.5}\\ \mathbf{elif}\;F \leq 3.15 \cdot 10^{-27}:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{\sin B \cdot \left(F + \frac{1}{F}\right)} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -0.0044)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F -2.1e-120)
       (* (/ F (sin B)) (sqrt 0.5))
       (if (<= F 3.15e-27)
         (- (* (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))) (/ F B)) t_0)
         (- (/ F (* (sin B) (+ F (/ 1.0 F)))) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -0.0044) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= -2.1e-120) {
		tmp = (F / sin(B)) * sqrt(0.5);
	} else if (F <= 3.15e-27) {
		tmp = (sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - t_0;
	} else {
		tmp = (F / (sin(B) * (F + (1.0 / F)))) - 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 <= (-0.0044d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= (-2.1d-120)) then
        tmp = (f / sin(b)) * sqrt(0.5d0)
    else if (f <= 3.15d-27) then
        tmp = (sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))) * (f / b)) - t_0
    else
        tmp = (f / (sin(b) * (f + (1.0d0 / f)))) - 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 <= -0.0044) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= -2.1e-120) {
		tmp = (F / Math.sin(B)) * Math.sqrt(0.5);
	} else if (F <= 3.15e-27) {
		tmp = (Math.sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - t_0;
	} else {
		tmp = (F / (Math.sin(B) * (F + (1.0 / F)))) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -0.0044:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= -2.1e-120:
		tmp = (F / math.sin(B)) * math.sqrt(0.5)
	elif F <= 3.15e-27:
		tmp = (math.sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - t_0
	else:
		tmp = (F / (math.sin(B) * (F + (1.0 / F)))) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -0.0044)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= -2.1e-120)
		tmp = Float64(Float64(F / sin(B)) * sqrt(0.5));
	elseif (F <= 3.15e-27)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0)))) * Float64(F / B)) - t_0);
	else
		tmp = Float64(Float64(F / Float64(sin(B) * Float64(F + Float64(1.0 / F)))) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -0.0044)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= -2.1e-120)
		tmp = (F / sin(B)) * sqrt(0.5);
	elseif (F <= 3.15e-27)
		tmp = (sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - t_0;
	else
		tmp = (F / (sin(B) * (F + (1.0 / F)))) - 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, -0.0044], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -2.1e-120], N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.15e-27], 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[(F / N[(N[Sin[B], $MachinePrecision] * N[(F + N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.00440000000000000027

   1. Initial program 74.9%

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

      1. distribute-lft-neg-in 74.9%

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

      2. +-commutative 74.9%

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

      3. cancel-sign-sub-inv 74.9%

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

   3. Simplified 82.8%

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

      1. clear-num 82.7%

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

      2. inv-pow 82.7%

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

      3. fma-def 82.7%

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

      4. fma-udef 82.7%

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

      5. *-commutative 82.7%

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

      6. fma-def 82.7%

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

      7. fma-def 82.7%

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

   5. Applied egg-rr 82.7%

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

      1. unpow-1 82.7%

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

   7. Simplified 82.7%

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

   8. Taylor expanded in F around -inf 99.8%

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

   if -0.00440000000000000027 < F < -2.1e-120

   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. Step-by-step derivation

      1. distribute-lft-neg-in 99.6%

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

      2. +-commutative 99.6%

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

      3. cancel-sign-sub-inv 99.6%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in F around 0 99.4%

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

   5. Taylor expanded in x around 0 83.9%

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

      1. div-inv 83.8%

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

      2. *-commutative 83.8%

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

      3. associate-*l* 83.9%

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

      4. div-inv 84.2%

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

   7. Applied egg-rr 84.2%

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

   if -2.1e-120 < F < 3.15000000000000005e-27

   1. Initial program 99.5%

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

      1. distribute-lft-neg-in 99.5%

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

      2. +-commutative 99.5%

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

      3. cancel-sign-sub-inv 99.5%

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

   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. Taylor expanded in F around 0 99.7%

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

   5. Taylor expanded in B around 0 93.0%

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

   if 3.15000000000000005e-27 < F

   1. Initial program 63.3%

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

      1. distribute-lft-neg-in 63.3%

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

      2. +-commutative 63.3%

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

      3. cancel-sign-sub-inv 63.3%

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

   3. Simplified 76.4%

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

      1. clear-num 76.3%

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

      2. inv-pow 76.3%

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

      3. fma-def 76.3%

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

      4. fma-udef 76.3%

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

      5. *-commutative 76.3%

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

      6. fma-def 76.3%

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

      7. fma-def 76.3%

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

   5. Applied egg-rr 76.3%

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

      1. unpow-1 76.3%

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

   7. Simplified 76.3%

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

      1. expm1-log1p-u 64.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(F \cdot \frac{1}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}\right)\right)} - \frac{x}{\tan B} \]

      2. expm1-udef 63.8%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(F \cdot \frac{1}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}\right)} - 1\right)} - \frac{x}{\tan B} \]

      3. un-div-inv 63.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      4. div-inv 63.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\color{blue}{\sin B \cdot \frac{1}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      5. pow-flip 63.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot \color{blue}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(--0.5\right)}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      6. metadata-eval 63.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{0.5}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      7. pow1/2 63.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot \color{blue}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}\right)} - 1\right) - \frac{x}{\tan B} \]

   9. Applied egg-rr 63.8%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\right)} - 1\right)} - \frac{x}{\tan B} \]
   10. Step-by-step derivation

       1. expm1-def 64.1%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\right)\right)} - \frac{x}{\tan B} \]

       2. expm1-log1p 76.4%

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

       3. associate-/r* 63.4%

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

   11. Simplified 63.4%

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

   12. Taylor expanded in F around inf 85.4%

       \[\leadsto \frac{\frac{F}{\sin B}}{\color{blue}{F + 0.5 \cdot \frac{2 + 2 \cdot x}{F}}} - \frac{x}{\tan B} \]

   13. Taylor expanded in x around 0 98.0%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.0044:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -2.1 \cdot 10^{-120}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{0.5}\\ \mathbf{elif}\;F \leq 3.15 \cdot 10^{-27}:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{\sin B \cdot \left(F + \frac{1}{F}\right)} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 11: 85.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ t_1 := 2 + x \cdot 2\\ \mathbf{if}\;F \leq -9.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq -8 \cdot 10^{-121}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{0.5}\\ \mathbf{elif}\;F \leq 1.05 \cdot 10^{-106}:\\ \;\;\;\;\frac{\frac{F}{B}}{F + \frac{0.5 \cdot t_1}{F}} - t_0\\ \mathbf{elif}\;F \leq 3.5 \cdot 10^{-28}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{t_1}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))) (t_1 (+ 2.0 (* x 2.0))))
   (if (<= F -9.5e-6)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F -8e-121)
       (* (/ F (sin B)) (sqrt 0.5))
       (if (<= F 1.05e-106)
         (- (/ (/ F B) (+ F (/ (* 0.5 t_1) F))) t_0)
         (if (<= F 3.5e-28)
           (/ (- (* F (sqrt (/ 1.0 t_1))) x) B)
           (- (/ 1.0 (sin B)) t_0)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double t_1 = 2.0 + (x * 2.0);
	double tmp;
	if (F <= -9.5e-6) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= -8e-121) {
		tmp = (F / sin(B)) * sqrt(0.5);
	} else if (F <= 1.05e-106) {
		tmp = ((F / B) / (F + ((0.5 * t_1) / F))) - t_0;
	} else if (F <= 3.5e-28) {
		tmp = ((F * sqrt((1.0 / t_1))) - x) / B;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x / tan(b)
    t_1 = 2.0d0 + (x * 2.0d0)
    if (f <= (-9.5d-6)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= (-8d-121)) then
        tmp = (f / sin(b)) * sqrt(0.5d0)
    else if (f <= 1.05d-106) then
        tmp = ((f / b) / (f + ((0.5d0 * t_1) / f))) - t_0
    else if (f <= 3.5d-28) then
        tmp = ((f * sqrt((1.0d0 / t_1))) - x) / b
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double t_1 = 2.0 + (x * 2.0);
	double tmp;
	if (F <= -9.5e-6) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= -8e-121) {
		tmp = (F / Math.sin(B)) * Math.sqrt(0.5);
	} else if (F <= 1.05e-106) {
		tmp = ((F / B) / (F + ((0.5 * t_1) / F))) - t_0;
	} else if (F <= 3.5e-28) {
		tmp = ((F * Math.sqrt((1.0 / t_1))) - x) / B;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	t_1 = 2.0 + (x * 2.0)
	tmp = 0
	if F <= -9.5e-6:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= -8e-121:
		tmp = (F / math.sin(B)) * math.sqrt(0.5)
	elif F <= 1.05e-106:
		tmp = ((F / B) / (F + ((0.5 * t_1) / F))) - t_0
	elif F <= 3.5e-28:
		tmp = ((F * math.sqrt((1.0 / t_1))) - x) / B
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	t_1 = Float64(2.0 + Float64(x * 2.0))
	tmp = 0.0
	if (F <= -9.5e-6)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= -8e-121)
		tmp = Float64(Float64(F / sin(B)) * sqrt(0.5));
	elseif (F <= 1.05e-106)
		tmp = Float64(Float64(Float64(F / B) / Float64(F + Float64(Float64(0.5 * t_1) / F))) - t_0);
	elseif (F <= 3.5e-28)
		tmp = Float64(Float64(Float64(F * sqrt(Float64(1.0 / t_1))) - x) / B);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	t_1 = 2.0 + (x * 2.0);
	tmp = 0.0;
	if (F <= -9.5e-6)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= -8e-121)
		tmp = (F / sin(B)) * sqrt(0.5);
	elseif (F <= 1.05e-106)
		tmp = ((F / B) / (F + ((0.5 * t_1) / F))) - t_0;
	elseif (F <= 3.5e-28)
		tmp = ((F * sqrt((1.0 / t_1))) - x) / B;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -9.5e-6], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -8e-121], N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.05e-106], N[(N[(N[(F / B), $MachinePrecision] / N[(F + N[(N[(0.5 * t$95$1), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 3.5e-28], N[(N[(N[(F * N[Sqrt[N[(1.0 / t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -9.5000000000000005e-6

   1. Initial program 74.9%

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

      1. distribute-lft-neg-in 74.9%

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

      2. +-commutative 74.9%

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

      3. cancel-sign-sub-inv 74.9%

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

   3. Simplified 82.8%

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

      1. clear-num 82.7%

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

      2. inv-pow 82.7%

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

      3. fma-def 82.7%

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

      4. fma-udef 82.7%

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

      5. *-commutative 82.7%

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

      6. fma-def 82.7%

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

      7. fma-def 82.7%

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

   5. Applied egg-rr 82.7%

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

      1. unpow-1 82.7%

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

   7. Simplified 82.7%

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

   8. Taylor expanded in F around -inf 99.8%

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

   if -9.5000000000000005e-6 < F < -7.9999999999999998e-121

   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. Step-by-step derivation

      1. distribute-lft-neg-in 99.6%

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

      2. +-commutative 99.6%

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

      3. cancel-sign-sub-inv 99.6%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in F around 0 99.4%

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

   5. Taylor expanded in x around 0 83.9%

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

      1. div-inv 83.8%

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

      2. *-commutative 83.8%

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

      3. associate-*l* 83.9%

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

      4. div-inv 84.2%

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

   7. Applied egg-rr 84.2%

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

   if -7.9999999999999998e-121 < F < 1.05000000000000002e-106

   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. Step-by-step derivation

      1. distribute-lft-neg-in 99.5%

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

      2. +-commutative 99.5%

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

      3. cancel-sign-sub-inv 99.5%

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

   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. Step-by-step derivation

      1. clear-num 99.7%

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

      2. inv-pow 99.7%

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

      3. fma-def 99.7%

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

      4. fma-udef 99.7%

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

      5. *-commutative 99.7%

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

      6. fma-def 99.7%

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

      7. fma-def 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. unpow-1 99.7%

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

   7. Simplified 99.7%

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

      1. expm1-log1p-u 90.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(F \cdot \frac{1}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}\right)\right)} - \frac{x}{\tan B} \]

      2. expm1-udef 81.3%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(F \cdot \frac{1}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}\right)} - 1\right)} - \frac{x}{\tan B} \]

      3. un-div-inv 81.3%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      4. div-inv 81.3%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\color{blue}{\sin B \cdot \frac{1}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      5. pow-flip 81.3%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot \color{blue}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(--0.5\right)}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      6. metadata-eval 81.3%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{0.5}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      7. pow1/2 81.3%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot \color{blue}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}\right)} - 1\right) - \frac{x}{\tan B} \]

   9. Applied egg-rr 81.3%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\right)} - 1\right)} - \frac{x}{\tan B} \]
   10. Step-by-step derivation

       1. expm1-def 90.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\right)\right)} - \frac{x}{\tan B} \]

       2. expm1-log1p 99.7%

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

       3. associate-/r* 99.7%

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

   11. Simplified 99.7%

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

   12. Taylor expanded in F around inf 84.5%

       \[\leadsto \frac{\frac{F}{\sin B}}{\color{blue}{F + 0.5 \cdot \frac{2 + 2 \cdot x}{F}}} - \frac{x}{\tan B} \]

   13. Taylor expanded in B around 0 84.5%

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

       1. associate-/r* 84.5%

          \[\leadsto \color{blue}{\frac{\frac{F}{B}}{F + 0.5 \cdot \frac{2 + 2 \cdot x}{F}}} - \frac{x}{\tan B} \]

       2. associate-*r/ 84.5%

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

   15. Simplified 84.5%

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

   if 1.05000000000000002e-106 < F < 3.5e-28

   1. Initial program 99.4%

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

      1. distribute-lft-neg-in 99.4%

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

      2. +-commutative 99.4%

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

      3. cancel-sign-sub-inv 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in F around 0 99.6%

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

   5. Taylor expanded in B around 0 77.3%

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

   if 3.5e-28 < F

   1. Initial program 63.3%

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

      1. distribute-lft-neg-in 63.3%

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

      2. +-commutative 63.3%

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

      3. cancel-sign-sub-inv 63.3%

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

   3. Simplified 76.4%

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

      1. clear-num 76.3%

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

      2. inv-pow 76.3%

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

      3. fma-def 76.3%

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

      4. fma-udef 76.3%

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

      5. *-commutative 76.3%

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

      6. fma-def 76.3%

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

      7. fma-def 76.3%

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

   5. Applied egg-rr 76.3%

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

      1. unpow-1 76.3%

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

   7. Simplified 76.3%

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

   8. Taylor expanded in F around inf 97.8%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -9.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -8 \cdot 10^{-121}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{0.5}\\ \mathbf{elif}\;F \leq 1.05 \cdot 10^{-106}:\\ \;\;\;\;\frac{\frac{F}{B}}{F + \frac{0.5 \cdot \left(2 + x \cdot 2\right)}{F}} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 3.5 \cdot 10^{-28}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 12: 70.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F}{\sin B}\\ t_1 := 2 + x \cdot 2\\ t_2 := \frac{x}{\tan B}\\ t_3 := \frac{-1}{B} - t_2\\ \mathbf{if}\;F \leq -2.7 \cdot 10^{+107}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;F \leq -1.45 \cdot 10^{+84}:\\ \;\;\;\;t_0 \cdot \frac{-1}{F} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -9.2 \cdot 10^{-7}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;F \leq -3.2 \cdot 10^{-121}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 2.1 \cdot 10^{-106}:\\ \;\;\;\;\frac{\frac{F}{B}}{F + \frac{0.5 \cdot t_1}{F}} - t_2\\ \mathbf{elif}\;F \leq 1.65 \cdot 10^{-6}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{t_1}} - x}{B}\\ \mathbf{elif}\;F \leq 7.4 \cdot 10^{+77} \lor \neg \left(F \leq 8 \cdot 10^{+163}\right) \land F \leq 2 \cdot 10^{+206}:\\ \;\;\;\;t_0 \cdot \frac{1}{F} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ F (sin B)))
        (t_1 (+ 2.0 (* x 2.0)))
        (t_2 (/ x (tan B)))
        (t_3 (- (/ -1.0 B) t_2)))
   (if (<= F -2.7e+107)
     t_3
     (if (<= F -1.45e+84)
       (- (* t_0 (/ -1.0 F)) (/ x B))
       (if (<= F -9.2e-7)
         t_3
         (if (<= F -3.2e-121)
           (* F (/ (sqrt 0.5) (sin B)))
           (if (<= F 2.1e-106)
             (- (/ (/ F B) (+ F (/ (* 0.5 t_1) F))) t_2)
             (if (<= F 1.65e-6)
               (/ (- (* F (sqrt (/ 1.0 t_1))) x) B)
               (if (or (<= F 7.4e+77) (and (not (<= F 8e+163)) (<= F 2e+206)))
                 (- (* t_0 (/ 1.0 F)) (/ x B))
                 (- (/ 1.0 B) (* x (/ 1.0 (tan B)))))))))))))

C

double code(double F, double B, double x) {
	double t_0 = F / sin(B);
	double t_1 = 2.0 + (x * 2.0);
	double t_2 = x / tan(B);
	double t_3 = (-1.0 / B) - t_2;
	double tmp;
	if (F <= -2.7e+107) {
		tmp = t_3;
	} else if (F <= -1.45e+84) {
		tmp = (t_0 * (-1.0 / F)) - (x / B);
	} else if (F <= -9.2e-7) {
		tmp = t_3;
	} else if (F <= -3.2e-121) {
		tmp = F * (sqrt(0.5) / sin(B));
	} else if (F <= 2.1e-106) {
		tmp = ((F / B) / (F + ((0.5 * t_1) / F))) - t_2;
	} else if (F <= 1.65e-6) {
		tmp = ((F * sqrt((1.0 / t_1))) - x) / B;
	} else if ((F <= 7.4e+77) || (!(F <= 8e+163) && (F <= 2e+206))) {
		tmp = (t_0 * (1.0 / F)) - (x / B);
	} else {
		tmp = (1.0 / B) - (x * (1.0 / 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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = f / sin(b)
    t_1 = 2.0d0 + (x * 2.0d0)
    t_2 = x / tan(b)
    t_3 = ((-1.0d0) / b) - t_2
    if (f <= (-2.7d+107)) then
        tmp = t_3
    else if (f <= (-1.45d+84)) then
        tmp = (t_0 * ((-1.0d0) / f)) - (x / b)
    else if (f <= (-9.2d-7)) then
        tmp = t_3
    else if (f <= (-3.2d-121)) then
        tmp = f * (sqrt(0.5d0) / sin(b))
    else if (f <= 2.1d-106) then
        tmp = ((f / b) / (f + ((0.5d0 * t_1) / f))) - t_2
    else if (f <= 1.65d-6) then
        tmp = ((f * sqrt((1.0d0 / t_1))) - x) / b
    else if ((f <= 7.4d+77) .or. (.not. (f <= 8d+163)) .and. (f <= 2d+206)) then
        tmp = (t_0 * (1.0d0 / f)) - (x / b)
    else
        tmp = (1.0d0 / b) - (x * (1.0d0 / tan(b)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = F / Math.sin(B);
	double t_1 = 2.0 + (x * 2.0);
	double t_2 = x / Math.tan(B);
	double t_3 = (-1.0 / B) - t_2;
	double tmp;
	if (F <= -2.7e+107) {
		tmp = t_3;
	} else if (F <= -1.45e+84) {
		tmp = (t_0 * (-1.0 / F)) - (x / B);
	} else if (F <= -9.2e-7) {
		tmp = t_3;
	} else if (F <= -3.2e-121) {
		tmp = F * (Math.sqrt(0.5) / Math.sin(B));
	} else if (F <= 2.1e-106) {
		tmp = ((F / B) / (F + ((0.5 * t_1) / F))) - t_2;
	} else if (F <= 1.65e-6) {
		tmp = ((F * Math.sqrt((1.0 / t_1))) - x) / B;
	} else if ((F <= 7.4e+77) || (!(F <= 8e+163) && (F <= 2e+206))) {
		tmp = (t_0 * (1.0 / F)) - (x / B);
	} else {
		tmp = (1.0 / B) - (x * (1.0 / Math.tan(B)));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = F / math.sin(B)
	t_1 = 2.0 + (x * 2.0)
	t_2 = x / math.tan(B)
	t_3 = (-1.0 / B) - t_2
	tmp = 0
	if F <= -2.7e+107:
		tmp = t_3
	elif F <= -1.45e+84:
		tmp = (t_0 * (-1.0 / F)) - (x / B)
	elif F <= -9.2e-7:
		tmp = t_3
	elif F <= -3.2e-121:
		tmp = F * (math.sqrt(0.5) / math.sin(B))
	elif F <= 2.1e-106:
		tmp = ((F / B) / (F + ((0.5 * t_1) / F))) - t_2
	elif F <= 1.65e-6:
		tmp = ((F * math.sqrt((1.0 / t_1))) - x) / B
	elif (F <= 7.4e+77) or (not (F <= 8e+163) and (F <= 2e+206)):
		tmp = (t_0 * (1.0 / F)) - (x / B)
	else:
		tmp = (1.0 / B) - (x * (1.0 / math.tan(B)))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(F / sin(B))
	t_1 = Float64(2.0 + Float64(x * 2.0))
	t_2 = Float64(x / tan(B))
	t_3 = Float64(Float64(-1.0 / B) - t_2)
	tmp = 0.0
	if (F <= -2.7e+107)
		tmp = t_3;
	elseif (F <= -1.45e+84)
		tmp = Float64(Float64(t_0 * Float64(-1.0 / F)) - Float64(x / B));
	elseif (F <= -9.2e-7)
		tmp = t_3;
	elseif (F <= -3.2e-121)
		tmp = Float64(F * Float64(sqrt(0.5) / sin(B)));
	elseif (F <= 2.1e-106)
		tmp = Float64(Float64(Float64(F / B) / Float64(F + Float64(Float64(0.5 * t_1) / F))) - t_2);
	elseif (F <= 1.65e-6)
		tmp = Float64(Float64(Float64(F * sqrt(Float64(1.0 / t_1))) - x) / B);
	elseif ((F <= 7.4e+77) || (!(F <= 8e+163) && (F <= 2e+206)))
		tmp = Float64(Float64(t_0 * Float64(1.0 / F)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x * Float64(1.0 / tan(B))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = F / sin(B);
	t_1 = 2.0 + (x * 2.0);
	t_2 = x / tan(B);
	t_3 = (-1.0 / B) - t_2;
	tmp = 0.0;
	if (F <= -2.7e+107)
		tmp = t_3;
	elseif (F <= -1.45e+84)
		tmp = (t_0 * (-1.0 / F)) - (x / B);
	elseif (F <= -9.2e-7)
		tmp = t_3;
	elseif (F <= -3.2e-121)
		tmp = F * (sqrt(0.5) / sin(B));
	elseif (F <= 2.1e-106)
		tmp = ((F / B) / (F + ((0.5 * t_1) / F))) - t_2;
	elseif (F <= 1.65e-6)
		tmp = ((F * sqrt((1.0 / t_1))) - x) / B;
	elseif ((F <= 7.4e+77) || (~((F <= 8e+163)) && (F <= 2e+206)))
		tmp = (t_0 * (1.0 / F)) - (x / B);
	else
		tmp = (1.0 / B) - (x * (1.0 / tan(B)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(-1.0 / B), $MachinePrecision] - t$95$2), $MachinePrecision]}, If[LessEqual[F, -2.7e+107], t$95$3, If[LessEqual[F, -1.45e+84], N[(N[(t$95$0 * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -9.2e-7], t$95$3, If[LessEqual[F, -3.2e-121], N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.1e-106], N[(N[(N[(F / B), $MachinePrecision] / N[(F + N[(N[(0.5 * t$95$1), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision], If[LessEqual[F, 1.65e-6], N[(N[(N[(F * N[Sqrt[N[(1.0 / t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[Or[LessEqual[F, 7.4e+77], And[N[Not[LessEqual[F, 8e+163]], $MachinePrecision], LessEqual[F, 2e+206]]], N[(N[(t$95$0 * N[(1.0 / F), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if F < -2.7000000000000001e107 or -1.44999999999999994e84 < F < -9.1999999999999998e-7

   1. Initial program 71.3%

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

      1. distribute-lft-neg-in 71.3%

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

      2. +-commutative 71.3%

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

      3. cancel-sign-sub-inv 71.3%

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

   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. Taylor expanded in F around -inf 99.6%

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

   5. Taylor expanded in B around 0 82.2%

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

   if -2.7000000000000001e107 < F < -1.44999999999999994e84

   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. Taylor expanded in F around -inf 99.6%

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

   3. Taylor expanded in B around 0 87.8%

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

   if -9.1999999999999998e-7 < F < -3.20000000000000019e-121

   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. Step-by-step derivation

      1. distribute-lft-neg-in 99.6%

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

      2. +-commutative 99.6%

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

      3. cancel-sign-sub-inv 99.6%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in F around 0 99.4%

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

   5. Taylor expanded in x around 0 83.9%

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

      1. expm1-log1p-u 66.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{F \cdot \sqrt{0.5}}{\sin B}\right)\right)} \]

      2. expm1-udef 18.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{F \cdot \sqrt{0.5}}{\sin B}\right)} - 1} \]

      3. associate-/l* 18.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{F}{\frac{\sin B}{\sqrt{0.5}}}}\right)} - 1 \]

   7. Applied egg-rr 18.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{F}{\frac{\sin B}{\sqrt{0.5}}}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 66.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{F}{\frac{\sin B}{\sqrt{0.5}}}\right)\right)} \]

      2. expm1-log1p 84.0%

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

      3. associate-/r/ 84.2%

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

      4. *-commutative 84.2%

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

      5. associate-*r/ 83.9%

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

      6. *-commutative 83.9%

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

      7. associate-*r/ 84.0%

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

   9. Simplified 84.0%

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

   if -3.20000000000000019e-121 < F < 2.10000000000000003e-106

   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. Step-by-step derivation

      1. distribute-lft-neg-in 99.5%

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

      2. +-commutative 99.5%

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

      3. cancel-sign-sub-inv 99.5%

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

   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. Step-by-step derivation

      1. clear-num 99.7%

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

      2. inv-pow 99.7%

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

      3. fma-def 99.7%

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

      4. fma-udef 99.7%

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

      5. *-commutative 99.7%

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

      6. fma-def 99.7%

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

      7. fma-def 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. unpow-1 99.7%

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

   7. Simplified 99.7%

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

      1. expm1-log1p-u 90.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(F \cdot \frac{1}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}\right)\right)} - \frac{x}{\tan B} \]

      2. expm1-udef 81.3%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(F \cdot \frac{1}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}\right)} - 1\right)} - \frac{x}{\tan B} \]

      3. un-div-inv 81.3%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      4. div-inv 81.3%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\color{blue}{\sin B \cdot \frac{1}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      5. pow-flip 81.3%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot \color{blue}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(--0.5\right)}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      6. metadata-eval 81.3%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{0.5}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      7. pow1/2 81.3%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot \color{blue}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}\right)} - 1\right) - \frac{x}{\tan B} \]

   9. Applied egg-rr 81.3%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\right)} - 1\right)} - \frac{x}{\tan B} \]
   10. Step-by-step derivation

       1. expm1-def 90.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\right)\right)} - \frac{x}{\tan B} \]

       2. expm1-log1p 99.7%

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

       3. associate-/r* 99.7%

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

   11. Simplified 99.7%

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

   12. Taylor expanded in F around inf 84.5%

       \[\leadsto \frac{\frac{F}{\sin B}}{\color{blue}{F + 0.5 \cdot \frac{2 + 2 \cdot x}{F}}} - \frac{x}{\tan B} \]

   13. Taylor expanded in B around 0 84.5%

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

       1. associate-/r* 84.5%

          \[\leadsto \color{blue}{\frac{\frac{F}{B}}{F + 0.5 \cdot \frac{2 + 2 \cdot x}{F}}} - \frac{x}{\tan B} \]

       2. associate-*r/ 84.5%

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

   15. Simplified 84.5%

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

   if 2.10000000000000003e-106 < F < 1.65000000000000008e-6

   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. Step-by-step derivation

      1. distribute-lft-neg-in 99.5%

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

      2. +-commutative 99.5%

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

      3. cancel-sign-sub-inv 99.5%

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

   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. Taylor expanded in F around 0 99.7%

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

   5. Taylor expanded in B around 0 75.6%

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

   if 1.65000000000000008e-6 < F < 7.3999999999999999e77 or 7.9999999999999995e163 < F < 2.0000000000000001e206

   1. Initial program 61.8%

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

   2. Taylor expanded in F around inf 88.7%

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

   3. Taylor expanded in B around 0 84.4%

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

   if 7.3999999999999999e77 < F < 7.9999999999999995e163 or 2.0000000000000001e206 < F

   1. Initial program 63.2%

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

   2. Taylor expanded in F around inf 83.0%

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

   3. Taylor expanded in B around 0 85.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{B}} \]
3. Recombined 7 regimes into one program.

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.7 \cdot 10^{+107}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.45 \cdot 10^{+84}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \frac{-1}{F} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -9.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -3.2 \cdot 10^{-121}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 2.1 \cdot 10^{-106}:\\ \;\;\;\;\frac{\frac{F}{B}}{F + \frac{0.5 \cdot \left(2 + x \cdot 2\right)}{F}} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.65 \cdot 10^{-6}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{elif}\;F \leq 7.4 \cdot 10^{+77} \lor \neg \left(F \leq 8 \cdot 10^{+163}\right) \land F \leq 2 \cdot 10^{+206}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \frac{1}{F} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \end{array} \]

Alternative 13: 70.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F}{\sin B}\\ t_1 := 2 + x \cdot 2\\ t_2 := \frac{x}{\tan B}\\ t_3 := \frac{-1}{B} - t_2\\ \mathbf{if}\;F \leq -2.1 \cdot 10^{+103}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;F \leq -2 \cdot 10^{+82}:\\ \;\;\;\;t_0 \cdot \frac{-1}{F} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -0.175:\\ \;\;\;\;t_3\\ \mathbf{elif}\;F \leq -2.1 \cdot 10^{-123}:\\ \;\;\;\;t_0 \cdot \sqrt{0.5}\\ \mathbf{elif}\;F \leq 3.5 \cdot 10^{-106}:\\ \;\;\;\;\frac{\frac{F}{B}}{F + \frac{0.5 \cdot t_1}{F}} - t_2\\ \mathbf{elif}\;F \leq 1.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{t_1}} - x}{B}\\ \mathbf{elif}\;F \leq 7.4 \cdot 10^{+81} \lor \neg \left(F \leq 9 \cdot 10^{+163}\right) \land F \leq 3.9 \cdot 10^{+206}:\\ \;\;\;\;t_0 \cdot \frac{1}{F} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ F (sin B)))
        (t_1 (+ 2.0 (* x 2.0)))
        (t_2 (/ x (tan B)))
        (t_3 (- (/ -1.0 B) t_2)))
   (if (<= F -2.1e+103)
     t_3
     (if (<= F -2e+82)
       (- (* t_0 (/ -1.0 F)) (/ x B))
       (if (<= F -0.175)
         t_3
         (if (<= F -2.1e-123)
           (* t_0 (sqrt 0.5))
           (if (<= F 3.5e-106)
             (- (/ (/ F B) (+ F (/ (* 0.5 t_1) F))) t_2)
             (if (<= F 1.6e-6)
               (/ (- (* F (sqrt (/ 1.0 t_1))) x) B)
               (if (or (<= F 7.4e+81)
                       (and (not (<= F 9e+163)) (<= F 3.9e+206)))
                 (- (* t_0 (/ 1.0 F)) (/ x B))
                 (- (/ 1.0 B) (* x (/ 1.0 (tan B)))))))))))))

C

double code(double F, double B, double x) {
	double t_0 = F / sin(B);
	double t_1 = 2.0 + (x * 2.0);
	double t_2 = x / tan(B);
	double t_3 = (-1.0 / B) - t_2;
	double tmp;
	if (F <= -2.1e+103) {
		tmp = t_3;
	} else if (F <= -2e+82) {
		tmp = (t_0 * (-1.0 / F)) - (x / B);
	} else if (F <= -0.175) {
		tmp = t_3;
	} else if (F <= -2.1e-123) {
		tmp = t_0 * sqrt(0.5);
	} else if (F <= 3.5e-106) {
		tmp = ((F / B) / (F + ((0.5 * t_1) / F))) - t_2;
	} else if (F <= 1.6e-6) {
		tmp = ((F * sqrt((1.0 / t_1))) - x) / B;
	} else if ((F <= 7.4e+81) || (!(F <= 9e+163) && (F <= 3.9e+206))) {
		tmp = (t_0 * (1.0 / F)) - (x / B);
	} else {
		tmp = (1.0 / B) - (x * (1.0 / 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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = f / sin(b)
    t_1 = 2.0d0 + (x * 2.0d0)
    t_2 = x / tan(b)
    t_3 = ((-1.0d0) / b) - t_2
    if (f <= (-2.1d+103)) then
        tmp = t_3
    else if (f <= (-2d+82)) then
        tmp = (t_0 * ((-1.0d0) / f)) - (x / b)
    else if (f <= (-0.175d0)) then
        tmp = t_3
    else if (f <= (-2.1d-123)) then
        tmp = t_0 * sqrt(0.5d0)
    else if (f <= 3.5d-106) then
        tmp = ((f / b) / (f + ((0.5d0 * t_1) / f))) - t_2
    else if (f <= 1.6d-6) then
        tmp = ((f * sqrt((1.0d0 / t_1))) - x) / b
    else if ((f <= 7.4d+81) .or. (.not. (f <= 9d+163)) .and. (f <= 3.9d+206)) then
        tmp = (t_0 * (1.0d0 / f)) - (x / b)
    else
        tmp = (1.0d0 / b) - (x * (1.0d0 / tan(b)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = F / Math.sin(B);
	double t_1 = 2.0 + (x * 2.0);
	double t_2 = x / Math.tan(B);
	double t_3 = (-1.0 / B) - t_2;
	double tmp;
	if (F <= -2.1e+103) {
		tmp = t_3;
	} else if (F <= -2e+82) {
		tmp = (t_0 * (-1.0 / F)) - (x / B);
	} else if (F <= -0.175) {
		tmp = t_3;
	} else if (F <= -2.1e-123) {
		tmp = t_0 * Math.sqrt(0.5);
	} else if (F <= 3.5e-106) {
		tmp = ((F / B) / (F + ((0.5 * t_1) / F))) - t_2;
	} else if (F <= 1.6e-6) {
		tmp = ((F * Math.sqrt((1.0 / t_1))) - x) / B;
	} else if ((F <= 7.4e+81) || (!(F <= 9e+163) && (F <= 3.9e+206))) {
		tmp = (t_0 * (1.0 / F)) - (x / B);
	} else {
		tmp = (1.0 / B) - (x * (1.0 / Math.tan(B)));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = F / math.sin(B)
	t_1 = 2.0 + (x * 2.0)
	t_2 = x / math.tan(B)
	t_3 = (-1.0 / B) - t_2
	tmp = 0
	if F <= -2.1e+103:
		tmp = t_3
	elif F <= -2e+82:
		tmp = (t_0 * (-1.0 / F)) - (x / B)
	elif F <= -0.175:
		tmp = t_3
	elif F <= -2.1e-123:
		tmp = t_0 * math.sqrt(0.5)
	elif F <= 3.5e-106:
		tmp = ((F / B) / (F + ((0.5 * t_1) / F))) - t_2
	elif F <= 1.6e-6:
		tmp = ((F * math.sqrt((1.0 / t_1))) - x) / B
	elif (F <= 7.4e+81) or (not (F <= 9e+163) and (F <= 3.9e+206)):
		tmp = (t_0 * (1.0 / F)) - (x / B)
	else:
		tmp = (1.0 / B) - (x * (1.0 / math.tan(B)))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(F / sin(B))
	t_1 = Float64(2.0 + Float64(x * 2.0))
	t_2 = Float64(x / tan(B))
	t_3 = Float64(Float64(-1.0 / B) - t_2)
	tmp = 0.0
	if (F <= -2.1e+103)
		tmp = t_3;
	elseif (F <= -2e+82)
		tmp = Float64(Float64(t_0 * Float64(-1.0 / F)) - Float64(x / B));
	elseif (F <= -0.175)
		tmp = t_3;
	elseif (F <= -2.1e-123)
		tmp = Float64(t_0 * sqrt(0.5));
	elseif (F <= 3.5e-106)
		tmp = Float64(Float64(Float64(F / B) / Float64(F + Float64(Float64(0.5 * t_1) / F))) - t_2);
	elseif (F <= 1.6e-6)
		tmp = Float64(Float64(Float64(F * sqrt(Float64(1.0 / t_1))) - x) / B);
	elseif ((F <= 7.4e+81) || (!(F <= 9e+163) && (F <= 3.9e+206)))
		tmp = Float64(Float64(t_0 * Float64(1.0 / F)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x * Float64(1.0 / tan(B))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = F / sin(B);
	t_1 = 2.0 + (x * 2.0);
	t_2 = x / tan(B);
	t_3 = (-1.0 / B) - t_2;
	tmp = 0.0;
	if (F <= -2.1e+103)
		tmp = t_3;
	elseif (F <= -2e+82)
		tmp = (t_0 * (-1.0 / F)) - (x / B);
	elseif (F <= -0.175)
		tmp = t_3;
	elseif (F <= -2.1e-123)
		tmp = t_0 * sqrt(0.5);
	elseif (F <= 3.5e-106)
		tmp = ((F / B) / (F + ((0.5 * t_1) / F))) - t_2;
	elseif (F <= 1.6e-6)
		tmp = ((F * sqrt((1.0 / t_1))) - x) / B;
	elseif ((F <= 7.4e+81) || (~((F <= 9e+163)) && (F <= 3.9e+206)))
		tmp = (t_0 * (1.0 / F)) - (x / B);
	else
		tmp = (1.0 / B) - (x * (1.0 / tan(B)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(-1.0 / B), $MachinePrecision] - t$95$2), $MachinePrecision]}, If[LessEqual[F, -2.1e+103], t$95$3, If[LessEqual[F, -2e+82], N[(N[(t$95$0 * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -0.175], t$95$3, If[LessEqual[F, -2.1e-123], N[(t$95$0 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.5e-106], N[(N[(N[(F / B), $MachinePrecision] / N[(F + N[(N[(0.5 * t$95$1), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision], If[LessEqual[F, 1.6e-6], N[(N[(N[(F * N[Sqrt[N[(1.0 / t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[Or[LessEqual[F, 7.4e+81], And[N[Not[LessEqual[F, 9e+163]], $MachinePrecision], LessEqual[F, 3.9e+206]]], N[(N[(t$95$0 * N[(1.0 / F), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if F < -2.1000000000000002e103 or -1.9999999999999999e82 < F < -0.17499999999999999

   1. Initial program 71.3%

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

      1. distribute-lft-neg-in 71.3%

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

      2. +-commutative 71.3%

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

      3. cancel-sign-sub-inv 71.3%

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

   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. Taylor expanded in F around -inf 99.6%

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

   5. Taylor expanded in B around 0 82.2%

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

   if -2.1000000000000002e103 < F < -1.9999999999999999e82

   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. Taylor expanded in F around -inf 99.6%

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

   3. Taylor expanded in B around 0 87.8%

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

   if -0.17499999999999999 < F < -2.0999999999999999e-123

   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. Step-by-step derivation

      1. distribute-lft-neg-in 99.6%

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

      2. +-commutative 99.6%

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

      3. cancel-sign-sub-inv 99.6%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in F around 0 99.4%

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

   5. Taylor expanded in x around 0 83.9%

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

      1. div-inv 83.8%

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

      2. *-commutative 83.8%

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

      3. associate-*l* 83.9%

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

      4. div-inv 84.2%

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

   7. Applied egg-rr 84.2%

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

   if -2.0999999999999999e-123 < F < 3.5e-106

   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. Step-by-step derivation

      1. distribute-lft-neg-in 99.5%

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

      2. +-commutative 99.5%

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

      3. cancel-sign-sub-inv 99.5%

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

   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. Step-by-step derivation

      1. clear-num 99.7%

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

      2. inv-pow 99.7%

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

      3. fma-def 99.7%

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

      4. fma-udef 99.7%

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

      5. *-commutative 99.7%

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

      6. fma-def 99.7%

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

      7. fma-def 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. unpow-1 99.7%

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

   7. Simplified 99.7%

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

      1. expm1-log1p-u 90.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(F \cdot \frac{1}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}\right)\right)} - \frac{x}{\tan B} \]

      2. expm1-udef 81.3%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(F \cdot \frac{1}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}\right)} - 1\right)} - \frac{x}{\tan B} \]

      3. un-div-inv 81.3%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      4. div-inv 81.3%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\color{blue}{\sin B \cdot \frac{1}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      5. pow-flip 81.3%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot \color{blue}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(--0.5\right)}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      6. metadata-eval 81.3%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{0.5}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      7. pow1/2 81.3%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot \color{blue}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}\right)} - 1\right) - \frac{x}{\tan B} \]

   9. Applied egg-rr 81.3%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\right)} - 1\right)} - \frac{x}{\tan B} \]
   10. Step-by-step derivation

       1. expm1-def 90.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\right)\right)} - \frac{x}{\tan B} \]

       2. expm1-log1p 99.7%

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

       3. associate-/r* 99.7%

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

   11. Simplified 99.7%

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

   12. Taylor expanded in F around inf 84.5%

       \[\leadsto \frac{\frac{F}{\sin B}}{\color{blue}{F + 0.5 \cdot \frac{2 + 2 \cdot x}{F}}} - \frac{x}{\tan B} \]

   13. Taylor expanded in B around 0 84.5%

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

       1. associate-/r* 84.5%

          \[\leadsto \color{blue}{\frac{\frac{F}{B}}{F + 0.5 \cdot \frac{2 + 2 \cdot x}{F}}} - \frac{x}{\tan B} \]

       2. associate-*r/ 84.5%

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

   15. Simplified 84.5%

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

   if 3.5e-106 < F < 1.5999999999999999e-6

   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. Step-by-step derivation

      1. distribute-lft-neg-in 99.5%

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

      2. +-commutative 99.5%

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

      3. cancel-sign-sub-inv 99.5%

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

   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. Taylor expanded in F around 0 99.7%

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

   5. Taylor expanded in B around 0 75.6%

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

   if 1.5999999999999999e-6 < F < 7.4000000000000001e81 or 8.99999999999999976e163 < F < 3.9e206

   1. Initial program 61.8%

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

   2. Taylor expanded in F around inf 88.7%

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

   3. Taylor expanded in B around 0 84.4%

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

   if 7.4000000000000001e81 < F < 8.99999999999999976e163 or 3.9e206 < F

   1. Initial program 63.2%

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

   2. Taylor expanded in F around inf 83.0%

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

   3. Taylor expanded in B around 0 85.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{B}} \]
3. Recombined 7 regimes into one program.

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.1 \cdot 10^{+103}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -2 \cdot 10^{+82}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \frac{-1}{F} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -0.175:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -2.1 \cdot 10^{-123}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{0.5}\\ \mathbf{elif}\;F \leq 3.5 \cdot 10^{-106}:\\ \;\;\;\;\frac{\frac{F}{B}}{F + \frac{0.5 \cdot \left(2 + x \cdot 2\right)}{F}} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{elif}\;F \leq 7.4 \cdot 10^{+81} \lor \neg \left(F \leq 9 \cdot 10^{+163}\right) \land F \leq 3.9 \cdot 10^{+206}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \frac{1}{F} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \end{array} \]

Alternative 14: 77.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F}{\sin B}\\ t_1 := \frac{x}{\tan B}\\ t_2 := 2 + x \cdot 2\\ \mathbf{if}\;F \leq -0.00041:\\ \;\;\;\;\frac{-1}{\sin B} - t_1\\ \mathbf{elif}\;F \leq -1.45 \cdot 10^{-125}:\\ \;\;\;\;t_0 \cdot \sqrt{0.5}\\ \mathbf{elif}\;F \leq 5.8 \cdot 10^{-106}:\\ \;\;\;\;\frac{\frac{F}{B}}{F + \frac{0.5 \cdot t_2}{F}} - t_1\\ \mathbf{elif}\;F \leq 1.65 \cdot 10^{-6}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{t_2}} - x}{B}\\ \mathbf{elif}\;F \leq 1.32 \cdot 10^{+80} \lor \neg \left(F \leq 6.5 \cdot 10^{+163}\right) \land F \leq 2.6 \cdot 10^{+206}:\\ \;\;\;\;t_0 \cdot \frac{1}{F} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ F (sin B))) (t_1 (/ x (tan B))) (t_2 (+ 2.0 (* x 2.0))))
   (if (<= F -0.00041)
     (- (/ -1.0 (sin B)) t_1)
     (if (<= F -1.45e-125)
       (* t_0 (sqrt 0.5))
       (if (<= F 5.8e-106)
         (- (/ (/ F B) (+ F (/ (* 0.5 t_2) F))) t_1)
         (if (<= F 1.65e-6)
           (/ (- (* F (sqrt (/ 1.0 t_2))) x) B)
           (if (or (<= F 1.32e+80) (and (not (<= F 6.5e+163)) (<= F 2.6e+206)))
             (- (* t_0 (/ 1.0 F)) (/ x B))
             (- (/ 1.0 B) (* x (/ 1.0 (tan B)))))))))))

C

double code(double F, double B, double x) {
	double t_0 = F / sin(B);
	double t_1 = x / tan(B);
	double t_2 = 2.0 + (x * 2.0);
	double tmp;
	if (F <= -0.00041) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= -1.45e-125) {
		tmp = t_0 * sqrt(0.5);
	} else if (F <= 5.8e-106) {
		tmp = ((F / B) / (F + ((0.5 * t_2) / F))) - t_1;
	} else if (F <= 1.65e-6) {
		tmp = ((F * sqrt((1.0 / t_2))) - x) / B;
	} else if ((F <= 1.32e+80) || (!(F <= 6.5e+163) && (F <= 2.6e+206))) {
		tmp = (t_0 * (1.0 / F)) - (x / B);
	} else {
		tmp = (1.0 / B) - (x * (1.0 / 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) :: t_2
    real(8) :: tmp
    t_0 = f / sin(b)
    t_1 = x / tan(b)
    t_2 = 2.0d0 + (x * 2.0d0)
    if (f <= (-0.00041d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_1
    else if (f <= (-1.45d-125)) then
        tmp = t_0 * sqrt(0.5d0)
    else if (f <= 5.8d-106) then
        tmp = ((f / b) / (f + ((0.5d0 * t_2) / f))) - t_1
    else if (f <= 1.65d-6) then
        tmp = ((f * sqrt((1.0d0 / t_2))) - x) / b
    else if ((f <= 1.32d+80) .or. (.not. (f <= 6.5d+163)) .and. (f <= 2.6d+206)) then
        tmp = (t_0 * (1.0d0 / f)) - (x / b)
    else
        tmp = (1.0d0 / b) - (x * (1.0d0 / tan(b)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = F / Math.sin(B);
	double t_1 = x / Math.tan(B);
	double t_2 = 2.0 + (x * 2.0);
	double tmp;
	if (F <= -0.00041) {
		tmp = (-1.0 / Math.sin(B)) - t_1;
	} else if (F <= -1.45e-125) {
		tmp = t_0 * Math.sqrt(0.5);
	} else if (F <= 5.8e-106) {
		tmp = ((F / B) / (F + ((0.5 * t_2) / F))) - t_1;
	} else if (F <= 1.65e-6) {
		tmp = ((F * Math.sqrt((1.0 / t_2))) - x) / B;
	} else if ((F <= 1.32e+80) || (!(F <= 6.5e+163) && (F <= 2.6e+206))) {
		tmp = (t_0 * (1.0 / F)) - (x / B);
	} else {
		tmp = (1.0 / B) - (x * (1.0 / Math.tan(B)));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = F / math.sin(B)
	t_1 = x / math.tan(B)
	t_2 = 2.0 + (x * 2.0)
	tmp = 0
	if F <= -0.00041:
		tmp = (-1.0 / math.sin(B)) - t_1
	elif F <= -1.45e-125:
		tmp = t_0 * math.sqrt(0.5)
	elif F <= 5.8e-106:
		tmp = ((F / B) / (F + ((0.5 * t_2) / F))) - t_1
	elif F <= 1.65e-6:
		tmp = ((F * math.sqrt((1.0 / t_2))) - x) / B
	elif (F <= 1.32e+80) or (not (F <= 6.5e+163) and (F <= 2.6e+206)):
		tmp = (t_0 * (1.0 / F)) - (x / B)
	else:
		tmp = (1.0 / B) - (x * (1.0 / math.tan(B)))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(F / sin(B))
	t_1 = Float64(x / tan(B))
	t_2 = Float64(2.0 + Float64(x * 2.0))
	tmp = 0.0
	if (F <= -0.00041)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= -1.45e-125)
		tmp = Float64(t_0 * sqrt(0.5));
	elseif (F <= 5.8e-106)
		tmp = Float64(Float64(Float64(F / B) / Float64(F + Float64(Float64(0.5 * t_2) / F))) - t_1);
	elseif (F <= 1.65e-6)
		tmp = Float64(Float64(Float64(F * sqrt(Float64(1.0 / t_2))) - x) / B);
	elseif ((F <= 1.32e+80) || (!(F <= 6.5e+163) && (F <= 2.6e+206)))
		tmp = Float64(Float64(t_0 * Float64(1.0 / F)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x * Float64(1.0 / tan(B))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = F / sin(B);
	t_1 = x / tan(B);
	t_2 = 2.0 + (x * 2.0);
	tmp = 0.0;
	if (F <= -0.00041)
		tmp = (-1.0 / sin(B)) - t_1;
	elseif (F <= -1.45e-125)
		tmp = t_0 * sqrt(0.5);
	elseif (F <= 5.8e-106)
		tmp = ((F / B) / (F + ((0.5 * t_2) / F))) - t_1;
	elseif (F <= 1.65e-6)
		tmp = ((F * sqrt((1.0 / t_2))) - x) / B;
	elseif ((F <= 1.32e+80) || (~((F <= 6.5e+163)) && (F <= 2.6e+206)))
		tmp = (t_0 * (1.0 / F)) - (x / B);
	else
		tmp = (1.0 / B) - (x * (1.0 / tan(B)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.00041], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, -1.45e-125], N[(t$95$0 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.8e-106], N[(N[(N[(F / B), $MachinePrecision] / N[(F + N[(N[(0.5 * t$95$2), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, 1.65e-6], N[(N[(N[(F * N[Sqrt[N[(1.0 / t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[Or[LessEqual[F, 1.32e+80], And[N[Not[LessEqual[F, 6.5e+163]], $MachinePrecision], LessEqual[F, 2.6e+206]]], N[(N[(t$95$0 * N[(1.0 / F), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if F < -4.0999999999999999e-4

   1. Initial program 74.9%

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

      1. distribute-lft-neg-in 74.9%

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

      2. +-commutative 74.9%

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

      3. cancel-sign-sub-inv 74.9%

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

   3. Simplified 82.8%

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

      1. clear-num 82.7%

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

      2. inv-pow 82.7%

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

      3. fma-def 82.7%

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

      4. fma-udef 82.7%

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

      5. *-commutative 82.7%

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

      6. fma-def 82.7%

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

      7. fma-def 82.7%

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

   5. Applied egg-rr 82.7%

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

      1. unpow-1 82.7%

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

   7. Simplified 82.7%

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

   8. Taylor expanded in F around -inf 99.8%

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

   if -4.0999999999999999e-4 < F < -1.4500000000000001e-125

   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. Step-by-step derivation

      1. distribute-lft-neg-in 99.6%

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

      2. +-commutative 99.6%

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

      3. cancel-sign-sub-inv 99.6%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in F around 0 99.4%

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

   5. Taylor expanded in x around 0 83.9%

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

      1. div-inv 83.8%

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

      2. *-commutative 83.8%

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

      3. associate-*l* 83.9%

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

      4. div-inv 84.2%

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

   7. Applied egg-rr 84.2%

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

   if -1.4500000000000001e-125 < F < 5.8000000000000001e-106

   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. Step-by-step derivation

      1. distribute-lft-neg-in 99.5%

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

      2. +-commutative 99.5%

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

      3. cancel-sign-sub-inv 99.5%

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

   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. Step-by-step derivation

      1. clear-num 99.7%

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

      2. inv-pow 99.7%

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

      3. fma-def 99.7%

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

      4. fma-udef 99.7%

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

      5. *-commutative 99.7%

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

      6. fma-def 99.7%

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

      7. fma-def 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. unpow-1 99.7%

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

   7. Simplified 99.7%

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

      1. expm1-log1p-u 90.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(F \cdot \frac{1}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}\right)\right)} - \frac{x}{\tan B} \]

      2. expm1-udef 81.3%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(F \cdot \frac{1}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}\right)} - 1\right)} - \frac{x}{\tan B} \]

      3. un-div-inv 81.3%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      4. div-inv 81.3%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\color{blue}{\sin B \cdot \frac{1}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      5. pow-flip 81.3%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot \color{blue}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(--0.5\right)}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      6. metadata-eval 81.3%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{0.5}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      7. pow1/2 81.3%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot \color{blue}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}\right)} - 1\right) - \frac{x}{\tan B} \]

   9. Applied egg-rr 81.3%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\right)} - 1\right)} - \frac{x}{\tan B} \]
   10. Step-by-step derivation

       1. expm1-def 90.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\right)\right)} - \frac{x}{\tan B} \]

       2. expm1-log1p 99.7%

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

       3. associate-/r* 99.7%

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

   11. Simplified 99.7%

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

   12. Taylor expanded in F around inf 84.5%

       \[\leadsto \frac{\frac{F}{\sin B}}{\color{blue}{F + 0.5 \cdot \frac{2 + 2 \cdot x}{F}}} - \frac{x}{\tan B} \]

   13. Taylor expanded in B around 0 84.5%

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

       1. associate-/r* 84.5%

          \[\leadsto \color{blue}{\frac{\frac{F}{B}}{F + 0.5 \cdot \frac{2 + 2 \cdot x}{F}}} - \frac{x}{\tan B} \]

       2. associate-*r/ 84.5%

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

   15. Simplified 84.5%

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

   if 5.8000000000000001e-106 < F < 1.65000000000000008e-6

   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. Step-by-step derivation

      1. distribute-lft-neg-in 99.5%

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

      2. +-commutative 99.5%

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

      3. cancel-sign-sub-inv 99.5%

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

   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. Taylor expanded in F around 0 99.7%

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

   5. Taylor expanded in B around 0 75.6%

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

   if 1.65000000000000008e-6 < F < 1.3200000000000001e80 or 6.4999999999999998e163 < F < 2.59999999999999989e206

   1. Initial program 61.8%

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

   2. Taylor expanded in F around inf 88.7%

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

   3. Taylor expanded in B around 0 84.4%

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

   if 1.3200000000000001e80 < F < 6.4999999999999998e163 or 2.59999999999999989e206 < F

   1. Initial program 63.2%

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

   2. Taylor expanded in F around inf 83.0%

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

   3. Taylor expanded in B around 0 85.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{B}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.00041:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.45 \cdot 10^{-125}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{0.5}\\ \mathbf{elif}\;F \leq 5.8 \cdot 10^{-106}:\\ \;\;\;\;\frac{\frac{F}{B}}{F + \frac{0.5 \cdot \left(2 + x \cdot 2\right)}{F}} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.65 \cdot 10^{-6}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{elif}\;F \leq 1.32 \cdot 10^{+80} \lor \neg \left(F \leq 6.5 \cdot 10^{+163}\right) \land F \leq 2.6 \cdot 10^{+206}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \frac{1}{F} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \end{array} \]

Alternative 15: 70.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ t_1 := \frac{-1}{B} - t_0\\ t_2 := \frac{F}{\sin B}\\ t_3 := 2 + x \cdot 2\\ t_4 := \frac{F \cdot \sqrt{\frac{1}{t_3}} - x}{B}\\ \mathbf{if}\;F \leq -4.8 \cdot 10^{+105}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq -1.65 \cdot 10^{+82}:\\ \;\;\;\;t_2 \cdot \frac{-1}{F} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -600000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq -3.5 \cdot 10^{-83}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;F \leq 6.5 \cdot 10^{-107}:\\ \;\;\;\;\frac{\frac{F}{B}}{F + \frac{0.5 \cdot t_3}{F}} - t_0\\ \mathbf{elif}\;F \leq 1.65 \cdot 10^{-6}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;F \leq 2.2 \cdot 10^{+81} \lor \neg \left(F \leq 2.1 \cdot 10^{+163}\right) \land F \leq 9 \cdot 10^{+206}:\\ \;\;\;\;t_2 \cdot \frac{1}{F} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B)))
        (t_1 (- (/ -1.0 B) t_0))
        (t_2 (/ F (sin B)))
        (t_3 (+ 2.0 (* x 2.0)))
        (t_4 (/ (- (* F (sqrt (/ 1.0 t_3))) x) B)))
   (if (<= F -4.8e+105)
     t_1
     (if (<= F -1.65e+82)
       (- (* t_2 (/ -1.0 F)) (/ x B))
       (if (<= F -600000.0)
         t_1
         (if (<= F -3.5e-83)
           t_4
           (if (<= F 6.5e-107)
             (- (/ (/ F B) (+ F (/ (* 0.5 t_3) F))) t_0)
             (if (<= F 1.65e-6)
               t_4
               (if (or (<= F 2.2e+81)
                       (and (not (<= F 2.1e+163)) (<= F 9e+206)))
                 (- (* t_2 (/ 1.0 F)) (/ x B))
                 (- (/ 1.0 B) (* x (/ 1.0 (tan B)))))))))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double t_1 = (-1.0 / B) - t_0;
	double t_2 = F / sin(B);
	double t_3 = 2.0 + (x * 2.0);
	double t_4 = ((F * sqrt((1.0 / t_3))) - x) / B;
	double tmp;
	if (F <= -4.8e+105) {
		tmp = t_1;
	} else if (F <= -1.65e+82) {
		tmp = (t_2 * (-1.0 / F)) - (x / B);
	} else if (F <= -600000.0) {
		tmp = t_1;
	} else if (F <= -3.5e-83) {
		tmp = t_4;
	} else if (F <= 6.5e-107) {
		tmp = ((F / B) / (F + ((0.5 * t_3) / F))) - t_0;
	} else if (F <= 1.65e-6) {
		tmp = t_4;
	} else if ((F <= 2.2e+81) || (!(F <= 2.1e+163) && (F <= 9e+206))) {
		tmp = (t_2 * (1.0 / F)) - (x / B);
	} else {
		tmp = (1.0 / B) - (x * (1.0 / 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) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = x / tan(b)
    t_1 = ((-1.0d0) / b) - t_0
    t_2 = f / sin(b)
    t_3 = 2.0d0 + (x * 2.0d0)
    t_4 = ((f * sqrt((1.0d0 / t_3))) - x) / b
    if (f <= (-4.8d+105)) then
        tmp = t_1
    else if (f <= (-1.65d+82)) then
        tmp = (t_2 * ((-1.0d0) / f)) - (x / b)
    else if (f <= (-600000.0d0)) then
        tmp = t_1
    else if (f <= (-3.5d-83)) then
        tmp = t_4
    else if (f <= 6.5d-107) then
        tmp = ((f / b) / (f + ((0.5d0 * t_3) / f))) - t_0
    else if (f <= 1.65d-6) then
        tmp = t_4
    else if ((f <= 2.2d+81) .or. (.not. (f <= 2.1d+163)) .and. (f <= 9d+206)) then
        tmp = (t_2 * (1.0d0 / f)) - (x / b)
    else
        tmp = (1.0d0 / b) - (x * (1.0d0 / tan(b)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double t_1 = (-1.0 / B) - t_0;
	double t_2 = F / Math.sin(B);
	double t_3 = 2.0 + (x * 2.0);
	double t_4 = ((F * Math.sqrt((1.0 / t_3))) - x) / B;
	double tmp;
	if (F <= -4.8e+105) {
		tmp = t_1;
	} else if (F <= -1.65e+82) {
		tmp = (t_2 * (-1.0 / F)) - (x / B);
	} else if (F <= -600000.0) {
		tmp = t_1;
	} else if (F <= -3.5e-83) {
		tmp = t_4;
	} else if (F <= 6.5e-107) {
		tmp = ((F / B) / (F + ((0.5 * t_3) / F))) - t_0;
	} else if (F <= 1.65e-6) {
		tmp = t_4;
	} else if ((F <= 2.2e+81) || (!(F <= 2.1e+163) && (F <= 9e+206))) {
		tmp = (t_2 * (1.0 / F)) - (x / B);
	} else {
		tmp = (1.0 / B) - (x * (1.0 / Math.tan(B)));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	t_1 = (-1.0 / B) - t_0
	t_2 = F / math.sin(B)
	t_3 = 2.0 + (x * 2.0)
	t_4 = ((F * math.sqrt((1.0 / t_3))) - x) / B
	tmp = 0
	if F <= -4.8e+105:
		tmp = t_1
	elif F <= -1.65e+82:
		tmp = (t_2 * (-1.0 / F)) - (x / B)
	elif F <= -600000.0:
		tmp = t_1
	elif F <= -3.5e-83:
		tmp = t_4
	elif F <= 6.5e-107:
		tmp = ((F / B) / (F + ((0.5 * t_3) / F))) - t_0
	elif F <= 1.65e-6:
		tmp = t_4
	elif (F <= 2.2e+81) or (not (F <= 2.1e+163) and (F <= 9e+206)):
		tmp = (t_2 * (1.0 / F)) - (x / B)
	else:
		tmp = (1.0 / B) - (x * (1.0 / math.tan(B)))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	t_1 = Float64(Float64(-1.0 / B) - t_0)
	t_2 = Float64(F / sin(B))
	t_3 = Float64(2.0 + Float64(x * 2.0))
	t_4 = Float64(Float64(Float64(F * sqrt(Float64(1.0 / t_3))) - x) / B)
	tmp = 0.0
	if (F <= -4.8e+105)
		tmp = t_1;
	elseif (F <= -1.65e+82)
		tmp = Float64(Float64(t_2 * Float64(-1.0 / F)) - Float64(x / B));
	elseif (F <= -600000.0)
		tmp = t_1;
	elseif (F <= -3.5e-83)
		tmp = t_4;
	elseif (F <= 6.5e-107)
		tmp = Float64(Float64(Float64(F / B) / Float64(F + Float64(Float64(0.5 * t_3) / F))) - t_0);
	elseif (F <= 1.65e-6)
		tmp = t_4;
	elseif ((F <= 2.2e+81) || (!(F <= 2.1e+163) && (F <= 9e+206)))
		tmp = Float64(Float64(t_2 * Float64(1.0 / F)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x * Float64(1.0 / tan(B))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	t_1 = (-1.0 / B) - t_0;
	t_2 = F / sin(B);
	t_3 = 2.0 + (x * 2.0);
	t_4 = ((F * sqrt((1.0 / t_3))) - x) / B;
	tmp = 0.0;
	if (F <= -4.8e+105)
		tmp = t_1;
	elseif (F <= -1.65e+82)
		tmp = (t_2 * (-1.0 / F)) - (x / B);
	elseif (F <= -600000.0)
		tmp = t_1;
	elseif (F <= -3.5e-83)
		tmp = t_4;
	elseif (F <= 6.5e-107)
		tmp = ((F / B) / (F + ((0.5 * t_3) / F))) - t_0;
	elseif (F <= 1.65e-6)
		tmp = t_4;
	elseif ((F <= 2.2e+81) || (~((F <= 2.1e+163)) && (F <= 9e+206)))
		tmp = (t_2 * (1.0 / F)) - (x / B);
	else
		tmp = (1.0 / B) - (x * (1.0 / tan(B)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(F * N[Sqrt[N[(1.0 / t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]}, If[LessEqual[F, -4.8e+105], t$95$1, If[LessEqual[F, -1.65e+82], N[(N[(t$95$2 * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -600000.0], t$95$1, If[LessEqual[F, -3.5e-83], t$95$4, If[LessEqual[F, 6.5e-107], N[(N[(N[(F / B), $MachinePrecision] / N[(F + N[(N[(0.5 * t$95$3), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.65e-6], t$95$4, If[Or[LessEqual[F, 2.2e+81], And[N[Not[LessEqual[F, 2.1e+163]], $MachinePrecision], LessEqual[F, 9e+206]]], N[(N[(t$95$2 * N[(1.0 / F), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if F < -4.7999999999999995e105 or -1.6499999999999999e82 < F < -6e5

   1. Initial program 70.8%

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

      1. distribute-lft-neg-in 70.8%

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

      2. +-commutative 70.8%

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

      3. cancel-sign-sub-inv 70.8%

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

   3. Simplified 80.1%

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

   4. Taylor expanded in F around -inf 99.6%

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

   5. Taylor expanded in B around 0 81.8%

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

   if -4.7999999999999995e105 < F < -1.6499999999999999e82

   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. Taylor expanded in F around -inf 99.6%

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

   3. Taylor expanded in B around 0 87.8%

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

   if -6e5 < F < -3.5000000000000003e-83 or 6.5000000000000002e-107 < F < 1.65000000000000008e-6

   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. Step-by-step derivation

      1. distribute-lft-neg-in 99.5%

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

      2. +-commutative 99.5%

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

      3. cancel-sign-sub-inv 99.5%

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

   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. Taylor expanded in F around 0 99.5%

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

   5. Taylor expanded in B around 0 71.1%

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

   if -3.5000000000000003e-83 < F < 6.5000000000000002e-107

   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. Step-by-step derivation

      1. distribute-lft-neg-in 99.5%

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

      2. +-commutative 99.5%

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

      3. cancel-sign-sub-inv 99.5%

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

   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. Step-by-step derivation

      1. clear-num 99.6%

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

      2. inv-pow 99.6%

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

      3. fma-def 99.6%

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

      4. fma-udef 99.6%

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

      5. *-commutative 99.6%

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

      6. fma-def 99.6%

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

      7. fma-def 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. unpow-1 99.6%

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

   7. Simplified 99.6%

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

      1. expm1-log1p-u 90.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(F \cdot \frac{1}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}\right)\right)} - \frac{x}{\tan B} \]

      2. expm1-udef 77.9%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(F \cdot \frac{1}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}\right)} - 1\right)} - \frac{x}{\tan B} \]

      3. un-div-inv 77.9%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      4. div-inv 77.9%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\color{blue}{\sin B \cdot \frac{1}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      5. pow-flip 77.9%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot \color{blue}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(--0.5\right)}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      6. metadata-eval 77.9%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{0.5}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      7. pow1/2 77.9%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot \color{blue}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}\right)} - 1\right) - \frac{x}{\tan B} \]

   9. Applied egg-rr 77.9%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\right)} - 1\right)} - \frac{x}{\tan B} \]
   10. Step-by-step derivation

       1. expm1-def 90.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\right)\right)} - \frac{x}{\tan B} \]

       2. expm1-log1p 99.7%

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

       3. associate-/r* 99.6%

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

   11. Simplified 99.6%

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

   12. Taylor expanded in F around inf 81.0%

       \[\leadsto \frac{\frac{F}{\sin B}}{\color{blue}{F + 0.5 \cdot \frac{2 + 2 \cdot x}{F}}} - \frac{x}{\tan B} \]

   13. Taylor expanded in B around 0 81.0%

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

       1. associate-/r* 81.0%

          \[\leadsto \color{blue}{\frac{\frac{F}{B}}{F + 0.5 \cdot \frac{2 + 2 \cdot x}{F}}} - \frac{x}{\tan B} \]

       2. associate-*r/ 81.0%

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

   15. Simplified 81.0%

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

   if 1.65000000000000008e-6 < F < 2.19999999999999987e81 or 2.1e163 < F < 9.00000000000000035e206

   1. Initial program 61.8%

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

   2. Taylor expanded in F around inf 88.7%

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

   3. Taylor expanded in B around 0 84.4%

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

   if 2.19999999999999987e81 < F < 2.1e163 or 9.00000000000000035e206 < F

   1. Initial program 63.2%

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

   2. Taylor expanded in F around inf 83.0%

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

   3. Taylor expanded in B around 0 85.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{B}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.8 \cdot 10^{+105}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.65 \cdot 10^{+82}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \frac{-1}{F} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -600000:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -3.5 \cdot 10^{-83}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{elif}\;F \leq 6.5 \cdot 10^{-107}:\\ \;\;\;\;\frac{\frac{F}{B}}{F + \frac{0.5 \cdot \left(2 + x \cdot 2\right)}{F}} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.65 \cdot 10^{-6}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{elif}\;F \leq 2.2 \cdot 10^{+81} \lor \neg \left(F \leq 2.1 \cdot 10^{+163}\right) \land F \leq 9 \cdot 10^{+206}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \frac{1}{F} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \end{array} \]

Alternative 16: 62.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F}{\sin B}\\ t_1 := \frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -5.5 \cdot 10^{+105}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq -3 \cdot 10^{+82}:\\ \;\;\;\;t_0 \cdot \frac{-1}{F} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -600000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 1.65 \cdot 10^{-6}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{+81} \lor \neg \left(F \leq 2.85 \cdot 10^{+163}\right) \land F \leq 9 \cdot 10^{+206}:\\ \;\;\;\;t_0 \cdot \frac{1}{F} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ F (sin B))) (t_1 (- (/ -1.0 B) (/ x (tan B)))))
   (if (<= F -5.5e+105)
     t_1
     (if (<= F -3e+82)
       (- (* t_0 (/ -1.0 F)) (/ x B))
       (if (<= F -600000.0)
         t_1
         (if (<= F 1.65e-6)
           (/ (- (* F (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))) x) B)
           (if (or (<= F 1.4e+81) (and (not (<= F 2.85e+163)) (<= F 9e+206)))
             (- (* t_0 (/ 1.0 F)) (/ x B))
             (- (/ 1.0 B) (* x (/ 1.0 (tan B)))))))))))

C

double code(double F, double B, double x) {
	double t_0 = F / sin(B);
	double t_1 = (-1.0 / B) - (x / tan(B));
	double tmp;
	if (F <= -5.5e+105) {
		tmp = t_1;
	} else if (F <= -3e+82) {
		tmp = (t_0 * (-1.0 / F)) - (x / B);
	} else if (F <= -600000.0) {
		tmp = t_1;
	} else if (F <= 1.65e-6) {
		tmp = ((F * sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B;
	} else if ((F <= 1.4e+81) || (!(F <= 2.85e+163) && (F <= 9e+206))) {
		tmp = (t_0 * (1.0 / F)) - (x / B);
	} else {
		tmp = (1.0 / B) - (x * (1.0 / 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 / sin(b)
    t_1 = ((-1.0d0) / b) - (x / tan(b))
    if (f <= (-5.5d+105)) then
        tmp = t_1
    else if (f <= (-3d+82)) then
        tmp = (t_0 * ((-1.0d0) / f)) - (x / b)
    else if (f <= (-600000.0d0)) then
        tmp = t_1
    else if (f <= 1.65d-6) then
        tmp = ((f * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))) - x) / b
    else if ((f <= 1.4d+81) .or. (.not. (f <= 2.85d+163)) .and. (f <= 9d+206)) then
        tmp = (t_0 * (1.0d0 / f)) - (x / b)
    else
        tmp = (1.0d0 / b) - (x * (1.0d0 / tan(b)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = F / Math.sin(B);
	double t_1 = (-1.0 / B) - (x / Math.tan(B));
	double tmp;
	if (F <= -5.5e+105) {
		tmp = t_1;
	} else if (F <= -3e+82) {
		tmp = (t_0 * (-1.0 / F)) - (x / B);
	} else if (F <= -600000.0) {
		tmp = t_1;
	} else if (F <= 1.65e-6) {
		tmp = ((F * Math.sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B;
	} else if ((F <= 1.4e+81) || (!(F <= 2.85e+163) && (F <= 9e+206))) {
		tmp = (t_0 * (1.0 / F)) - (x / B);
	} else {
		tmp = (1.0 / B) - (x * (1.0 / Math.tan(B)));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = F / math.sin(B)
	t_1 = (-1.0 / B) - (x / math.tan(B))
	tmp = 0
	if F <= -5.5e+105:
		tmp = t_1
	elif F <= -3e+82:
		tmp = (t_0 * (-1.0 / F)) - (x / B)
	elif F <= -600000.0:
		tmp = t_1
	elif F <= 1.65e-6:
		tmp = ((F * math.sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B
	elif (F <= 1.4e+81) or (not (F <= 2.85e+163) and (F <= 9e+206)):
		tmp = (t_0 * (1.0 / F)) - (x / B)
	else:
		tmp = (1.0 / B) - (x * (1.0 / math.tan(B)))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(F / sin(B))
	t_1 = Float64(Float64(-1.0 / B) - Float64(x / tan(B)))
	tmp = 0.0
	if (F <= -5.5e+105)
		tmp = t_1;
	elseif (F <= -3e+82)
		tmp = Float64(Float64(t_0 * Float64(-1.0 / F)) - Float64(x / B));
	elseif (F <= -600000.0)
		tmp = t_1;
	elseif (F <= 1.65e-6)
		tmp = Float64(Float64(Float64(F * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))) - x) / B);
	elseif ((F <= 1.4e+81) || (!(F <= 2.85e+163) && (F <= 9e+206)))
		tmp = Float64(Float64(t_0 * Float64(1.0 / F)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x * Float64(1.0 / tan(B))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = F / sin(B);
	t_1 = (-1.0 / B) - (x / tan(B));
	tmp = 0.0;
	if (F <= -5.5e+105)
		tmp = t_1;
	elseif (F <= -3e+82)
		tmp = (t_0 * (-1.0 / F)) - (x / B);
	elseif (F <= -600000.0)
		tmp = t_1;
	elseif (F <= 1.65e-6)
		tmp = ((F * sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B;
	elseif ((F <= 1.4e+81) || (~((F <= 2.85e+163)) && (F <= 9e+206)))
		tmp = (t_0 * (1.0 / F)) - (x / B);
	else
		tmp = (1.0 / B) - (x * (1.0 / tan(B)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -5.5e+105], t$95$1, If[LessEqual[F, -3e+82], N[(N[(t$95$0 * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -600000.0], t$95$1, If[LessEqual[F, 1.65e-6], N[(N[(N[(F * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[Or[LessEqual[F, 1.4e+81], And[N[Not[LessEqual[F, 2.85e+163]], $MachinePrecision], LessEqual[F, 9e+206]]], N[(N[(t$95$0 * N[(1.0 / F), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -5.49999999999999979e105 or -2.99999999999999989e82 < F < -6e5

   1. Initial program 70.8%

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

      1. distribute-lft-neg-in 70.8%

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

      2. +-commutative 70.8%

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

      3. cancel-sign-sub-inv 70.8%

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

   3. Simplified 80.1%

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

   4. Taylor expanded in F around -inf 99.6%

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

   5. Taylor expanded in B around 0 81.8%

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

   if -5.49999999999999979e105 < F < -2.99999999999999989e82

   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. Taylor expanded in F around -inf 99.6%

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

   3. Taylor expanded in B around 0 87.8%

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

   if -6e5 < F < 1.65000000000000008e-6

   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. Step-by-step derivation

      1. distribute-lft-neg-in 99.5%

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

      2. +-commutative 99.5%

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

      3. cancel-sign-sub-inv 99.5%

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

   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. Taylor expanded in F around 0 99.7%

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

   5. Taylor expanded in B around 0 59.1%

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

   if 1.65000000000000008e-6 < F < 1.39999999999999997e81 or 2.8499999999999999e163 < F < 9.00000000000000035e206

   1. Initial program 61.8%

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

   2. Taylor expanded in F around inf 88.7%

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

   3. Taylor expanded in B around 0 84.4%

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

   if 1.39999999999999997e81 < F < 2.8499999999999999e163 or 9.00000000000000035e206 < F

   1. Initial program 63.2%

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

   2. Taylor expanded in F around inf 83.0%

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

   3. Taylor expanded in B around 0 85.2%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5.5 \cdot 10^{+105}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -3 \cdot 10^{+82}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \frac{-1}{F} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -600000:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.65 \cdot 10^{-6}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{+81} \lor \neg \left(F \leq 2.85 \cdot 10^{+163}\right) \land F \leq 9 \cdot 10^{+206}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \frac{1}{F} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \end{array} \]

Alternative 17: 60.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{B} - \frac{x}{\tan B}\\ t_1 := F + 0.5 \cdot \frac{2 + x \cdot 2}{F}\\ \mathbf{if}\;B \leq -1.06 \cdot 10^{+220}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;B \leq -1.28 \cdot 10^{+188}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \frac{-1}{F} - \frac{x}{B}\\ \mathbf{elif}\;B \leq -0.0028:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;B \leq 2.15 \cdot 10^{-11}:\\ \;\;\;\;\left(B \cdot \left(x \cdot 0.3333333333333333 - -0.16666666666666666 \cdot \frac{F}{t_1}\right) + \frac{F}{B \cdot t_1}\right) - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ -1.0 B) (/ x (tan B))))
        (t_1 (+ F (* 0.5 (/ (+ 2.0 (* x 2.0)) F)))))
   (if (<= B -1.06e+220)
     t_0
     (if (<= B -1.28e+188)
       (- (* (/ F (sin B)) (/ -1.0 F)) (/ x B))
       (if (<= B -0.0028)
         (- (/ 1.0 B) (* x (/ 1.0 (tan B))))
         (if (<= B 2.15e-11)
           (-
            (+
             (*
              B
              (- (* x 0.3333333333333333) (* -0.16666666666666666 (/ F t_1))))
             (/ F (* B t_1)))
            (/ x B))
           t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 / B) - (x / tan(B));
	double t_1 = F + (0.5 * ((2.0 + (x * 2.0)) / F));
	double tmp;
	if (B <= -1.06e+220) {
		tmp = t_0;
	} else if (B <= -1.28e+188) {
		tmp = ((F / sin(B)) * (-1.0 / F)) - (x / B);
	} else if (B <= -0.0028) {
		tmp = (1.0 / B) - (x * (1.0 / tan(B)));
	} else if (B <= 2.15e-11) {
		tmp = ((B * ((x * 0.3333333333333333) - (-0.16666666666666666 * (F / t_1)))) + (F / (B * t_1))) - (x / B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((-1.0d0) / b) - (x / tan(b))
    t_1 = f + (0.5d0 * ((2.0d0 + (x * 2.0d0)) / f))
    if (b <= (-1.06d+220)) then
        tmp = t_0
    else if (b <= (-1.28d+188)) then
        tmp = ((f / sin(b)) * ((-1.0d0) / f)) - (x / b)
    else if (b <= (-0.0028d0)) then
        tmp = (1.0d0 / b) - (x * (1.0d0 / tan(b)))
    else if (b <= 2.15d-11) then
        tmp = ((b * ((x * 0.3333333333333333d0) - ((-0.16666666666666666d0) * (f / t_1)))) + (f / (b * t_1))) - (x / b)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (-1.0 / B) - (x / Math.tan(B));
	double t_1 = F + (0.5 * ((2.0 + (x * 2.0)) / F));
	double tmp;
	if (B <= -1.06e+220) {
		tmp = t_0;
	} else if (B <= -1.28e+188) {
		tmp = ((F / Math.sin(B)) * (-1.0 / F)) - (x / B);
	} else if (B <= -0.0028) {
		tmp = (1.0 / B) - (x * (1.0 / Math.tan(B)));
	} else if (B <= 2.15e-11) {
		tmp = ((B * ((x * 0.3333333333333333) - (-0.16666666666666666 * (F / t_1)))) + (F / (B * t_1))) - (x / B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (-1.0 / B) - (x / math.tan(B))
	t_1 = F + (0.5 * ((2.0 + (x * 2.0)) / F))
	tmp = 0
	if B <= -1.06e+220:
		tmp = t_0
	elif B <= -1.28e+188:
		tmp = ((F / math.sin(B)) * (-1.0 / F)) - (x / B)
	elif B <= -0.0028:
		tmp = (1.0 / B) - (x * (1.0 / math.tan(B)))
	elif B <= 2.15e-11:
		tmp = ((B * ((x * 0.3333333333333333) - (-0.16666666666666666 * (F / t_1)))) + (F / (B * t_1))) - (x / B)
	else:
		tmp = t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 / B) - Float64(x / tan(B)))
	t_1 = Float64(F + Float64(0.5 * Float64(Float64(2.0 + Float64(x * 2.0)) / F)))
	tmp = 0.0
	if (B <= -1.06e+220)
		tmp = t_0;
	elseif (B <= -1.28e+188)
		tmp = Float64(Float64(Float64(F / sin(B)) * Float64(-1.0 / F)) - Float64(x / B));
	elseif (B <= -0.0028)
		tmp = Float64(Float64(1.0 / B) - Float64(x * Float64(1.0 / tan(B))));
	elseif (B <= 2.15e-11)
		tmp = Float64(Float64(Float64(B * Float64(Float64(x * 0.3333333333333333) - Float64(-0.16666666666666666 * Float64(F / t_1)))) + Float64(F / Float64(B * t_1))) - Float64(x / B));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (-1.0 / B) - (x / tan(B));
	t_1 = F + (0.5 * ((2.0 + (x * 2.0)) / F));
	tmp = 0.0;
	if (B <= -1.06e+220)
		tmp = t_0;
	elseif (B <= -1.28e+188)
		tmp = ((F / sin(B)) * (-1.0 / F)) - (x / B);
	elseif (B <= -0.0028)
		tmp = (1.0 / B) - (x * (1.0 / tan(B)));
	elseif (B <= 2.15e-11)
		tmp = ((B * ((x * 0.3333333333333333) - (-0.16666666666666666 * (F / t_1)))) + (F / (B * t_1))) - (x / B);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(F + N[(0.5 * N[(N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, -1.06e+220], t$95$0, If[LessEqual[B, -1.28e+188], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, -0.0028], N[(N[(1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 2.15e-11], N[(N[(N[(B * N[(N[(x * 0.3333333333333333), $MachinePrecision] - N[(-0.16666666666666666 * N[(F / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(F / N[(B * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < -1.05999999999999997e220 or 2.15000000000000001e-11 < B

   1. Initial program 89.0%

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

      1. distribute-lft-neg-in 89.0%

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

      2. +-commutative 89.0%

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

      3. cancel-sign-sub-inv 89.0%

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

   3. Simplified 89.1%

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

   4. Taylor expanded in F around -inf 49.9%

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

   5. Taylor expanded in B around 0 58.9%

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

   if -1.05999999999999997e220 < B < -1.27999999999999997e188

   1. Initial program 46.7%

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

   2. Taylor expanded in F around -inf 56.6%

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

   3. Taylor expanded in B around 0 52.2%

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

   if -1.27999999999999997e188 < B < -0.00279999999999999997

   1. Initial program 95.0%

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

   2. Taylor expanded in F around inf 51.6%

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

   3. Taylor expanded in B around 0 55.8%

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

   if -0.00279999999999999997 < B < 2.15000000000000001e-11

   1. Initial program 77.3%

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

      1. distribute-lft-neg-in 77.3%

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

      2. +-commutative 77.3%

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

      3. cancel-sign-sub-inv 77.3%

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

   3. Simplified 89.2%

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

      1. clear-num 89.1%

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

      2. inv-pow 89.1%

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

      3. fma-def 89.1%

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

      4. fma-udef 89.1%

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

      5. *-commutative 89.1%

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

      6. fma-def 89.1%

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

      7. fma-def 89.1%

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

   5. Applied egg-rr 89.1%

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

      1. unpow-1 89.1%

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

   7. Simplified 89.1%

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

      1. expm1-log1p-u 66.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(F \cdot \frac{1}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}\right)\right)} - \frac{x}{\tan B} \]

      2. expm1-udef 60.1%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(F \cdot \frac{1}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}\right)} - 1\right)} - \frac{x}{\tan B} \]

      3. un-div-inv 60.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      4. div-inv 60.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\color{blue}{\sin B \cdot \frac{1}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      5. pow-flip 60.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot \color{blue}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(--0.5\right)}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      6. metadata-eval 60.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{0.5}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      7. pow1/2 60.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot \color{blue}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}\right)} - 1\right) - \frac{x}{\tan B} \]

   9. Applied egg-rr 60.1%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\right)} - 1\right)} - \frac{x}{\tan B} \]
   10. Step-by-step derivation

       1. expm1-def 66.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\right)\right)} - \frac{x}{\tan B} \]

       2. expm1-log1p 89.2%

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

       3. associate-/r* 77.4%

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

   11. Simplified 77.4%

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

   12. Taylor expanded in F around inf 60.2%

       \[\leadsto \frac{\frac{F}{\sin B}}{\color{blue}{F + 0.5 \cdot \frac{2 + 2 \cdot x}{F}}} - \frac{x}{\tan B} \]

   13. Taylor expanded in B around 0 69.2%

       \[\leadsto \color{blue}{\left(B \cdot \left(0.3333333333333333 \cdot x - -0.16666666666666666 \cdot \frac{F}{F + 0.5 \cdot \frac{2 + 2 \cdot x}{F}}\right) + \frac{F}{B \cdot \left(F + 0.5 \cdot \frac{2 + 2 \cdot x}{F}\right)}\right) - \frac{x}{B}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -1.06 \cdot 10^{+220}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;B \leq -1.28 \cdot 10^{+188}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \frac{-1}{F} - \frac{x}{B}\\ \mathbf{elif}\;B \leq -0.0028:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;B \leq 2.15 \cdot 10^{-11}:\\ \;\;\;\;\left(B \cdot \left(x \cdot 0.3333333333333333 - -0.16666666666666666 \cdot \frac{F}{F + 0.5 \cdot \frac{2 + x \cdot 2}{F}}\right) + \frac{F}{B \cdot \left(F + 0.5 \cdot \frac{2 + x \cdot 2}{F}\right)}\right) - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 18: 61.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := F + 0.5 \cdot \frac{2 + x \cdot 2}{F}\\ \mathbf{if}\;B \leq -0.0028 \lor \neg \left(B \leq 1.95 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\left(B \cdot \left(x \cdot 0.3333333333333333 - -0.16666666666666666 \cdot \frac{F}{t_0}\right) + \frac{F}{B \cdot t_0}\right) - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (+ F (* 0.5 (/ (+ 2.0 (* x 2.0)) F)))))
   (if (or (<= B -0.0028) (not (<= B 1.95e-11)))
     (- (/ -1.0 B) (/ x (tan B)))
     (-
      (+
       (* B (- (* x 0.3333333333333333) (* -0.16666666666666666 (/ F t_0))))
       (/ F (* B t_0)))
      (/ x B)))))

C

double code(double F, double B, double x) {
	double t_0 = F + (0.5 * ((2.0 + (x * 2.0)) / F));
	double tmp;
	if ((B <= -0.0028) || !(B <= 1.95e-11)) {
		tmp = (-1.0 / B) - (x / tan(B));
	} else {
		tmp = ((B * ((x * 0.3333333333333333) - (-0.16666666666666666 * (F / t_0)))) + (F / (B * t_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 = f + (0.5d0 * ((2.0d0 + (x * 2.0d0)) / f))
    if ((b <= (-0.0028d0)) .or. (.not. (b <= 1.95d-11))) then
        tmp = ((-1.0d0) / b) - (x / tan(b))
    else
        tmp = ((b * ((x * 0.3333333333333333d0) - ((-0.16666666666666666d0) * (f / t_0)))) + (f / (b * t_0))) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = F + (0.5 * ((2.0 + (x * 2.0)) / F));
	double tmp;
	if ((B <= -0.0028) || !(B <= 1.95e-11)) {
		tmp = (-1.0 / B) - (x / Math.tan(B));
	} else {
		tmp = ((B * ((x * 0.3333333333333333) - (-0.16666666666666666 * (F / t_0)))) + (F / (B * t_0))) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = F + (0.5 * ((2.0 + (x * 2.0)) / F))
	tmp = 0
	if (B <= -0.0028) or not (B <= 1.95e-11):
		tmp = (-1.0 / B) - (x / math.tan(B))
	else:
		tmp = ((B * ((x * 0.3333333333333333) - (-0.16666666666666666 * (F / t_0)))) + (F / (B * t_0))) - (x / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(F + Float64(0.5 * Float64(Float64(2.0 + Float64(x * 2.0)) / F)))
	tmp = 0.0
	if ((B <= -0.0028) || !(B <= 1.95e-11))
		tmp = Float64(Float64(-1.0 / B) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(Float64(B * Float64(Float64(x * 0.3333333333333333) - Float64(-0.16666666666666666 * Float64(F / t_0)))) + Float64(F / Float64(B * t_0))) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = F + (0.5 * ((2.0 + (x * 2.0)) / F));
	tmp = 0.0;
	if ((B <= -0.0028) || ~((B <= 1.95e-11)))
		tmp = (-1.0 / B) - (x / tan(B));
	else
		tmp = ((B * ((x * 0.3333333333333333) - (-0.16666666666666666 * (F / t_0)))) + (F / (B * t_0))) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(F + N[(0.5 * N[(N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[B, -0.0028], N[Not[LessEqual[B, 1.95e-11]], $MachinePrecision]], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(B * N[(N[(x * 0.3333333333333333), $MachinePrecision] - N[(-0.16666666666666666 * N[(F / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(F / N[(B * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < -0.00279999999999999997 or 1.95000000000000005e-11 < B

   1. Initial program 88.0%

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

      1. distribute-lft-neg-in 88.0%

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

      2. +-commutative 88.0%

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

      3. cancel-sign-sub-inv 88.0%

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

   3. Simplified 88.0%

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

   4. Taylor expanded in F around -inf 51.7%

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

   5. Taylor expanded in B around 0 54.0%

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

   if -0.00279999999999999997 < B < 1.95000000000000005e-11

   1. Initial program 77.3%

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

      1. distribute-lft-neg-in 77.3%

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

      2. +-commutative 77.3%

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

      3. cancel-sign-sub-inv 77.3%

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

   3. Simplified 89.2%

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

      1. clear-num 89.1%

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

      2. inv-pow 89.1%

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

      3. fma-def 89.1%

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

      4. fma-udef 89.1%

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

      5. *-commutative 89.1%

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

      6. fma-def 89.1%

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

      7. fma-def 89.1%

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

   5. Applied egg-rr 89.1%

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

      1. unpow-1 89.1%

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

   7. Simplified 89.1%

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

      1. expm1-log1p-u 66.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(F \cdot \frac{1}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}\right)\right)} - \frac{x}{\tan B} \]

      2. expm1-udef 60.1%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(F \cdot \frac{1}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}\right)} - 1\right)} - \frac{x}{\tan B} \]

      3. un-div-inv 60.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      4. div-inv 60.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\color{blue}{\sin B \cdot \frac{1}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      5. pow-flip 60.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot \color{blue}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(--0.5\right)}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      6. metadata-eval 60.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{0.5}}}\right)} - 1\right) - \frac{x}{\tan B} \]

      7. pow1/2 60.1%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot \color{blue}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}\right)} - 1\right) - \frac{x}{\tan B} \]

   9. Applied egg-rr 60.1%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\right)} - 1\right)} - \frac{x}{\tan B} \]
   10. Step-by-step derivation

       1. expm1-def 66.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\right)\right)} - \frac{x}{\tan B} \]

       2. expm1-log1p 89.2%

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

       3. associate-/r* 77.4%

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

   11. Simplified 77.4%

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

   12. Taylor expanded in F around inf 60.2%

       \[\leadsto \frac{\frac{F}{\sin B}}{\color{blue}{F + 0.5 \cdot \frac{2 + 2 \cdot x}{F}}} - \frac{x}{\tan B} \]

   13. Taylor expanded in B around 0 69.2%

       \[\leadsto \color{blue}{\left(B \cdot \left(0.3333333333333333 \cdot x - -0.16666666666666666 \cdot \frac{F}{F + 0.5 \cdot \frac{2 + 2 \cdot x}{F}}\right) + \frac{F}{B \cdot \left(F + 0.5 \cdot \frac{2 + 2 \cdot x}{F}\right)}\right) - \frac{x}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -0.0028 \lor \neg \left(B \leq 1.95 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\left(B \cdot \left(x \cdot 0.3333333333333333 - -0.16666666666666666 \cdot \frac{F}{F + 0.5 \cdot \frac{2 + x \cdot 2}{F}}\right) + \frac{F}{B \cdot \left(F + 0.5 \cdot \frac{2 + x \cdot 2}{F}\right)}\right) - \frac{x}{B}\\ \end{array} \]

Alternative 19: 36.5% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{F}{F + \frac{0.5 \cdot \left(2 + x \cdot 2\right)}{F}} - x}{B} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (/ (- (/ F (+ F (/ (* 0.5 (+ 2.0 (* x 2.0))) F))) x) B))

C

double code(double F, double B, double x) {
	return ((F / (F + ((0.5 * (2.0 + (x * 2.0))) / F))) - 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 = ((f / (f + ((0.5d0 * (2.0d0 + (x * 2.0d0))) / f))) - x) / b
end function

Java

public static double code(double F, double B, double x) {
	return ((F / (F + ((0.5 * (2.0 + (x * 2.0))) / F))) - x) / B;
}

Python

def code(F, B, x):
	return ((F / (F + ((0.5 * (2.0 + (x * 2.0))) / F))) - x) / B

Julia

function code(F, B, x)
	return Float64(Float64(Float64(F / Float64(F + Float64(Float64(0.5 * Float64(2.0 + Float64(x * 2.0))) / F))) - x) / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = ((F / (F + ((0.5 * (2.0 + (x * 2.0))) / F))) - x) / B;
end

Wolfram

code[F_, B_, x_] := N[(N[(N[(F / N[(F + N[(N[(0.5 * N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]

Derivation

1. Initial program 82.7%

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

   1. distribute-lft-neg-in 82.7%

      \[\leadsto \color{blue}{\left(-x\right) \cdot \frac{1}{\tan B}} + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. +-commutative 82.7%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x\right) \cdot \frac{1}{\tan B}} \]

   3. cancel-sign-sub-inv 82.7%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

3. Simplified 88.6%

   \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
4. Step-by-step derivation

   1. clear-num 88.6%

      \[\leadsto F \cdot \color{blue}{\frac{1}{\frac{\sin B}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}} - \frac{x}{\tan B} \]

   2. inv-pow 88.6%

      \[\leadsto F \cdot \color{blue}{{\left(\frac{\sin B}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}\right)}^{-1}} - \frac{x}{\tan B} \]

   3. fma-def 88.6%

      \[\leadsto F \cdot {\left(\frac{\sin B}{{\left(\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F + 2}\right)\right)}^{-0.5}}\right)}^{-1} - \frac{x}{\tan B} \]

   4. fma-udef 88.6%

      \[\leadsto F \cdot {\left(\frac{\sin B}{{\color{blue}{\left(x \cdot 2 + \left(F \cdot F + 2\right)\right)}}^{-0.5}}\right)}^{-1} - \frac{x}{\tan B} \]

   5. *-commutative 88.6%

      \[\leadsto F \cdot {\left(\frac{\sin B}{{\left(\color{blue}{2 \cdot x} + \left(F \cdot F + 2\right)\right)}^{-0.5}}\right)}^{-1} - \frac{x}{\tan B} \]

   6. fma-def 88.6%

      \[\leadsto F \cdot {\left(\frac{\sin B}{{\color{blue}{\left(\mathsf{fma}\left(2, x, F \cdot F + 2\right)\right)}}^{-0.5}}\right)}^{-1} - \frac{x}{\tan B} \]

   7. fma-def 88.6%

      \[\leadsto F \cdot {\left(\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)\right)}^{-0.5}}\right)}^{-1} - \frac{x}{\tan B} \]

5. Applied egg-rr 88.6%

   \[\leadsto F \cdot \color{blue}{{\left(\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}\right)}^{-1}} - \frac{x}{\tan B} \]
6. Step-by-step derivation

   1. unpow-1 88.6%

      \[\leadsto F \cdot \color{blue}{\frac{1}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}} - \frac{x}{\tan B} \]

7. Simplified 88.6%

   \[\leadsto F \cdot \color{blue}{\frac{1}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}} - \frac{x}{\tan B} \]
8. Step-by-step derivation

   1. expm1-log1p-u 71.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(F \cdot \frac{1}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}\right)\right)} - \frac{x}{\tan B} \]

   2. expm1-udef 63.1%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(F \cdot \frac{1}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}\right)} - 1\right)} - \frac{x}{\tan B} \]

   3. un-div-inv 63.1%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{F}{\frac{\sin B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}}\right)} - 1\right) - \frac{x}{\tan B} \]

   4. div-inv 63.1%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\color{blue}{\sin B \cdot \frac{1}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}}\right)} - 1\right) - \frac{x}{\tan B} \]

   5. pow-flip 63.1%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot \color{blue}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(--0.5\right)}}}\right)} - 1\right) - \frac{x}{\tan B} \]

   6. metadata-eval 63.1%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{0.5}}}\right)} - 1\right) - \frac{x}{\tan B} \]

   7. pow1/2 63.1%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot \color{blue}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}\right)} - 1\right) - \frac{x}{\tan B} \]

9. Applied egg-rr 63.1%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\right)} - 1\right)} - \frac{x}{\tan B} \]
10. Step-by-step derivation

    1. expm1-def 71.6%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\right)\right)} - \frac{x}{\tan B} \]

    2. expm1-log1p 88.6%

       \[\leadsto \color{blue}{\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}} - \frac{x}{\tan B} \]

    3. associate-/r* 82.8%

       \[\leadsto \color{blue}{\frac{\frac{F}{\sin B}}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}} - \frac{x}{\tan B} \]

11. Simplified 82.8%

    \[\leadsto \color{blue}{\frac{\frac{F}{\sin B}}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}} - \frac{x}{\tan B} \]

12. Taylor expanded in F around inf 67.7%

    \[\leadsto \frac{\frac{F}{\sin B}}{\color{blue}{F + 0.5 \cdot \frac{2 + 2 \cdot x}{F}}} - \frac{x}{\tan B} \]

13. Taylor expanded in B around 0 37.1%

    \[\leadsto \color{blue}{\frac{\frac{F}{F + 0.5 \cdot \frac{2 + 2 \cdot x}{F}} - x}{B}} \]
14. Step-by-step derivation

    1. associate-*r/ 37.1%

       \[\leadsto \frac{\frac{F}{F + \color{blue}{\frac{0.5 \cdot \left(2 + 2 \cdot x\right)}{F}}} - x}{B} \]

15. Simplified 37.1%

    \[\leadsto \color{blue}{\frac{\frac{F}{F + \frac{0.5 \cdot \left(2 + 2 \cdot x\right)}{F}} - x}{B}} \]

16. Final simplification 37.1%

    \[\leadsto \frac{\frac{F}{F + \frac{0.5 \cdot \left(2 + x \cdot 2\right)}{F}} - x}{B} \]

Alternative 20: 30.0% accurate, 36.0× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \cdot \left(B \cdot x\right) - \frac{x}{B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (- (* 0.3333333333333333 (* B x)) (/ x B)))

C

double code(double F, double B, double x) {
	return (0.3333333333333333 * (B * x)) - (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 = (0.3333333333333333d0 * (b * x)) - (x / b)
end function

Java

public static double code(double F, double B, double x) {
	return (0.3333333333333333 * (B * x)) - (x / B);
}

Python

def code(F, B, x):
	return (0.3333333333333333 * (B * x)) - (x / B)

Julia

function code(F, B, x)
	return Float64(Float64(0.3333333333333333 * Float64(B * x)) - Float64(x / B))
end

MATLAB

function tmp = code(F, B, x)
	tmp = (0.3333333333333333 * (B * x)) - (x / B);
end

Wolfram

code[F_, B_, x_] := N[(N[(0.3333333333333333 * N[(B * x), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.7%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
2. Step-by-step derivation

   1. distribute-lft-neg-in 82.7%

      \[\leadsto \color{blue}{\left(-x\right) \cdot \frac{1}{\tan B}} + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. +-commutative 82.7%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x\right) \cdot \frac{1}{\tan B}} \]

   3. fma-def 82.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{F}{\sin B}, {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}, \left(-x\right) \cdot \frac{1}{\tan B}\right)} \]

   4. +-commutative 82.7%

      \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\color{blue}{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)}, \left(-x\right) \cdot \frac{1}{\tan B}\right) \]

   5. *-commutative 82.7%

      \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\color{blue}{x \cdot 2} + \left(F \cdot F + 2\right)\right)}^{\left(-\frac{1}{2}\right)}, \left(-x\right) \cdot \frac{1}{\tan B}\right) \]

   6. fma-def 82.7%

      \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\color{blue}{\left(\mathsf{fma}\left(x, 2, F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)}, \left(-x\right) \cdot \frac{1}{\tan B}\right) \]

   7. fma-def 82.7%

      \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)\right)}^{\left(-\frac{1}{2}\right)}, \left(-x\right) \cdot \frac{1}{\tan B}\right) \]

   8. metadata-eval 82.7%

      \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(-\color{blue}{0.5}\right)}, \left(-x\right) \cdot \frac{1}{\tan B}\right) \]

   9. metadata-eval 82.7%

      \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{-0.5}}, \left(-x\right) \cdot \frac{1}{\tan B}\right) \]

   10. associate-*r/ 82.8%

       \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \color{blue}{\frac{\left(-x\right) \cdot 1}{\tan B}}\right) \]

   11. *-rgt-identity 82.8%

       \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{\color{blue}{-x}}{\tan B}\right) \]

3. Simplified 82.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{-x}{\tan B}\right)} \]

4. Taylor expanded in F around 0 58.7%

   \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
5. Step-by-step derivation

   1. mul-1-neg 58.7%

      \[\leadsto \color{blue}{-\frac{x \cdot \cos B}{\sin B}} \]

   2. associate-*l/ 58.7%

      \[\leadsto -\color{blue}{\frac{x}{\sin B} \cdot \cos B} \]

   3. *-commutative 58.7%

      \[\leadsto -\color{blue}{\cos B \cdot \frac{x}{\sin B}} \]

   4. distribute-rgt-neg-in 58.7%

      \[\leadsto \color{blue}{\cos B \cdot \left(-\frac{x}{\sin B}\right)} \]

6. Simplified 58.7%

   \[\leadsto \color{blue}{\cos B \cdot \left(-\frac{x}{\sin B}\right)} \]
7. Step-by-step derivation

   1. clear-num 58.6%

      \[\leadsto \cos B \cdot \left(-\color{blue}{\frac{1}{\frac{\sin B}{x}}}\right) \]

   2. inv-pow 58.6%

      \[\leadsto \cos B \cdot \left(-\color{blue}{{\left(\frac{\sin B}{x}\right)}^{-1}}\right) \]

8. Applied egg-rr 58.6%

   \[\leadsto \cos B \cdot \left(-\color{blue}{{\left(\frac{\sin B}{x}\right)}^{-1}}\right) \]
9. Step-by-step derivation

   1. unpow-1 58.6%

      \[\leadsto \cos B \cdot \left(-\color{blue}{\frac{1}{\frac{\sin B}{x}}}\right) \]

10. Simplified 58.6%

    \[\leadsto \cos B \cdot \left(-\color{blue}{\frac{1}{\frac{\sin B}{x}}}\right) \]

11. Taylor expanded in B around 0 31.6%

    \[\leadsto \color{blue}{-1 \cdot \left(B \cdot \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right)\right) + -1 \cdot \frac{x}{B}} \]
12. Step-by-step derivation

    1. mul-1-neg 31.6%

       \[\leadsto -1 \cdot \left(B \cdot \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right)\right) + \color{blue}{\left(-\frac{x}{B}\right)} \]

    2. unsub-neg 31.6%

       \[\leadsto \color{blue}{-1 \cdot \left(B \cdot \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right)\right) - \frac{x}{B}} \]

    3. distribute-rgt-out-- 31.6%

       \[\leadsto -1 \cdot \left(B \cdot \color{blue}{\left(x \cdot \left(-0.5 - -0.16666666666666666\right)\right)}\right) - \frac{x}{B} \]

    4. metadata-eval 31.6%

       \[\leadsto -1 \cdot \left(B \cdot \left(x \cdot \color{blue}{-0.3333333333333333}\right)\right) - \frac{x}{B} \]

    5. associate-*r* 31.6%

       \[\leadsto -1 \cdot \color{blue}{\left(\left(B \cdot x\right) \cdot -0.3333333333333333\right)} - \frac{x}{B} \]

    6. *-commutative 31.6%

       \[\leadsto -1 \cdot \color{blue}{\left(-0.3333333333333333 \cdot \left(B \cdot x\right)\right)} - \frac{x}{B} \]

    7. associate-*r* 31.6%

       \[\leadsto \color{blue}{\left(-1 \cdot -0.3333333333333333\right) \cdot \left(B \cdot x\right)} - \frac{x}{B} \]

    8. metadata-eval 31.6%

       \[\leadsto \color{blue}{0.3333333333333333} \cdot \left(B \cdot x\right) - \frac{x}{B} \]

13. Simplified 31.6%

    \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(B \cdot x\right) - \frac{x}{B}} \]

14. Final simplification 31.6%

    \[\leadsto 0.3333333333333333 \cdot \left(B \cdot x\right) - \frac{x}{B} \]

Alternative 21: 29.7% 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 82.7%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
2. Step-by-step derivation

   1. distribute-lft-neg-in 82.7%

      \[\leadsto \color{blue}{\left(-x\right) \cdot \frac{1}{\tan B}} + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. +-commutative 82.7%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x\right) \cdot \frac{1}{\tan B}} \]

   3. fma-def 82.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{F}{\sin B}, {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}, \left(-x\right) \cdot \frac{1}{\tan B}\right)} \]

   4. +-commutative 82.7%

      \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\color{blue}{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)}, \left(-x\right) \cdot \frac{1}{\tan B}\right) \]

   5. *-commutative 82.7%

      \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\color{blue}{x \cdot 2} + \left(F \cdot F + 2\right)\right)}^{\left(-\frac{1}{2}\right)}, \left(-x\right) \cdot \frac{1}{\tan B}\right) \]

   6. fma-def 82.7%

      \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\color{blue}{\left(\mathsf{fma}\left(x, 2, F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)}, \left(-x\right) \cdot \frac{1}{\tan B}\right) \]

   7. fma-def 82.7%

      \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)\right)}^{\left(-\frac{1}{2}\right)}, \left(-x\right) \cdot \frac{1}{\tan B}\right) \]

   8. metadata-eval 82.7%

      \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(-\color{blue}{0.5}\right)}, \left(-x\right) \cdot \frac{1}{\tan B}\right) \]

   9. metadata-eval 82.7%

      \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{-0.5}}, \left(-x\right) \cdot \frac{1}{\tan B}\right) \]

   10. associate-*r/ 82.8%

       \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \color{blue}{\frac{\left(-x\right) \cdot 1}{\tan B}}\right) \]

   11. *-rgt-identity 82.8%

       \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{\color{blue}{-x}}{\tan B}\right) \]

3. Simplified 82.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{-x}{\tan B}\right)} \]

4. Taylor expanded in F around 0 58.7%

   \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
5. Step-by-step derivation

   1. mul-1-neg 58.7%

      \[\leadsto \color{blue}{-\frac{x \cdot \cos B}{\sin B}} \]

   2. associate-*l/ 58.7%

      \[\leadsto -\color{blue}{\frac{x}{\sin B} \cdot \cos B} \]

   3. *-commutative 58.7%

      \[\leadsto -\color{blue}{\cos B \cdot \frac{x}{\sin B}} \]

   4. distribute-rgt-neg-in 58.7%

      \[\leadsto \color{blue}{\cos B \cdot \left(-\frac{x}{\sin B}\right)} \]

6. Simplified 58.7%

   \[\leadsto \color{blue}{\cos B \cdot \left(-\frac{x}{\sin B}\right)} \]

7. Taylor expanded in B around 0 31.5%

   \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
8. Step-by-step derivation

   1. associate-*r/ 31.5%

      \[\leadsto \color{blue}{\frac{-1 \cdot x}{B}} \]

   2. neg-mul-1 31.5%

      \[\leadsto \frac{\color{blue}{-x}}{B} \]

9. Simplified 31.5%

   \[\leadsto \color{blue}{\frac{-x}{B}} \]

10. Final simplification 31.5%

    \[\leadsto \frac{-x}{B} \]

Reproduce

herbie shell --seed 2023272 
(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))))))