VandenBroeck and Keller, Equation (23)

Percentage Accurate: 77.1% → 99.4%

Time: 21.1s

Alternatives: 26

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2.85 \cdot 10^{+78}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 300000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-0.5} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -2.85e+78)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 300000000.0)
       (- (* (/ F (sin B)) (pow (fma F F (fma 2.0 x 2.0)) -0.5)) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -2.85e+78) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 300000000.0) {
		tmp = ((F / sin(B)) * pow(fma(F, F, fma(2.0, x, 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 <= -2.85e+78)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 300000000.0)
		tmp = Float64(Float64(Float64(F / sin(B)) * (fma(F, F, fma(2.0, x, 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, -2.85e+78], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 300000000.0], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.84999999999999993e78

   1. Initial program 43.6%

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

      1. distribute-lft-neg-in 43.6%

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

      2. +-commutative 43.6%

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

      3. cancel-sign-sub-inv 43.6%

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

   3. Simplified 63.3%

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

      1. clear-num 63.3%

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

      2. inv-pow 63.3%

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

      3. fma-def 63.3%

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

      4. fma-udef 63.3%

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

      5. *-commutative 63.3%

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

      6. fma-def 63.3%

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

      7. fma-def 63.3%

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

   5. Applied egg-rr 63.3%

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

      1. unpow-1 63.3%

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

      2. fma-udef 63.3%

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

      3. fma-udef 63.3%

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

      4. unpow2 63.3%

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

      5. +-commutative 63.3%

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

      6. associate-+r+ 63.3%

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

      7. +-commutative 63.3%

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

      8. +-commutative 63.3%

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

      9. unpow2 63.3%

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

      10. fma-def 63.3%

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

      11. +-commutative 63.3%

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

      12. fma-def 63.3%

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

   7. Simplified 63.3%

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

   8. Taylor expanded in F around -inf 99.9%

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

   if -2.84999999999999993e78 < F < 3e8

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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/ 99.6%

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

      11. *-rgt-identity 99.6%

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

   3. Simplified 99.6%

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

      1. fma-udef 99.6%

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

      2. associate-*l/ 99.5%

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

      3. associate-*r/ 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. distribute-frac-neg 99.6%

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

      5. unsub-neg 99.6%

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

      6. associate-*r/ 99.5%

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

      7. associate-*l/ 99.6%

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

      8. fma-def 99.6%

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

      9. fma-udef 99.6%

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

      10. *-commutative 99.6%

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

      11. fma-def 99.6%

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

      12. fma-def 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. fma-udef 99.6%

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

      2. fma-udef 99.6%

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

      3. unpow2 99.6%

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

      4. +-commutative 99.6%

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

      5. associate-+r+ 99.6%

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

      6. +-commutative 99.6%

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

      7. +-commutative 99.6%

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

      8. unpow2 99.6%

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

      9. fma-def 99.6%

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

      10. +-commutative 99.6%

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

      11. fma-def 99.6%

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

   7. Simplified 99.6%

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

   if 3e8 < F

   1. Initial program 54.0%

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

      1. distribute-lft-neg-in 54.0%

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

      2. +-commutative 54.0%

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

      3. cancel-sign-sub-inv 54.0%

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

   3. Simplified 68.1%

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

      1. clear-num 68.1%

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

      2. inv-pow 68.1%

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

      3. fma-def 68.1%

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

      4. fma-udef 68.1%

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

      5. *-commutative 68.1%

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

      6. fma-def 68.1%

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

      7. fma-def 68.1%

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

   5. Applied egg-rr 68.1%

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

      1. unpow-1 68.1%

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

      2. fma-udef 68.1%

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

      3. fma-udef 68.1%

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

      4. unpow2 68.1%

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

      5. +-commutative 68.1%

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

      6. associate-+r+ 68.1%

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

      7. +-commutative 68.1%

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

      8. +-commutative 68.1%

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

      9. unpow2 68.1%

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

      10. fma-def 68.1%

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

      11. +-commutative 68.1%

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

      12. fma-def 68.1%

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

   7. Simplified 68.1%

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

   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 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.85 \cdot 10^{+78}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 300000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-0.5} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 2: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3 \cdot 10^{+78}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 60000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{-0.5} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -3e+78)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 60000000.0)
       (- (* (/ F (sin B)) (pow (+ (* x 2.0) (+ 2.0 (* F F))) -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 <= -3e+78) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 60000000.0) {
		tmp = ((F / sin(B)) * pow(((x * 2.0) + (2.0 + (F * F))), -0.5)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-3d+78)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 60000000.0d0) then
        tmp = ((f / sin(b)) * (((x * 2.0d0) + (2.0d0 + (f * f))) ** (-0.5d0))) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -3e+78) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 60000000.0) {
		tmp = ((F / Math.sin(B)) * Math.pow(((x * 2.0) + (2.0 + (F * F))), -0.5)) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -3e+78:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 60000000.0:
		tmp = ((F / math.sin(B)) * math.pow(((x * 2.0) + (2.0 + (F * F))), -0.5)) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -3e+78)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 60000000.0)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(x * 2.0) + Float64(2.0 + Float64(F * F))) ^ -0.5)) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -3e+78)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 60000000.0)
		tmp = ((F / sin(B)) * (((x * 2.0) + (2.0 + (F * F))) ^ -0.5)) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -3e+78], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 60000000.0], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(x * 2.0), $MachinePrecision] + N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.99999999999999982e78

   1. Initial program 43.6%

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

      1. distribute-lft-neg-in 43.6%

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

      2. +-commutative 43.6%

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

      3. cancel-sign-sub-inv 43.6%

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

   3. Simplified 63.3%

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

      1. clear-num 63.3%

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

      2. inv-pow 63.3%

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

      3. fma-def 63.3%

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

      4. fma-udef 63.3%

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

      5. *-commutative 63.3%

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

      6. fma-def 63.3%

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

      7. fma-def 63.3%

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

   5. Applied egg-rr 63.3%

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

      1. unpow-1 63.3%

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

      2. fma-udef 63.3%

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

      3. fma-udef 63.3%

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

      4. unpow2 63.3%

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

      5. +-commutative 63.3%

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

      6. associate-+r+ 63.3%

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

      7. +-commutative 63.3%

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

      8. +-commutative 63.3%

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

      9. unpow2 63.3%

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

      10. fma-def 63.3%

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

      11. +-commutative 63.3%

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

      12. fma-def 63.3%

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

   7. Simplified 63.3%

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

   8. Taylor expanded in F around -inf 99.9%

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

   if -2.99999999999999982e78 < F < 6e7

   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. div-inv 99.6%

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

      2. expm1-log1p-u 80.2%

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

      3. expm1-udef 58.4%

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

   3. Applied egg-rr 58.4%

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

      1. expm1-def 80.2%

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

      2. expm1-log1p 99.6%

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

   5. Simplified 99.6%

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

   if 6e7 < F

   1. Initial program 54.0%

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

      1. distribute-lft-neg-in 54.0%

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

      2. +-commutative 54.0%

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

      3. cancel-sign-sub-inv 54.0%

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

   3. Simplified 68.1%

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

      1. clear-num 68.1%

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

      2. inv-pow 68.1%

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

      3. fma-def 68.1%

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

      4. fma-udef 68.1%

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

      5. *-commutative 68.1%

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

      6. fma-def 68.1%

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

      7. fma-def 68.1%

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

   5. Applied egg-rr 68.1%

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

      1. unpow-1 68.1%

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

      2. fma-udef 68.1%

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

      3. fma-udef 68.1%

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

      4. unpow2 68.1%

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

      5. +-commutative 68.1%

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

      6. associate-+r+ 68.1%

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

      7. +-commutative 68.1%

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

      8. +-commutative 68.1%

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

      9. unpow2 68.1%

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

      10. fma-def 68.1%

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

      11. +-commutative 68.1%

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

      12. fma-def 68.1%

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

   7. Simplified 68.1%

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

   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 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3 \cdot 10^{+78}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 60000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{-0.5} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 3: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + x \cdot 2\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -5.3 \cdot 10^{+25}:\\ \;\;\;\;\frac{-1}{\sin B} - t_1\\ \mathbf{elif}\;F \leq 0.044:\\ \;\;\;\;F \cdot \frac{1}{\sin B \cdot \sqrt{t_0}} - t_1\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{\sin B \cdot \left(F + 0.5 \cdot \frac{t_0}{F}\right)} - t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (+ 2.0 (* x 2.0))) (t_1 (/ x (tan B))))
   (if (<= F -5.3e+25)
     (- (/ -1.0 (sin B)) t_1)
     (if (<= F 0.044)
       (- (* F (/ 1.0 (* (sin B) (sqrt t_0)))) t_1)
       (- (* F (/ 1.0 (* (sin B) (+ F (* 0.5 (/ t_0 F)))))) t_1)))))

C

double code(double F, double B, double x) {
	double t_0 = 2.0 + (x * 2.0);
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -5.3e+25) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= 0.044) {
		tmp = (F * (1.0 / (sin(B) * sqrt(t_0)))) - t_1;
	} else {
		tmp = (F * (1.0 / (sin(B) * (F + (0.5 * (t_0 / F)))))) - t_1;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 + (x * 2.0d0)
    t_1 = x / tan(b)
    if (f <= (-5.3d+25)) then
        tmp = ((-1.0d0) / sin(b)) - t_1
    else if (f <= 0.044d0) then
        tmp = (f * (1.0d0 / (sin(b) * sqrt(t_0)))) - t_1
    else
        tmp = (f * (1.0d0 / (sin(b) * (f + (0.5d0 * (t_0 / f)))))) - t_1
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = 2.0 + (x * 2.0);
	double t_1 = x / Math.tan(B);
	double tmp;
	if (F <= -5.3e+25) {
		tmp = (-1.0 / Math.sin(B)) - t_1;
	} else if (F <= 0.044) {
		tmp = (F * (1.0 / (Math.sin(B) * Math.sqrt(t_0)))) - t_1;
	} else {
		tmp = (F * (1.0 / (Math.sin(B) * (F + (0.5 * (t_0 / F)))))) - t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = 2.0 + (x * 2.0)
	t_1 = x / math.tan(B)
	tmp = 0
	if F <= -5.3e+25:
		tmp = (-1.0 / math.sin(B)) - t_1
	elif F <= 0.044:
		tmp = (F * (1.0 / (math.sin(B) * math.sqrt(t_0)))) - t_1
	else:
		tmp = (F * (1.0 / (math.sin(B) * (F + (0.5 * (t_0 / F)))))) - t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(2.0 + Float64(x * 2.0))
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -5.3e+25)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= 0.044)
		tmp = Float64(Float64(F * Float64(1.0 / Float64(sin(B) * sqrt(t_0)))) - t_1);
	else
		tmp = Float64(Float64(F * Float64(1.0 / Float64(sin(B) * Float64(F + Float64(0.5 * Float64(t_0 / F)))))) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = 2.0 + (x * 2.0);
	t_1 = x / tan(B);
	tmp = 0.0;
	if (F <= -5.3e+25)
		tmp = (-1.0 / sin(B)) - t_1;
	elseif (F <= 0.044)
		tmp = (F * (1.0 / (sin(B) * sqrt(t_0)))) - t_1;
	else
		tmp = (F * (1.0 / (sin(B) * (F + (0.5 * (t_0 / F)))))) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -5.3e+25], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, 0.044], N[(N[(F * N[(1.0 / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(N[(F * N[(1.0 / N[(N[Sin[B], $MachinePrecision] * N[(F + N[(0.5 * N[(t$95$0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.29999999999999986e25

   1. Initial program 51.6%

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

      1. distribute-lft-neg-in 51.6%

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

      2. +-commutative 51.6%

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

      3. cancel-sign-sub-inv 51.6%

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

   3. Simplified 68.5%

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

      1. clear-num 68.5%

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

      2. inv-pow 68.5%

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

      3. fma-def 68.5%

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

      4. fma-udef 68.5%

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

      5. *-commutative 68.5%

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

      6. fma-def 68.5%

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

      7. fma-def 68.5%

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

   5. Applied egg-rr 68.5%

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

      1. unpow-1 68.5%

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

      2. fma-udef 68.5%

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

      3. fma-udef 68.5%

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

      4. unpow2 68.5%

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

      5. +-commutative 68.5%

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

      6. associate-+r+ 68.5%

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

      7. +-commutative 68.5%

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

      8. +-commutative 68.5%

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

      9. unpow2 68.5%

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

      10. fma-def 68.5%

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

      11. +-commutative 68.5%

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

      12. fma-def 68.5%

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

   7. Simplified 68.5%

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

   8. Taylor expanded in F around -inf 99.9%

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

   if -5.29999999999999986e25 < F < 0.043999999999999997

   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. 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%

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

      2. fma-udef 99.6%

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

      3. fma-udef 99.6%

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

      4. unpow2 99.6%

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

      5. +-commutative 99.6%

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

      6. associate-+r+ 99.6%

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

      7. +-commutative 99.6%

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

      8. +-commutative 99.6%

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

      9. unpow2 99.6%

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

      10. fma-def 99.6%

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

      11. +-commutative 99.6%

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

      12. fma-def 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in F around 0 97.8%

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

   if 0.043999999999999997 < F

   1. Initial program 55.6%

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

      1. distribute-lft-neg-in 55.6%

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

      2. +-commutative 55.6%

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

      3. cancel-sign-sub-inv 55.6%

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

   3. Simplified 69.2%

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

      1. clear-num 69.3%

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

      2. inv-pow 69.3%

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

      3. fma-def 69.3%

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

      4. fma-udef 69.3%

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

      5. *-commutative 69.3%

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

      6. fma-def 69.3%

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

      7. fma-def 69.3%

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

   5. Applied egg-rr 69.3%

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

      1. unpow-1 69.3%

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

      2. fma-udef 69.3%

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

      3. fma-udef 69.3%

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

      4. unpow2 69.3%

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

      5. +-commutative 69.3%

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

      6. associate-+r+ 69.3%

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

      7. +-commutative 69.3%

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

      8. +-commutative 69.3%

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

      9. unpow2 69.3%

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

      10. fma-def 69.3%

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

      11. +-commutative 69.3%

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

      12. fma-def 69.3%

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

   7. Simplified 69.3%

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

   8. Taylor expanded in B around inf 69.2%

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

   9. Taylor expanded in F around inf 99.2%

      \[\leadsto F \cdot \frac{1}{\sin B \cdot \color{blue}{\left(F + 0.5 \cdot \frac{2 + 2 \cdot x}{F}\right)}} - \frac{x}{\tan B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5.3 \cdot 10^{+25}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.044:\\ \;\;\;\;F \cdot \frac{1}{\sin B \cdot \sqrt{2 + x \cdot 2}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{\sin B \cdot \left(F + 0.5 \cdot \frac{2 + x \cdot 2}{F}\right)} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 4: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2.8 \cdot 10^{+93}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq 22000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + F \cdot F}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -2.8e+93)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 22000000.0)
       (- (* (/ F (sin B)) (sqrt (/ 1.0 (+ 2.0 (* F F))))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -2.8e+93) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 22000000.0) {
		tmp = ((F / sin(B)) * sqrt((1.0 / (2.0 + (F * F))))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-2.8d+93)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 22000000.0d0) then
        tmp = ((f / sin(b)) * sqrt((1.0d0 / (2.0d0 + (f * f))))) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -2.8e+93) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 22000000.0) {
		tmp = ((F / Math.sin(B)) * Math.sqrt((1.0 / (2.0 + (F * F))))) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -2.8e+93:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 22000000.0:
		tmp = ((F / math.sin(B)) * math.sqrt((1.0 / (2.0 + (F * F))))) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -2.8e+93)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 22000000.0)
		tmp = Float64(Float64(Float64(F / sin(B)) * sqrt(Float64(1.0 / Float64(2.0 + Float64(F * F))))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -2.8e+93)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 22000000.0)
		tmp = ((F / sin(B)) * sqrt((1.0 / (2.0 + (F * F))))) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.8e+93], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 22000000.0], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.79999999999999989e93

   1. Initial program 38.6%

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

      1. distribute-lft-neg-in 38.6%

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

      2. +-commutative 38.6%

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

      3. cancel-sign-sub-inv 38.6%

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

   3. Simplified 60.0%

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

      1. clear-num 60.0%

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

      2. inv-pow 60.0%

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

      3. fma-def 60.0%

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

      4. fma-udef 60.0%

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

      5. *-commutative 60.0%

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

      6. fma-def 60.0%

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

      7. fma-def 60.0%

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

   5. Applied egg-rr 60.0%

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

      1. unpow-1 60.0%

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

      2. fma-udef 60.0%

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

      3. fma-udef 60.0%

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

      4. unpow2 60.0%

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

      5. +-commutative 60.0%

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

      6. associate-+r+ 60.0%

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

      7. +-commutative 60.0%

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

      8. +-commutative 60.0%

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

      9. unpow2 60.0%

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

      10. fma-def 60.0%

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

      11. +-commutative 60.0%

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

      12. fma-def 60.0%

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

   7. Simplified 60.0%

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

   8. Taylor expanded in F around -inf 99.9%

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

   if -2.79999999999999989e93 < F < 2.2e7

   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%

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

      2. fma-udef 99.6%

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

      3. fma-udef 99.6%

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

      4. unpow2 99.6%

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

      5. +-commutative 99.6%

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

      6. associate-+r+ 99.6%

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

      7. +-commutative 99.6%

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

      8. +-commutative 99.6%

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

      9. unpow2 99.6%

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

      10. fma-def 99.6%

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

      11. +-commutative 99.6%

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

      12. fma-def 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in x around 0 99.2%

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

      1. *-commutative 99.2%

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

      2. +-commutative 99.2%

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

      3. unpow2 99.2%

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

   10. Simplified 99.2%

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

   if 2.2e7 < F

   1. Initial program 54.0%

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

      1. distribute-lft-neg-in 54.0%

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

      2. +-commutative 54.0%

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

      3. cancel-sign-sub-inv 54.0%

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

   3. Simplified 68.1%

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

      1. clear-num 68.1%

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

      2. inv-pow 68.1%

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

      3. fma-def 68.1%

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

      4. fma-udef 68.1%

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

      5. *-commutative 68.1%

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

      6. fma-def 68.1%

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

      7. fma-def 68.1%

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

   5. Applied egg-rr 68.1%

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

      1. unpow-1 68.1%

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

      2. fma-udef 68.1%

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

      3. fma-udef 68.1%

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

      4. unpow2 68.1%

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

      5. +-commutative 68.1%

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

      6. associate-+r+ 68.1%

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

      7. +-commutative 68.1%

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

      8. +-commutative 68.1%

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

      9. unpow2 68.1%

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

      10. fma-def 68.1%

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

      11. +-commutative 68.1%

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

      12. fma-def 68.1%

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

   7. Simplified 68.1%

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

   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 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.8 \cdot 10^{+93}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 22000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + F \cdot F}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 5: 88.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + x \cdot 2\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.65 \cdot 10^{+24}:\\ \;\;\;\;\frac{-1}{\sin B} - t_1\\ \mathbf{elif}\;F \leq -4.7 \cdot 10^{-160}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{-100} \lor \neg \left(F \leq 3.3 \cdot 10^{-7}\right):\\ \;\;\;\;F \cdot \frac{1}{\sin B \cdot \left(F + 0.5 \cdot \frac{t_0}{F}\right)} - t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{t_0}} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (+ 2.0 (* x 2.0))) (t_1 (/ x (tan B))))
   (if (<= F -1.65e+24)
     (- (/ -1.0 (sin B)) t_1)
     (if (<= F -4.7e-160)
       (- (* F (/ (sqrt 0.5) (sin B))) (/ x B))
       (if (or (<= F 7.5e-100) (not (<= F 3.3e-7)))
         (- (* F (/ 1.0 (* (sin B) (+ F (* 0.5 (/ t_0 F)))))) t_1)
         (- (* (/ F (sin B)) (sqrt (/ 1.0 t_0))) (/ x B)))))))

C

double code(double F, double B, double x) {
	double t_0 = 2.0 + (x * 2.0);
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -1.65e+24) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= -4.7e-160) {
		tmp = (F * (sqrt(0.5) / sin(B))) - (x / B);
	} else if ((F <= 7.5e-100) || !(F <= 3.3e-7)) {
		tmp = (F * (1.0 / (sin(B) * (F + (0.5 * (t_0 / F)))))) - t_1;
	} else {
		tmp = ((F / sin(B)) * sqrt((1.0 / 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) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 + (x * 2.0d0)
    t_1 = x / tan(b)
    if (f <= (-1.65d+24)) then
        tmp = ((-1.0d0) / sin(b)) - t_1
    else if (f <= (-4.7d-160)) then
        tmp = (f * (sqrt(0.5d0) / sin(b))) - (x / b)
    else if ((f <= 7.5d-100) .or. (.not. (f <= 3.3d-7))) then
        tmp = (f * (1.0d0 / (sin(b) * (f + (0.5d0 * (t_0 / f)))))) - t_1
    else
        tmp = ((f / sin(b)) * sqrt((1.0d0 / t_0))) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = 2.0 + (x * 2.0);
	double t_1 = x / Math.tan(B);
	double tmp;
	if (F <= -1.65e+24) {
		tmp = (-1.0 / Math.sin(B)) - t_1;
	} else if (F <= -4.7e-160) {
		tmp = (F * (Math.sqrt(0.5) / Math.sin(B))) - (x / B);
	} else if ((F <= 7.5e-100) || !(F <= 3.3e-7)) {
		tmp = (F * (1.0 / (Math.sin(B) * (F + (0.5 * (t_0 / F)))))) - t_1;
	} else {
		tmp = ((F / Math.sin(B)) * Math.sqrt((1.0 / t_0))) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = 2.0 + (x * 2.0)
	t_1 = x / math.tan(B)
	tmp = 0
	if F <= -1.65e+24:
		tmp = (-1.0 / math.sin(B)) - t_1
	elif F <= -4.7e-160:
		tmp = (F * (math.sqrt(0.5) / math.sin(B))) - (x / B)
	elif (F <= 7.5e-100) or not (F <= 3.3e-7):
		tmp = (F * (1.0 / (math.sin(B) * (F + (0.5 * (t_0 / F)))))) - t_1
	else:
		tmp = ((F / math.sin(B)) * math.sqrt((1.0 / t_0))) - (x / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(2.0 + Float64(x * 2.0))
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.65e+24)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= -4.7e-160)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / sin(B))) - Float64(x / B));
	elseif ((F <= 7.5e-100) || !(F <= 3.3e-7))
		tmp = Float64(Float64(F * Float64(1.0 / Float64(sin(B) * Float64(F + Float64(0.5 * Float64(t_0 / F)))))) - t_1);
	else
		tmp = Float64(Float64(Float64(F / sin(B)) * sqrt(Float64(1.0 / t_0))) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = 2.0 + (x * 2.0);
	t_1 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.65e+24)
		tmp = (-1.0 / sin(B)) - t_1;
	elseif (F <= -4.7e-160)
		tmp = (F * (sqrt(0.5) / sin(B))) - (x / B);
	elseif ((F <= 7.5e-100) || ~((F <= 3.3e-7)))
		tmp = (F * (1.0 / (sin(B) * (F + (0.5 * (t_0 / F)))))) - t_1;
	else
		tmp = ((F / sin(B)) * sqrt((1.0 / t_0))) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.65e+24], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, -4.7e-160], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[F, 7.5e-100], N[Not[LessEqual[F, 3.3e-7]], $MachinePrecision]], N[(N[(F * N[(1.0 / N[(N[Sin[B], $MachinePrecision] * N[(F + N[(0.5 * N[(t$95$0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.6499999999999999e24

   1. Initial program 52.3%

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

      1. distribute-lft-neg-in 52.3%

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

      2. +-commutative 52.3%

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

      3. cancel-sign-sub-inv 52.3%

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

   3. Simplified 69.0%

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

      1. clear-num 68.9%

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

      2. inv-pow 68.9%

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

      3. fma-def 68.9%

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

      4. fma-udef 68.9%

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

      5. *-commutative 68.9%

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

      6. fma-def 68.9%

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

      7. fma-def 68.9%

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

   5. Applied egg-rr 68.9%

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

      1. unpow-1 68.9%

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

      2. fma-udef 68.9%

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

      3. fma-udef 68.9%

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

      4. unpow2 68.9%

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

      5. +-commutative 68.9%

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

      6. associate-+r+ 68.9%

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

      7. +-commutative 68.9%

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

      8. +-commutative 68.9%

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

      9. unpow2 68.9%

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

      10. fma-def 68.9%

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

      11. +-commutative 68.9%

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

      12. fma-def 68.9%

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

   7. Simplified 68.9%

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

   8. Taylor expanded in F around -inf 99.9%

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

   if -1.6499999999999999e24 < F < -4.6999999999999998e-160

   1. Initial program 99.3%

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

   2. Taylor expanded in B around 0 83.9%

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

   3. Taylor expanded in F around 0 82.2%

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

   4. Taylor expanded in x around 0 82.2%

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

      1. associate-/l* 82.4%

         \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{\sqrt{0.5}}{\frac{\sin B}{F}}} \]

      2. associate-/r/ 82.4%

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

   6. Simplified 82.4%

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

   if -4.6999999999999998e-160 < F < 7.50000000000000015e-100 or 3.3000000000000002e-7 < F

   1. Initial program 75.2%

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

      1. distribute-lft-neg-in 75.2%

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

      2. +-commutative 75.2%

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

      3. cancel-sign-sub-inv 75.2%

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

   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 82.8%

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

      2. inv-pow 82.8%

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

      3. fma-def 82.8%

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

      4. fma-udef 82.8%

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

      5. *-commutative 82.8%

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

      6. fma-def 82.8%

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

      7. fma-def 82.8%

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

   5. Applied egg-rr 82.8%

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

      1. unpow-1 82.8%

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

      2. fma-udef 82.8%

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

      3. fma-udef 82.8%

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

      4. unpow2 82.8%

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

      5. +-commutative 82.8%

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

      6. associate-+r+ 82.8%

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

      7. +-commutative 82.8%

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

      8. +-commutative 82.8%

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

      9. unpow2 82.8%

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

      10. fma-def 82.8%

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

      11. +-commutative 82.8%

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

      12. fma-def 82.8%

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

   7. Simplified 82.8%

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

   8. Taylor expanded in B around inf 82.8%

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

   9. Taylor expanded in F around inf 90.7%

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

   if 7.50000000000000015e-100 < F < 3.3000000000000002e-7

   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. Taylor expanded in B around 0 87.0%

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

   3. Taylor expanded in F around 0 87.0%

      \[\leadsto \left(-\frac{x}{B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\sqrt{\frac{1}{2 \cdot x + 2}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.65 \cdot 10^{+24}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -4.7 \cdot 10^{-160}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{-100} \lor \neg \left(F \leq 3.3 \cdot 10^{-7}\right):\\ \;\;\;\;F \cdot \frac{1}{\sin B \cdot \left(F + 0.5 \cdot \frac{2 + x \cdot 2}{F}\right)} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ \end{array} \]

Alternative 6: 89.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.65 \cdot 10^{+24}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq -8 \cdot 10^{-155} \lor \neg \left(F \leq 1.15 \cdot 10^{-98}\right) \land F \leq 0.044:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{\sin B \cdot \left(F + 0.5 \cdot \frac{2 + x \cdot 2}{F}\right)} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.65e+24)
     (- (/ -1.0 (sin B)) t_0)
     (if (or (<= F -8e-155) (and (not (<= F 1.15e-98)) (<= F 0.044)))
       (- (* (/ F (sin B)) (pow (+ (* x 2.0) (+ 2.0 (* F F))) -0.5)) (/ x B))
       (-
        (* F (/ 1.0 (* (sin B) (+ F (* 0.5 (/ (+ 2.0 (* x 2.0)) F))))))
        t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.65e+24) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if ((F <= -8e-155) || (!(F <= 1.15e-98) && (F <= 0.044))) {
		tmp = ((F / sin(B)) * pow(((x * 2.0) + (2.0 + (F * F))), -0.5)) - (x / B);
	} else {
		tmp = (F * (1.0 / (sin(B) * (F + (0.5 * ((2.0 + (x * 2.0)) / 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 <= (-1.65d+24)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if ((f <= (-8d-155)) .or. (.not. (f <= 1.15d-98)) .and. (f <= 0.044d0)) then
        tmp = ((f / sin(b)) * (((x * 2.0d0) + (2.0d0 + (f * f))) ** (-0.5d0))) - (x / b)
    else
        tmp = (f * (1.0d0 / (sin(b) * (f + (0.5d0 * ((2.0d0 + (x * 2.0d0)) / 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 <= -1.65e+24) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if ((F <= -8e-155) || (!(F <= 1.15e-98) && (F <= 0.044))) {
		tmp = ((F / Math.sin(B)) * Math.pow(((x * 2.0) + (2.0 + (F * F))), -0.5)) - (x / B);
	} else {
		tmp = (F * (1.0 / (Math.sin(B) * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))))) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.65e+24:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif (F <= -8e-155) or (not (F <= 1.15e-98) and (F <= 0.044)):
		tmp = ((F / math.sin(B)) * math.pow(((x * 2.0) + (2.0 + (F * F))), -0.5)) - (x / B)
	else:
		tmp = (F * (1.0 / (math.sin(B) * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))))) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.65e+24)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif ((F <= -8e-155) || (!(F <= 1.15e-98) && (F <= 0.044)))
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(x * 2.0) + Float64(2.0 + Float64(F * F))) ^ -0.5)) - Float64(x / B));
	else
		tmp = Float64(Float64(F * Float64(1.0 / Float64(sin(B) * Float64(F + Float64(0.5 * Float64(Float64(2.0 + Float64(x * 2.0)) / 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 <= -1.65e+24)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif ((F <= -8e-155) || (~((F <= 1.15e-98)) && (F <= 0.044)))
		tmp = ((F / sin(B)) * (((x * 2.0) + (2.0 + (F * F))) ^ -0.5)) - (x / B);
	else
		tmp = (F * (1.0 / (sin(B) * (F + (0.5 * ((2.0 + (x * 2.0)) / 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, -1.65e+24], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[Or[LessEqual[F, -8e-155], And[N[Not[LessEqual[F, 1.15e-98]], $MachinePrecision], LessEqual[F, 0.044]]], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(x * 2.0), $MachinePrecision] + N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(F * N[(1.0 / N[(N[Sin[B], $MachinePrecision] * N[(F + N[(0.5 * N[(N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.6499999999999999e24

   1. Initial program 52.3%

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

      1. distribute-lft-neg-in 52.3%

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

      2. +-commutative 52.3%

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

      3. cancel-sign-sub-inv 52.3%

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

   3. Simplified 69.0%

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

      1. clear-num 68.9%

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

      2. inv-pow 68.9%

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

      3. fma-def 68.9%

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

      4. fma-udef 68.9%

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

      5. *-commutative 68.9%

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

      6. fma-def 68.9%

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

      7. fma-def 68.9%

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

   5. Applied egg-rr 68.9%

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

      1. unpow-1 68.9%

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

      2. fma-udef 68.9%

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

      3. fma-udef 68.9%

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

      4. unpow2 68.9%

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

      5. +-commutative 68.9%

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

      6. associate-+r+ 68.9%

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

      7. +-commutative 68.9%

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

      8. +-commutative 68.9%

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

      9. unpow2 68.9%

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

      10. fma-def 68.9%

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

      11. +-commutative 68.9%

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

      12. fma-def 68.9%

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

   7. Simplified 68.9%

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

   8. Taylor expanded in F around -inf 99.9%

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

   if -1.6499999999999999e24 < F < -8.00000000000000011e-155 or 1.15e-98 < F < 0.043999999999999997

   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. Taylor expanded in B around 0 82.3%

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

   if -8.00000000000000011e-155 < F < 1.15e-98 or 0.043999999999999997 < F

   1. Initial program 74.4%

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

      1. distribute-lft-neg-in 74.4%

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

      2. +-commutative 74.4%

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

      3. cancel-sign-sub-inv 74.4%

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

   3. Simplified 82.2%

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

      1. clear-num 82.2%

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

      2. inv-pow 82.2%

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

      3. fma-def 82.2%

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

      4. fma-udef 82.2%

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

      5. *-commutative 82.2%

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

      6. fma-def 82.2%

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

      7. fma-def 82.2%

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

   5. Applied egg-rr 82.2%

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

      1. unpow-1 82.2%

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

      2. fma-udef 82.2%

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

      3. fma-udef 82.2%

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

      4. unpow2 82.2%

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

      5. +-commutative 82.2%

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

      6. associate-+r+ 82.2%

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

      7. +-commutative 82.2%

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

      8. +-commutative 82.2%

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

      9. unpow2 82.2%

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

      10. fma-def 82.2%

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

      11. +-commutative 82.2%

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

      12. fma-def 82.2%

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

   7. Simplified 82.2%

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

   8. Taylor expanded in B around inf 82.2%

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

   9. Taylor expanded in F around inf 92.2%

      \[\leadsto F \cdot \frac{1}{\sin B \cdot \color{blue}{\left(F + 0.5 \cdot \frac{2 + 2 \cdot x}{F}\right)}} - \frac{x}{\tan B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.65 \cdot 10^{+24}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -8 \cdot 10^{-155} \lor \neg \left(F \leq 1.15 \cdot 10^{-98}\right) \land F \leq 0.044:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{\sin B \cdot \left(F + 0.5 \cdot \frac{2 + x \cdot 2}{F}\right)} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 7: 91.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{-0.5}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.65 \cdot 10^{+24}:\\ \;\;\;\;\frac{-1}{\sin B} - t_1\\ \mathbf{elif}\;F \leq -5.6 \cdot 10^{-53}:\\ \;\;\;\;\frac{F}{\sin B} \cdot t_0 - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.00017:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + t_0 \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{\sin B \cdot \left(F + 0.5 \cdot \frac{2 + x \cdot 2}{F}\right)} - t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (pow (+ (* x 2.0) (+ 2.0 (* F F))) -0.5)) (t_1 (/ x (tan B))))
   (if (<= F -1.65e+24)
     (- (/ -1.0 (sin B)) t_1)
     (if (<= F -5.6e-53)
       (- (* (/ F (sin B)) t_0) (/ x B))
       (if (<= F 0.00017)
         (+ (* x (/ -1.0 (tan B))) (* t_0 (/ F B)))
         (-
          (* F (/ 1.0 (* (sin B) (+ F (* 0.5 (/ (+ 2.0 (* x 2.0)) F))))))
          t_1))))))

C

double code(double F, double B, double x) {
	double t_0 = pow(((x * 2.0) + (2.0 + (F * F))), -0.5);
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -1.65e+24) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= -5.6e-53) {
		tmp = ((F / sin(B)) * t_0) - (x / B);
	} else if (F <= 0.00017) {
		tmp = (x * (-1.0 / tan(B))) + (t_0 * (F / B));
	} else {
		tmp = (F * (1.0 / (sin(B) * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))))) - t_1;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((x * 2.0d0) + (2.0d0 + (f * f))) ** (-0.5d0)
    t_1 = x / tan(b)
    if (f <= (-1.65d+24)) then
        tmp = ((-1.0d0) / sin(b)) - t_1
    else if (f <= (-5.6d-53)) then
        tmp = ((f / sin(b)) * t_0) - (x / b)
    else if (f <= 0.00017d0) then
        tmp = (x * ((-1.0d0) / tan(b))) + (t_0 * (f / b))
    else
        tmp = (f * (1.0d0 / (sin(b) * (f + (0.5d0 * ((2.0d0 + (x * 2.0d0)) / f)))))) - t_1
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = Math.pow(((x * 2.0) + (2.0 + (F * F))), -0.5);
	double t_1 = x / Math.tan(B);
	double tmp;
	if (F <= -1.65e+24) {
		tmp = (-1.0 / Math.sin(B)) - t_1;
	} else if (F <= -5.6e-53) {
		tmp = ((F / Math.sin(B)) * t_0) - (x / B);
	} else if (F <= 0.00017) {
		tmp = (x * (-1.0 / Math.tan(B))) + (t_0 * (F / B));
	} else {
		tmp = (F * (1.0 / (Math.sin(B) * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))))) - t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = math.pow(((x * 2.0) + (2.0 + (F * F))), -0.5)
	t_1 = x / math.tan(B)
	tmp = 0
	if F <= -1.65e+24:
		tmp = (-1.0 / math.sin(B)) - t_1
	elif F <= -5.6e-53:
		tmp = ((F / math.sin(B)) * t_0) - (x / B)
	elif F <= 0.00017:
		tmp = (x * (-1.0 / math.tan(B))) + (t_0 * (F / B))
	else:
		tmp = (F * (1.0 / (math.sin(B) * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))))) - t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(x * 2.0) + Float64(2.0 + Float64(F * F))) ^ -0.5
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.65e+24)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= -5.6e-53)
		tmp = Float64(Float64(Float64(F / sin(B)) * t_0) - Float64(x / B));
	elseif (F <= 0.00017)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(t_0 * Float64(F / B)));
	else
		tmp = Float64(Float64(F * Float64(1.0 / Float64(sin(B) * Float64(F + Float64(0.5 * Float64(Float64(2.0 + Float64(x * 2.0)) / F)))))) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = ((x * 2.0) + (2.0 + (F * F))) ^ -0.5;
	t_1 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.65e+24)
		tmp = (-1.0 / sin(B)) - t_1;
	elseif (F <= -5.6e-53)
		tmp = ((F / sin(B)) * t_0) - (x / B);
	elseif (F <= 0.00017)
		tmp = (x * (-1.0 / tan(B))) + (t_0 * (F / B));
	else
		tmp = (F * (1.0 / (sin(B) * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))))) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[Power[N[(N[(x * 2.0), $MachinePrecision] + N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.65e+24], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, -5.6e-53], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.00017], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(F / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(F * N[(1.0 / N[(N[Sin[B], $MachinePrecision] * N[(F + N[(0.5 * N[(N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.6499999999999999e24

   1. Initial program 52.3%

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

      1. distribute-lft-neg-in 52.3%

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

      2. +-commutative 52.3%

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

      3. cancel-sign-sub-inv 52.3%

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

   3. Simplified 69.0%

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

      1. clear-num 68.9%

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

      2. inv-pow 68.9%

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

      3. fma-def 68.9%

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

      4. fma-udef 68.9%

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

      5. *-commutative 68.9%

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

      6. fma-def 68.9%

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

      7. fma-def 68.9%

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

   5. Applied egg-rr 68.9%

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

      1. unpow-1 68.9%

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

      2. fma-udef 68.9%

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

      3. fma-udef 68.9%

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

      4. unpow2 68.9%

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

      5. +-commutative 68.9%

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

      6. associate-+r+ 68.9%

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

      7. +-commutative 68.9%

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

      8. +-commutative 68.9%

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

      9. unpow2 68.9%

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

      10. fma-def 68.9%

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

      11. +-commutative 68.9%

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

      12. fma-def 68.9%

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

   7. Simplified 68.9%

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

   8. Taylor expanded in F around -inf 99.9%

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

   if -1.6499999999999999e24 < F < -5.59999999999999971e-53

   1. Initial program 99.7%

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

   2. Taylor expanded in B around 0 94.1%

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

   if -5.59999999999999971e-53 < F < 1.7e-4

   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. Taylor expanded in B around 0 84.7%

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

   if 1.7e-4 < F

   1. Initial program 57.2%

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

      1. distribute-lft-neg-in 57.2%

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

      2. +-commutative 57.2%

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

      3. cancel-sign-sub-inv 57.2%

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

   3. Simplified 70.3%

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

      1. clear-num 70.3%

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

      2. inv-pow 70.3%

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

      3. fma-def 70.3%

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

      4. fma-udef 70.3%

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

      5. *-commutative 70.3%

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

      6. fma-def 70.3%

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

      7. fma-def 70.3%

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

   5. Applied egg-rr 70.3%

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

      1. unpow-1 70.3%

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

      2. fma-udef 70.3%

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

      3. fma-udef 70.3%

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

      4. unpow2 70.3%

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

      5. +-commutative 70.3%

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

      6. associate-+r+ 70.3%

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

      7. +-commutative 70.3%

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

      8. +-commutative 70.3%

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

      9. unpow2 70.3%

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

      10. fma-def 70.3%

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

      11. +-commutative 70.3%

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

      12. fma-def 70.3%

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

   7. Simplified 70.3%

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

   8. Taylor expanded in B around inf 70.3%

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

   9. Taylor expanded in F around inf 96.2%

      \[\leadsto F \cdot \frac{1}{\sin B \cdot \color{blue}{\left(F + 0.5 \cdot \frac{2 + 2 \cdot x}{F}\right)}} - \frac{x}{\tan B} \]
3. Recombined 4 regimes into one program.

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.65 \cdot 10^{+24}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -5.6 \cdot 10^{-53}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.00017:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + {\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{-0.5} \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{\sin B \cdot \left(F + 0.5 \cdot \frac{2 + x \cdot 2}{F}\right)} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 8: 91.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{-0.5}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.65 \cdot 10^{+24}:\\ \;\;\;\;\frac{-1}{\sin B} - t_1\\ \mathbf{elif}\;F \leq -3.4 \cdot 10^{-57}:\\ \;\;\;\;t_0 \cdot \left(F \cdot \frac{1}{\sin B}\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.00017:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + t_0 \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{\sin B \cdot \left(F + 0.5 \cdot \frac{2 + x \cdot 2}{F}\right)} - t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (pow (+ (* x 2.0) (+ 2.0 (* F F))) -0.5)) (t_1 (/ x (tan B))))
   (if (<= F -1.65e+24)
     (- (/ -1.0 (sin B)) t_1)
     (if (<= F -3.4e-57)
       (- (* t_0 (* F (/ 1.0 (sin B)))) (/ x B))
       (if (<= F 0.00017)
         (+ (* x (/ -1.0 (tan B))) (* t_0 (/ F B)))
         (-
          (* F (/ 1.0 (* (sin B) (+ F (* 0.5 (/ (+ 2.0 (* x 2.0)) F))))))
          t_1))))))

C

double code(double F, double B, double x) {
	double t_0 = pow(((x * 2.0) + (2.0 + (F * F))), -0.5);
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -1.65e+24) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= -3.4e-57) {
		tmp = (t_0 * (F * (1.0 / sin(B)))) - (x / B);
	} else if (F <= 0.00017) {
		tmp = (x * (-1.0 / tan(B))) + (t_0 * (F / B));
	} else {
		tmp = (F * (1.0 / (sin(B) * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))))) - t_1;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((x * 2.0d0) + (2.0d0 + (f * f))) ** (-0.5d0)
    t_1 = x / tan(b)
    if (f <= (-1.65d+24)) then
        tmp = ((-1.0d0) / sin(b)) - t_1
    else if (f <= (-3.4d-57)) then
        tmp = (t_0 * (f * (1.0d0 / sin(b)))) - (x / b)
    else if (f <= 0.00017d0) then
        tmp = (x * ((-1.0d0) / tan(b))) + (t_0 * (f / b))
    else
        tmp = (f * (1.0d0 / (sin(b) * (f + (0.5d0 * ((2.0d0 + (x * 2.0d0)) / f)))))) - t_1
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = Math.pow(((x * 2.0) + (2.0 + (F * F))), -0.5);
	double t_1 = x / Math.tan(B);
	double tmp;
	if (F <= -1.65e+24) {
		tmp = (-1.0 / Math.sin(B)) - t_1;
	} else if (F <= -3.4e-57) {
		tmp = (t_0 * (F * (1.0 / Math.sin(B)))) - (x / B);
	} else if (F <= 0.00017) {
		tmp = (x * (-1.0 / Math.tan(B))) + (t_0 * (F / B));
	} else {
		tmp = (F * (1.0 / (Math.sin(B) * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))))) - t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = math.pow(((x * 2.0) + (2.0 + (F * F))), -0.5)
	t_1 = x / math.tan(B)
	tmp = 0
	if F <= -1.65e+24:
		tmp = (-1.0 / math.sin(B)) - t_1
	elif F <= -3.4e-57:
		tmp = (t_0 * (F * (1.0 / math.sin(B)))) - (x / B)
	elif F <= 0.00017:
		tmp = (x * (-1.0 / math.tan(B))) + (t_0 * (F / B))
	else:
		tmp = (F * (1.0 / (math.sin(B) * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))))) - t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(x * 2.0) + Float64(2.0 + Float64(F * F))) ^ -0.5
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.65e+24)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= -3.4e-57)
		tmp = Float64(Float64(t_0 * Float64(F * Float64(1.0 / sin(B)))) - Float64(x / B));
	elseif (F <= 0.00017)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(t_0 * Float64(F / B)));
	else
		tmp = Float64(Float64(F * Float64(1.0 / Float64(sin(B) * Float64(F + Float64(0.5 * Float64(Float64(2.0 + Float64(x * 2.0)) / F)))))) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = ((x * 2.0) + (2.0 + (F * F))) ^ -0.5;
	t_1 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.65e+24)
		tmp = (-1.0 / sin(B)) - t_1;
	elseif (F <= -3.4e-57)
		tmp = (t_0 * (F * (1.0 / sin(B)))) - (x / B);
	elseif (F <= 0.00017)
		tmp = (x * (-1.0 / tan(B))) + (t_0 * (F / B));
	else
		tmp = (F * (1.0 / (sin(B) * (F + (0.5 * ((2.0 + (x * 2.0)) / F)))))) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[Power[N[(N[(x * 2.0), $MachinePrecision] + N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.65e+24], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, -3.4e-57], N[(N[(t$95$0 * N[(F * N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.00017], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(F / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(F * N[(1.0 / N[(N[Sin[B], $MachinePrecision] * N[(F + N[(0.5 * N[(N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.6499999999999999e24

   1. Initial program 52.3%

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

      1. distribute-lft-neg-in 52.3%

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

      2. +-commutative 52.3%

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

      3. cancel-sign-sub-inv 52.3%

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

   3. Simplified 69.0%

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

      1. clear-num 68.9%

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

      2. inv-pow 68.9%

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

      3. fma-def 68.9%

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

      4. fma-udef 68.9%

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

      5. *-commutative 68.9%

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

      6. fma-def 68.9%

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

      7. fma-def 68.9%

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

   5. Applied egg-rr 68.9%

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

      1. unpow-1 68.9%

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

      2. fma-udef 68.9%

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

      3. fma-udef 68.9%

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

      4. unpow2 68.9%

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

      5. +-commutative 68.9%

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

      6. associate-+r+ 68.9%

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

      7. +-commutative 68.9%

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

      8. +-commutative 68.9%

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

      9. unpow2 68.9%

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

      10. fma-def 68.9%

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

      11. +-commutative 68.9%

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

      12. fma-def 68.9%

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

   7. Simplified 68.9%

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

   8. Taylor expanded in F around -inf 99.9%

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

   if -1.6499999999999999e24 < F < -3.40000000000000016e-57

   1. Initial program 99.7%

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

   2. Taylor expanded in B around 0 94.1%

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

      1. clear-num 94.1%

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

      2. associate-/r/ 94.1%

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

   4. Applied egg-rr 94.1%

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

   if -3.40000000000000016e-57 < F < 1.7e-4

   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. Taylor expanded in B around 0 84.7%

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

   if 1.7e-4 < F

   1. Initial program 57.2%

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

      1. distribute-lft-neg-in 57.2%

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

      2. +-commutative 57.2%

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

      3. cancel-sign-sub-inv 57.2%

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

   3. Simplified 70.3%

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

      1. clear-num 70.3%

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

      2. inv-pow 70.3%

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

      3. fma-def 70.3%

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

      4. fma-udef 70.3%

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

      5. *-commutative 70.3%

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

      6. fma-def 70.3%

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

      7. fma-def 70.3%

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

   5. Applied egg-rr 70.3%

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

      1. unpow-1 70.3%

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

      2. fma-udef 70.3%

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

      3. fma-udef 70.3%

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

      4. unpow2 70.3%

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

      5. +-commutative 70.3%

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

      6. associate-+r+ 70.3%

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

      7. +-commutative 70.3%

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

      8. +-commutative 70.3%

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

      9. unpow2 70.3%

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

      10. fma-def 70.3%

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

      11. +-commutative 70.3%

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

      12. fma-def 70.3%

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

   7. Simplified 70.3%

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

   8. Taylor expanded in B around inf 70.3%

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

   9. Taylor expanded in F around inf 96.2%

      \[\leadsto F \cdot \frac{1}{\sin B \cdot \color{blue}{\left(F + 0.5 \cdot \frac{2 + 2 \cdot x}{F}\right)}} - \frac{x}{\tan B} \]
3. Recombined 4 regimes into one program.

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.65 \cdot 10^{+24}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -3.4 \cdot 10^{-57}:\\ \;\;\;\;{\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{-0.5} \cdot \left(F \cdot \frac{1}{\sin B}\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.00017:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + {\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{-0.5} \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{\sin B \cdot \left(F + 0.5 \cdot \frac{2 + x \cdot 2}{F}\right)} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 9: 88.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.65 \cdot 10^{+24}:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq -2.5 \cdot 10^{-155}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.16 \cdot 10^{-95}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 1.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.65e+24)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F -2.5e-155)
       (- (* F (/ (sqrt 0.5) (sin B))) (/ x B))
       (if (<= F 1.16e-95)
         (/ (- x) (tan B))
         (if (<= F 1.2e-15)
           (- (* (/ F (sin B)) (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))) (/ x 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 <= -1.65e+24) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= -2.5e-155) {
		tmp = (F * (sqrt(0.5) / sin(B))) - (x / B);
	} else if (F <= 1.16e-95) {
		tmp = -x / tan(B);
	} else if (F <= 1.2e-15) {
		tmp = ((F / sin(B)) * sqrt((1.0 / (2.0 + (x * 2.0))))) - (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) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.65d+24)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= (-2.5d-155)) then
        tmp = (f * (sqrt(0.5d0) / sin(b))) - (x / b)
    else if (f <= 1.16d-95) then
        tmp = -x / tan(b)
    else if (f <= 1.2d-15) then
        tmp = ((f / sin(b)) * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))) - (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 tmp;
	if (F <= -1.65e+24) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= -2.5e-155) {
		tmp = (F * (Math.sqrt(0.5) / Math.sin(B))) - (x / B);
	} else if (F <= 1.16e-95) {
		tmp = -x / Math.tan(B);
	} else if (F <= 1.2e-15) {
		tmp = ((F / Math.sin(B)) * Math.sqrt((1.0 / (2.0 + (x * 2.0))))) - (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)
	tmp = 0
	if F <= -1.65e+24:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= -2.5e-155:
		tmp = (F * (math.sqrt(0.5) / math.sin(B))) - (x / B)
	elif F <= 1.16e-95:
		tmp = -x / math.tan(B)
	elif F <= 1.2e-15:
		tmp = ((F / math.sin(B)) * math.sqrt((1.0 / (2.0 + (x * 2.0))))) - (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))
	tmp = 0.0
	if (F <= -1.65e+24)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= -2.5e-155)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / sin(B))) - Float64(x / B));
	elseif (F <= 1.16e-95)
		tmp = Float64(Float64(-x) / tan(B));
	elseif (F <= 1.2e-15)
		tmp = Float64(Float64(Float64(F / sin(B)) * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))) - Float64(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);
	tmp = 0.0;
	if (F <= -1.65e+24)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= -2.5e-155)
		tmp = (F * (sqrt(0.5) / sin(B))) - (x / B);
	elseif (F <= 1.16e-95)
		tmp = -x / tan(B);
	elseif (F <= 1.2e-15)
		tmp = ((F / sin(B)) * sqrt((1.0 / (2.0 + (x * 2.0))))) - (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]}, If[LessEqual[F, -1.65e+24], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -2.5e-155], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.16e-95], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.2e-15], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -1.6499999999999999e24

   1. Initial program 52.3%

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

      1. distribute-lft-neg-in 52.3%

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

      2. +-commutative 52.3%

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

      3. cancel-sign-sub-inv 52.3%

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

   3. Simplified 69.0%

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

      1. clear-num 68.9%

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

      2. inv-pow 68.9%

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

      3. fma-def 68.9%

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

      4. fma-udef 68.9%

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

      5. *-commutative 68.9%

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

      6. fma-def 68.9%

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

      7. fma-def 68.9%

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

   5. Applied egg-rr 68.9%

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

      1. unpow-1 68.9%

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

      2. fma-udef 68.9%

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

      3. fma-udef 68.9%

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

      4. unpow2 68.9%

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

      5. +-commutative 68.9%

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

      6. associate-+r+ 68.9%

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

      7. +-commutative 68.9%

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

      8. +-commutative 68.9%

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

      9. unpow2 68.9%

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

      10. fma-def 68.9%

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

      11. +-commutative 68.9%

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

      12. fma-def 68.9%

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

   7. Simplified 68.9%

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

   8. Taylor expanded in F around -inf 99.9%

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

   if -1.6499999999999999e24 < F < -2.4999999999999999e-155

   1. Initial program 99.3%

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

   2. Taylor expanded in B around 0 83.9%

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

   3. Taylor expanded in F around 0 82.2%

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

   4. Taylor expanded in x around 0 82.2%

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

      1. associate-/l* 82.4%

         \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{\sqrt{0.5}}{\frac{\sin B}{F}}} \]

      2. associate-/r/ 82.4%

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

   6. Simplified 82.4%

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

   if -2.4999999999999999e-155 < F < 1.15999999999999997e-95

   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. 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%

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

   4. Taylor expanded in F around 0 82.7%

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

      1. associate-*r/ 82.7%

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

      2. *-commutative 82.7%

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

      3. associate-*r* 82.7%

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

      4. mul-1-neg 82.7%

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

   6. Simplified 82.7%

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

      1. expm1-log1p-u 56.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(-x\right) \cdot \cos B}{\sin B}\right)\right)} \]

      2. expm1-udef 23.8%

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

      3. associate-/l* 23.8%

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

      4. tan-quot 23.8%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{-x}{\color{blue}{\tan B}}\right)} - 1 \]

   8. Applied egg-rr 23.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-x}{\tan B}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 56.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-x}{\tan B}\right)\right)} \]

      2. expm1-log1p 82.7%

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

   10. Simplified 82.7%

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

   if 1.15999999999999997e-95 < F < 1.19999999999999997e-15

   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. Taylor expanded in B around 0 87.0%

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

   3. Taylor expanded in F around 0 87.0%

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

   if 1.19999999999999997e-15 < F

   1. Initial program 58.2%

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

      1. distribute-lft-neg-in 58.2%

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

      2. +-commutative 58.2%

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

      3. cancel-sign-sub-inv 58.2%

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

   3. Simplified 70.9%

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

      1. clear-num 71.0%

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

      2. inv-pow 71.0%

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

      3. fma-def 71.0%

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

      4. fma-udef 71.0%

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

      5. *-commutative 71.0%

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

      6. fma-def 71.0%

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

      7. fma-def 71.0%

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

   5. Applied egg-rr 71.0%

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

      1. unpow-1 71.0%

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

      2. fma-udef 71.0%

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

      3. fma-udef 71.0%

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

      4. unpow2 71.0%

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

      5. +-commutative 71.0%

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

      6. associate-+r+ 71.0%

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

      7. +-commutative 71.0%

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

      8. +-commutative 71.0%

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

      9. unpow2 71.0%

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

      10. fma-def 71.0%

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

      11. +-commutative 71.0%

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

      12. fma-def 71.0%

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

   7. Simplified 71.0%

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

   8. Taylor expanded in F around inf 96.1%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.65 \cdot 10^{+24}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -2.5 \cdot 10^{-155}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.16 \cdot 10^{-95}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 1.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 10: 88.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := F \cdot \frac{\sqrt{0.5}}{\sin B} - \frac{x}{B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.65 \cdot 10^{+24}:\\ \;\;\;\;\frac{-1}{\sin B} - t_1\\ \mathbf{elif}\;F \leq -4.25 \cdot 10^{-156}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 2.65 \cdot 10^{-95}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 3.3 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (* F (/ (sqrt 0.5) (sin B))) (/ x B))) (t_1 (/ x (tan B))))
   (if (<= F -1.65e+24)
     (- (/ -1.0 (sin B)) t_1)
     (if (<= F -4.25e-156)
       t_0
       (if (<= F 2.65e-95)
         (/ (- x) (tan B))
         (if (<= F 3.3e-7) t_0 (- (/ 1.0 (sin B)) t_1)))))))

C

double code(double F, double B, double x) {
	double t_0 = (F * (sqrt(0.5) / sin(B))) - (x / B);
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -1.65e+24) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= -4.25e-156) {
		tmp = t_0;
	} else if (F <= 2.65e-95) {
		tmp = -x / tan(B);
	} else if (F <= 3.3e-7) {
		tmp = t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_1;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (f * (sqrt(0.5d0) / sin(b))) - (x / b)
    t_1 = x / tan(b)
    if (f <= (-1.65d+24)) then
        tmp = ((-1.0d0) / sin(b)) - t_1
    else if (f <= (-4.25d-156)) then
        tmp = t_0
    else if (f <= 2.65d-95) then
        tmp = -x / tan(b)
    else if (f <= 3.3d-7) then
        tmp = t_0
    else
        tmp = (1.0d0 / sin(b)) - t_1
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (F * (Math.sqrt(0.5) / Math.sin(B))) - (x / B);
	double t_1 = x / Math.tan(B);
	double tmp;
	if (F <= -1.65e+24) {
		tmp = (-1.0 / Math.sin(B)) - t_1;
	} else if (F <= -4.25e-156) {
		tmp = t_0;
	} else if (F <= 2.65e-95) {
		tmp = -x / Math.tan(B);
	} else if (F <= 3.3e-7) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (F * (math.sqrt(0.5) / math.sin(B))) - (x / B)
	t_1 = x / math.tan(B)
	tmp = 0
	if F <= -1.65e+24:
		tmp = (-1.0 / math.sin(B)) - t_1
	elif F <= -4.25e-156:
		tmp = t_0
	elif F <= 2.65e-95:
		tmp = -x / math.tan(B)
	elif F <= 3.3e-7:
		tmp = t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(F * Float64(sqrt(0.5) / sin(B))) - Float64(x / B))
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.65e+24)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= -4.25e-156)
		tmp = t_0;
	elseif (F <= 2.65e-95)
		tmp = Float64(Float64(-x) / tan(B));
	elseif (F <= 3.3e-7)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (F * (sqrt(0.5) / sin(B))) - (x / B);
	t_1 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.65e+24)
		tmp = (-1.0 / sin(B)) - t_1;
	elseif (F <= -4.25e-156)
		tmp = t_0;
	elseif (F <= 2.65e-95)
		tmp = -x / tan(B);
	elseif (F <= 3.3e-7)
		tmp = t_0;
	else
		tmp = (1.0 / sin(B)) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.65e+24], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, -4.25e-156], t$95$0, If[LessEqual[F, 2.65e-95], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.3e-7], t$95$0, N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.6499999999999999e24

   1. Initial program 52.3%

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

      1. distribute-lft-neg-in 52.3%

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

      2. +-commutative 52.3%

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

      3. cancel-sign-sub-inv 52.3%

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

   3. Simplified 69.0%

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

      1. clear-num 68.9%

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

      2. inv-pow 68.9%

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

      3. fma-def 68.9%

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

      4. fma-udef 68.9%

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

      5. *-commutative 68.9%

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

      6. fma-def 68.9%

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

      7. fma-def 68.9%

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

   5. Applied egg-rr 68.9%

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

      1. unpow-1 68.9%

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

      2. fma-udef 68.9%

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

      3. fma-udef 68.9%

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

      4. unpow2 68.9%

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

      5. +-commutative 68.9%

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

      6. associate-+r+ 68.9%

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

      7. +-commutative 68.9%

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

      8. +-commutative 68.9%

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

      9. unpow2 68.9%

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

      10. fma-def 68.9%

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

      11. +-commutative 68.9%

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

      12. fma-def 68.9%

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

   7. Simplified 68.9%

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

   8. Taylor expanded in F around -inf 99.9%

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

   if -1.6499999999999999e24 < F < -4.25e-156 or 2.6499999999999999e-95 < F < 3.3000000000000002e-7

   1. Initial program 99.3%

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

   2. Taylor expanded in B around 0 84.6%

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

   3. Taylor expanded in F around 0 83.3%

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

   4. Taylor expanded in x around 0 83.1%

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

      1. associate-/l* 83.3%

         \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{\sqrt{0.5}}{\frac{\sin B}{F}}} \]

      2. associate-/r/ 83.3%

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

   6. Simplified 83.3%

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

   if -4.25e-156 < F < 2.6499999999999999e-95

   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. 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%

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

   4. Taylor expanded in F around 0 82.7%

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

      1. associate-*r/ 82.7%

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

      2. *-commutative 82.7%

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

      3. associate-*r* 82.7%

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

      4. mul-1-neg 82.7%

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

   6. Simplified 82.7%

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

      1. expm1-log1p-u 56.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(-x\right) \cdot \cos B}{\sin B}\right)\right)} \]

      2. expm1-udef 23.8%

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

      3. associate-/l* 23.8%

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

      4. tan-quot 23.8%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{-x}{\color{blue}{\tan B}}\right)} - 1 \]

   8. Applied egg-rr 23.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-x}{\tan B}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 56.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-x}{\tan B}\right)\right)} \]

      2. expm1-log1p 82.7%

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

   10. Simplified 82.7%

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

   if 3.3000000000000002e-7 < F

   1. Initial program 58.2%

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

      1. distribute-lft-neg-in 58.2%

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

      2. +-commutative 58.2%

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

      3. cancel-sign-sub-inv 58.2%

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

   3. Simplified 70.9%

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

      1. clear-num 71.0%

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

      2. inv-pow 71.0%

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

      3. fma-def 71.0%

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

      4. fma-udef 71.0%

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

      5. *-commutative 71.0%

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

      6. fma-def 71.0%

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

      7. fma-def 71.0%

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

   5. Applied egg-rr 71.0%

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

      1. unpow-1 71.0%

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

      2. fma-udef 71.0%

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

      3. fma-udef 71.0%

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

      4. unpow2 71.0%

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

      5. +-commutative 71.0%

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

      6. associate-+r+ 71.0%

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

      7. +-commutative 71.0%

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

      8. +-commutative 71.0%

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

      9. unpow2 71.0%

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

      10. fma-def 71.0%

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

      11. +-commutative 71.0%

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

      12. fma-def 71.0%

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

   7. Simplified 71.0%

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

   8. Taylor expanded in F around inf 96.1%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.65 \cdot 10^{+24}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -4.25 \cdot 10^{-156}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.65 \cdot 10^{-95}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 3.3 \cdot 10^{-7}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 11: 85.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.012:\\ \;\;\;\;\frac{-1}{\sin B} - t_0\\ \mathbf{elif}\;F \leq -3.2 \cdot 10^{-108}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\sin B}{F}}\\ \mathbf{elif}\;F \leq 1.85 \cdot 10^{-74}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -0.012)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F -3.2e-108)
       (/ (sqrt 0.5) (/ (sin B) F))
       (if (<= F 1.85e-74) (/ (- 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 <= -0.012) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= -3.2e-108) {
		tmp = sqrt(0.5) / (sin(B) / F);
	} else if (F <= 1.85e-74) {
		tmp = -x / tan(B);
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-0.012d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= (-3.2d-108)) then
        tmp = sqrt(0.5d0) / (sin(b) / f)
    else if (f <= 1.85d-74) then
        tmp = -x / tan(b)
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -0.012) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= -3.2e-108) {
		tmp = Math.sqrt(0.5) / (Math.sin(B) / F);
	} else if (F <= 1.85e-74) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -0.012:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= -3.2e-108:
		tmp = math.sqrt(0.5) / (math.sin(B) / F)
	elif F <= 1.85e-74:
		tmp = -x / math.tan(B)
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -0.012)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= -3.2e-108)
		tmp = Float64(sqrt(0.5) / Float64(sin(B) / F));
	elseif (F <= 1.85e-74)
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -0.012)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= -3.2e-108)
		tmp = sqrt(0.5) / (sin(B) / F);
	elseif (F <= 1.85e-74)
		tmp = -x / tan(B);
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.012], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -3.2e-108], N[(N[Sqrt[0.5], $MachinePrecision] / N[(N[Sin[B], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.85e-74], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.012

   1. Initial program 53.0%

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

      1. distribute-lft-neg-in 53.0%

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

      2. +-commutative 53.0%

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

      3. cancel-sign-sub-inv 53.0%

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

   3. Simplified 69.4%

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

      1. clear-num 69.4%

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

      2. inv-pow 69.4%

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

      3. fma-def 69.4%

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

      4. fma-udef 69.4%

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

      5. *-commutative 69.4%

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

      6. fma-def 69.4%

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

      7. fma-def 69.4%

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

   5. Applied egg-rr 69.4%

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

      1. unpow-1 69.4%

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

      2. fma-udef 69.4%

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

      3. fma-udef 69.4%

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

      4. unpow2 69.4%

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

      5. +-commutative 69.4%

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

      6. associate-+r+ 69.4%

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

      7. +-commutative 69.4%

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

      8. +-commutative 69.4%

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

      9. unpow2 69.4%

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

      10. fma-def 69.4%

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

      11. +-commutative 69.4%

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

      12. fma-def 69.4%

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

   7. Simplified 69.4%

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

   8. Taylor expanded in F around -inf 99.9%

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

   if -0.012 < F < -3.2e-108

   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. fma-def 99.6%

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

      4. +-commutative 99.6%

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

      5. *-commutative 99.6%

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

      6. fma-def 99.6%

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

      7. fma-def 99.6%

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

      8. metadata-eval 99.6%

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

      9. metadata-eval 99.6%

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

      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%

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

   4. Taylor expanded in F around 0 96.2%

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

   5. Taylor expanded in x around 0 71.9%

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

      1. associate-/l* 72.2%

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

   7. Simplified 72.2%

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

   if -3.2e-108 < F < 1.84999999999999997e-74

   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/ 99.6%

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

      11. *-rgt-identity 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in F around 0 76.2%

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

      1. associate-*r/ 76.2%

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

      2. *-commutative 76.2%

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

      3. associate-*r* 76.2%

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

      4. mul-1-neg 76.2%

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

   6. Simplified 76.2%

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

      1. expm1-log1p-u 53.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(-x\right) \cdot \cos B}{\sin B}\right)\right)} \]

      2. expm1-udef 22.0%

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

      3. associate-/l* 22.0%

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

      4. tan-quot 22.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{-x}{\color{blue}{\tan B}}\right)} - 1 \]

   8. Applied egg-rr 22.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-x}{\tan B}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 53.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-x}{\tan B}\right)\right)} \]

      2. expm1-log1p 76.3%

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

   10. Simplified 76.3%

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

   if 1.84999999999999997e-74 < F

   1. Initial program 61.2%

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

      1. distribute-lft-neg-in 61.2%

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

      2. +-commutative 61.2%

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

      3. cancel-sign-sub-inv 61.2%

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

   3. Simplified 73.0%

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

      1. clear-num 73.1%

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

      2. inv-pow 73.1%

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

      3. fma-def 73.1%

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

      4. fma-udef 73.1%

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

      5. *-commutative 73.1%

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

      6. fma-def 73.1%

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

      7. fma-def 73.1%

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

   5. Applied egg-rr 73.1%

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

      1. unpow-1 73.1%

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

      2. fma-udef 73.1%

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

      3. fma-udef 73.1%

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

      4. unpow2 73.1%

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

      5. +-commutative 73.1%

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

      6. associate-+r+ 73.1%

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

      7. +-commutative 73.1%

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

      8. +-commutative 73.1%

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

      9. unpow2 73.1%

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

      10. fma-def 73.1%

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

      11. +-commutative 73.1%

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

      12. fma-def 73.1%

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

   7. Simplified 73.1%

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

   8. Taylor expanded in F around inf 91.4%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.012:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -3.2 \cdot 10^{-108}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\sin B}{F}}\\ \mathbf{elif}\;F \leq 1.85 \cdot 10^{-74}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 12: 72.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.055:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -4.6 \cdot 10^{-107}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\sin B}{F}}\\ \mathbf{elif}\;F \leq 12:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.055)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F -4.6e-107)
     (/ (sqrt 0.5) (/ (sin B) F))
     (if (<= F 12.0) (/ (- x) (tan B)) (- (/ 1.0 (sin B)) (/ x B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.055) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -4.6e-107) {
		tmp = sqrt(0.5) / (sin(B) / F);
	} else if (F <= 12.0) {
		tmp = -x / tan(B);
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-0.055d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-4.6d-107)) then
        tmp = sqrt(0.5d0) / (sin(b) / f)
    else if (f <= 12.0d0) then
        tmp = -x / tan(b)
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.055) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -4.6e-107) {
		tmp = Math.sqrt(0.5) / (Math.sin(B) / F);
	} else if (F <= 12.0) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -0.055:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -4.6e-107:
		tmp = math.sqrt(0.5) / (math.sin(B) / F)
	elif F <= 12.0:
		tmp = -x / math.tan(B)
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.055)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -4.6e-107)
		tmp = Float64(sqrt(0.5) / Float64(sin(B) / F));
	elseif (F <= 12.0)
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -0.055)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -4.6e-107)
		tmp = sqrt(0.5) / (sin(B) / F);
	elseif (F <= 12.0)
		tmp = -x / tan(B);
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.055], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -4.6e-107], N[(N[Sqrt[0.5], $MachinePrecision] / N[(N[Sin[B], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 12.0], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.0550000000000000003

   1. Initial program 53.0%

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

   2. Taylor expanded in B around 0 32.7%

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

   3. Taylor expanded in F around -inf 78.5%

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

   if -0.0550000000000000003 < F < -4.60000000000000007e-107

   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. fma-def 99.6%

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

      4. +-commutative 99.6%

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

      5. *-commutative 99.6%

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

      6. fma-def 99.6%

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

      7. fma-def 99.6%

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

      8. metadata-eval 99.6%

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

      9. metadata-eval 99.6%

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

      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%

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

   4. Taylor expanded in F around 0 96.2%

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

   5. Taylor expanded in x around 0 71.9%

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

      1. associate-/l* 72.2%

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

   7. Simplified 72.2%

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

   if -4.60000000000000007e-107 < F < 12

   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/ 99.6%

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

      11. *-rgt-identity 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in F around 0 71.1%

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

      1. associate-*r/ 71.1%

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

      2. *-commutative 71.1%

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

      3. associate-*r* 71.1%

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

      4. mul-1-neg 71.1%

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

   6. Simplified 71.1%

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

      1. expm1-log1p-u 48.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(-x\right) \cdot \cos B}{\sin B}\right)\right)} \]

      2. expm1-udef 21.2%

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

      3. associate-/l* 21.2%

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

      4. tan-quot 21.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{-x}{\color{blue}{\tan B}}\right)} - 1 \]

   8. Applied egg-rr 21.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-x}{\tan B}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 48.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-x}{\tan B}\right)\right)} \]

      2. expm1-log1p 71.2%

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

   10. Simplified 71.2%

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

   if 12 < F

   1. Initial program 55.1%

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

   2. Taylor expanded in B around 0 34.7%

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

   3. Taylor expanded in F around inf 77.2%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.055:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -4.6 \cdot 10^{-107}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\sin B}{F}}\\ \mathbf{elif}\;F \leq 12:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 13: 79.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.0027:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.25 \cdot 10^{-106}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\sin B}{F}}\\ \mathbf{elif}\;F \leq 0.32:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.0027)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F -1.25e-106)
     (/ (sqrt 0.5) (/ (sin B) F))
     (if (<= F 0.32) (/ (- x) (tan B)) (- (/ 1.0 (sin B)) (/ x B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.0027) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= -1.25e-106) {
		tmp = sqrt(0.5) / (sin(B) / F);
	} else if (F <= 0.32) {
		tmp = -x / tan(B);
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-0.0027d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= (-1.25d-106)) then
        tmp = sqrt(0.5d0) / (sin(b) / f)
    else if (f <= 0.32d0) then
        tmp = -x / tan(b)
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.0027) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= -1.25e-106) {
		tmp = Math.sqrt(0.5) / (Math.sin(B) / F);
	} else if (F <= 0.32) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -0.0027:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= -1.25e-106:
		tmp = math.sqrt(0.5) / (math.sin(B) / F)
	elif F <= 0.32:
		tmp = -x / math.tan(B)
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.0027)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= -1.25e-106)
		tmp = Float64(sqrt(0.5) / Float64(sin(B) / F));
	elseif (F <= 0.32)
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -0.0027)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= -1.25e-106)
		tmp = sqrt(0.5) / (sin(B) / F);
	elseif (F <= 0.32)
		tmp = -x / tan(B);
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.0027], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.25e-106], N[(N[Sqrt[0.5], $MachinePrecision] / N[(N[Sin[B], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.32], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.0027000000000000001

   1. Initial program 53.0%

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

      1. distribute-lft-neg-in 53.0%

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

      2. +-commutative 53.0%

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

      3. cancel-sign-sub-inv 53.0%

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

   3. Simplified 69.4%

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

      1. clear-num 69.4%

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

      2. inv-pow 69.4%

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

      3. fma-def 69.4%

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

      4. fma-udef 69.4%

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

      5. *-commutative 69.4%

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

      6. fma-def 69.4%

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

      7. fma-def 69.4%

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

   5. Applied egg-rr 69.4%

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

      1. unpow-1 69.4%

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

      2. fma-udef 69.4%

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

      3. fma-udef 69.4%

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

      4. unpow2 69.4%

         \[\leadsto F \cdot \frac{1}{\frac{\sin B}{{\left(2 \cdot x + \left(\color{blue}{{F}^{2}} + 2\right)\right)}^{-0.5}}} - \frac{x}{\tan B} \]

      5. +-commutative 69.4%

         \[\leadsto F \cdot \frac{1}{\frac{\sin B}{{\left(2 \cdot x + \color{blue}{\left(2 + {F}^{2}\right)}\right)}^{-0.5}}} - \frac{x}{\tan B} \]

      6. associate-+r+ 69.4%

         \[\leadsto F \cdot \frac{1}{\frac{\sin B}{{\color{blue}{\left(\left(2 \cdot x + 2\right) + {F}^{2}\right)}}^{-0.5}}} - \frac{x}{\tan B} \]

      7. +-commutative 69.4%

         \[\leadsto F \cdot \frac{1}{\frac{\sin B}{{\left(\color{blue}{\left(2 + 2 \cdot x\right)} + {F}^{2}\right)}^{-0.5}}} - \frac{x}{\tan B} \]

      8. +-commutative 69.4%

         \[\leadsto F \cdot \frac{1}{\frac{\sin B}{{\color{blue}{\left({F}^{2} + \left(2 + 2 \cdot x\right)\right)}}^{-0.5}}} - \frac{x}{\tan B} \]

      9. unpow2 69.4%

         \[\leadsto F \cdot \frac{1}{\frac{\sin B}{{\left(\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)\right)}^{-0.5}}} - \frac{x}{\tan B} \]

      10. fma-def 69.4%

          \[\leadsto F \cdot \frac{1}{\frac{\sin B}{{\color{blue}{\left(\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)\right)}}^{-0.5}}} - \frac{x}{\tan B} \]

      11. +-commutative 69.4%

          \[\leadsto F \cdot \frac{1}{\frac{\sin B}{{\left(\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)\right)}^{-0.5}}} - \frac{x}{\tan B} \]

      12. fma-def 69.4%

          \[\leadsto F \cdot \frac{1}{\frac{\sin B}{{\left(\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)\right)}^{-0.5}}} - \frac{x}{\tan B} \]

   7. Simplified 69.4%

      \[\leadsto F \cdot \color{blue}{\frac{1}{\frac{\sin B}{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-0.5}}}} - \frac{x}{\tan B} \]

   8. Taylor expanded in F around -inf 99.9%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} - \frac{x}{\tan B} \]

   if -0.0027000000000000001 < F < -1.24999999999999996e-106

   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. fma-def 99.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{F}{\sin B}, {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}, \left(-x\right) \cdot \frac{1}{\tan B}\right)} \]

      4. +-commutative 99.6%

         \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\color{blue}{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)}, \left(-x\right) \cdot \frac{1}{\tan B}\right) \]

      5. *-commutative 99.6%

         \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\color{blue}{x \cdot 2} + \left(F \cdot F + 2\right)\right)}^{\left(-\frac{1}{2}\right)}, \left(-x\right) \cdot \frac{1}{\tan B}\right) \]

      6. fma-def 99.6%

         \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\color{blue}{\left(\mathsf{fma}\left(x, 2, F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)}, \left(-x\right) \cdot \frac{1}{\tan B}\right) \]

      7. fma-def 99.6%

         \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)\right)}^{\left(-\frac{1}{2}\right)}, \left(-x\right) \cdot \frac{1}{\tan B}\right) \]

      8. metadata-eval 99.6%

         \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(-\color{blue}{0.5}\right)}, \left(-x\right) \cdot \frac{1}{\tan B}\right) \]

      9. metadata-eval 99.6%

         \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{-0.5}}, \left(-x\right) \cdot \frac{1}{\tan B}\right) \]

      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%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{-x}{\tan B}\right)} \]

   4. Taylor expanded in F around 0 96.2%

      \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, \color{blue}{\sqrt{\frac{1}{2 + 2 \cdot x}}}, \frac{-x}{\tan B}\right) \]

   5. Taylor expanded in x around 0 71.9%

      \[\leadsto \color{blue}{\frac{\sqrt{0.5} \cdot F}{\sin B}} \]
   6. Step-by-step derivation

      1. associate-/l* 72.2%

         \[\leadsto \color{blue}{\frac{\sqrt{0.5}}{\frac{\sin B}{F}}} \]

   7. Simplified 72.2%

      \[\leadsto \color{blue}{\frac{\sqrt{0.5}}{\frac{\sin B}{F}}} \]

   if -1.24999999999999996e-106 < F < 0.320000000000000007

   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/ 99.6%

          \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \color{blue}{\frac{\left(-x\right) \cdot 1}{\tan B}}\right) \]

      11. *-rgt-identity 99.6%

          \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{\color{blue}{-x}}{\tan B}\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{-x}{\tan B}\right)} \]

   4. Taylor expanded in F around 0 71.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{\cos B \cdot x}{\sin B}} \]
   5. Step-by-step derivation

      1. associate-*r/ 71.1%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(\cos B \cdot x\right)}{\sin B}} \]

      2. *-commutative 71.1%

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(x \cdot \cos B\right)}}{\sin B} \]

      3. associate-*r* 71.1%

         \[\leadsto \frac{\color{blue}{\left(-1 \cdot x\right) \cdot \cos B}}{\sin B} \]

      4. mul-1-neg 71.1%

         \[\leadsto \frac{\color{blue}{\left(-x\right)} \cdot \cos B}{\sin B} \]

   6. Simplified 71.1%

      \[\leadsto \color{blue}{\frac{\left(-x\right) \cdot \cos B}{\sin B}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 48.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(-x\right) \cdot \cos B}{\sin B}\right)\right)} \]

      2. expm1-udef 21.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(-x\right) \cdot \cos B}{\sin B}\right)} - 1} \]

      3. associate-/l* 21.2%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{-x}{\frac{\sin B}{\cos B}}}\right)} - 1 \]

      4. tan-quot 21.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{-x}{\color{blue}{\tan B}}\right)} - 1 \]

   8. Applied egg-rr 21.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-x}{\tan B}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 48.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-x}{\tan B}\right)\right)} \]

      2. expm1-log1p 71.2%

         \[\leadsto \color{blue}{\frac{-x}{\tan B}} \]

   10. Simplified 71.2%

       \[\leadsto \color{blue}{\frac{-x}{\tan B}} \]

   if 0.320000000000000007 < F

   1. Initial program 55.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 34.7%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in F around inf 77.2%

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{B}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.0027:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.25 \cdot 10^{-106}:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\sin B}{F}}\\ \mathbf{elif}\;F \leq 0.32:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 14: 62.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{+80}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 6.3 \cdot 10^{-145}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2e+80)
   (- (/ -1.0 B) (/ x B))
   (if (<= F 6.3e-145) (/ (- x) (tan B)) (- (/ 1.0 B) (/ x (tan B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2e+80) {
		tmp = (-1.0 / B) - (x / B);
	} else if (F <= 6.3e-145) {
		tmp = -x / tan(B);
	} else {
		tmp = (1.0 / B) - (x / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-2d+80)) then
        tmp = ((-1.0d0) / b) - (x / b)
    else if (f <= 6.3d-145) then
        tmp = -x / tan(b)
    else
        tmp = (1.0d0 / b) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2e+80) {
		tmp = (-1.0 / B) - (x / B);
	} else if (F <= 6.3e-145) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2e+80:
		tmp = (-1.0 / B) - (x / B)
	elif F <= 6.3e-145:
		tmp = -x / math.tan(B)
	else:
		tmp = (1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2e+80)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / B));
	elseif (F <= 6.3e-145)
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2e+80)
		tmp = (-1.0 / B) - (x / B);
	elseif (F <= 6.3e-145)
		tmp = -x / tan(B);
	else
		tmp = (1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2e+80], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 6.3e-145], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2e80

   1. Initial program 43.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 37.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in B around 0 22.7%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Taylor expanded in F around -inf 54.1%

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{-1}{B}} \]

   if -2e80 < F < 6.3000000000000001e-145

   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/ 99.6%

          \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \color{blue}{\frac{\left(-x\right) \cdot 1}{\tan B}}\right) \]

      11. *-rgt-identity 99.6%

          \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{\color{blue}{-x}}{\tan B}\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{-x}{\tan B}\right)} \]

   4. Taylor expanded in F around 0 66.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{\cos B \cdot x}{\sin B}} \]
   5. Step-by-step derivation

      1. associate-*r/ 66.9%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(\cos B \cdot x\right)}{\sin B}} \]

      2. *-commutative 66.9%

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(x \cdot \cos B\right)}}{\sin B} \]

      3. associate-*r* 66.9%

         \[\leadsto \frac{\color{blue}{\left(-1 \cdot x\right) \cdot \cos B}}{\sin B} \]

      4. mul-1-neg 66.9%

         \[\leadsto \frac{\color{blue}{\left(-x\right)} \cdot \cos B}{\sin B} \]

   6. Simplified 66.9%

      \[\leadsto \color{blue}{\frac{\left(-x\right) \cdot \cos B}{\sin B}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 41.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(-x\right) \cdot \cos B}{\sin B}\right)\right)} \]

      2. expm1-udef 18.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(-x\right) \cdot \cos B}{\sin B}\right)} - 1} \]

      3. associate-/l* 18.2%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{-x}{\frac{\sin B}{\cos B}}}\right)} - 1 \]

      4. tan-quot 18.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{-x}{\color{blue}{\tan B}}\right)} - 1 \]

   8. Applied egg-rr 18.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-x}{\tan B}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 41.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-x}{\tan B}\right)\right)} \]

      2. expm1-log1p 66.9%

         \[\leadsto \color{blue}{\frac{-x}{\tan B}} \]

   10. Simplified 66.9%

       \[\leadsto \color{blue}{\frac{-x}{\tan B}} \]

   if 6.3000000000000001e-145 < F

   1. Initial program 65.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. distribute-lft-neg-in 65.3%

         \[\leadsto \color{blue}{\left(-x\right) \cdot \frac{1}{\tan B}} + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. +-commutative 65.3%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x\right) \cdot \frac{1}{\tan B}} \]

      3. cancel-sign-sub-inv 65.3%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

   3. Simplified 75.8%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]

   4. Taylor expanded in F around inf 84.9%

      \[\leadsto F \cdot \color{blue}{\frac{1}{\sin B \cdot F}} - \frac{x}{\tan B} \]

   5. Taylor expanded in B around 0 70.7%

      \[\leadsto \color{blue}{\frac{1}{B}} - \frac{x}{\tan B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{+80}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 6.3 \cdot 10^{-145}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 15: 64.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -6.8 \cdot 10^{+78}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.4:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -6.8e+78)
   (- (/ -1.0 B) (/ x B))
   (if (<= F 2.4) (/ (- x) (tan B)) (- (/ 1.0 (sin B)) (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -6.8e+78) {
		tmp = (-1.0 / B) - (x / B);
	} else if (F <= 2.4) {
		tmp = -x / tan(B);
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-6.8d+78)) then
        tmp = ((-1.0d0) / b) - (x / b)
    else if (f <= 2.4d0) then
        tmp = -x / tan(b)
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -6.8e+78) {
		tmp = (-1.0 / B) - (x / B);
	} else if (F <= 2.4) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -6.8e+78:
		tmp = (-1.0 / B) - (x / B)
	elif F <= 2.4:
		tmp = -x / math.tan(B)
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -6.8e+78)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / B));
	elseif (F <= 2.4)
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -6.8e+78)
		tmp = (-1.0 / B) - (x / B);
	elseif (F <= 2.4)
		tmp = -x / tan(B);
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -6.8e+78], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.4], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -6.80000000000000014e78

   1. Initial program 43.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 37.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in B around 0 22.7%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Taylor expanded in F around -inf 54.1%

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{-1}{B}} \]

   if -6.80000000000000014e78 < F < 2.39999999999999991

   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/ 99.6%

          \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \color{blue}{\frac{\left(-x\right) \cdot 1}{\tan B}}\right) \]

      11. *-rgt-identity 99.6%

          \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{\color{blue}{-x}}{\tan B}\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{-x}{\tan B}\right)} \]

   4. Taylor expanded in F around 0 64.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{\cos B \cdot x}{\sin B}} \]
   5. Step-by-step derivation

      1. associate-*r/ 64.7%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(\cos B \cdot x\right)}{\sin B}} \]

      2. *-commutative 64.7%

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(x \cdot \cos B\right)}}{\sin B} \]

      3. associate-*r* 64.7%

         \[\leadsto \frac{\color{blue}{\left(-1 \cdot x\right) \cdot \cos B}}{\sin B} \]

      4. mul-1-neg 64.7%

         \[\leadsto \frac{\color{blue}{\left(-x\right)} \cdot \cos B}{\sin B} \]

   6. Simplified 64.7%

      \[\leadsto \color{blue}{\frac{\left(-x\right) \cdot \cos B}{\sin B}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 40.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(-x\right) \cdot \cos B}{\sin B}\right)\right)} \]

      2. expm1-udef 19.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(-x\right) \cdot \cos B}{\sin B}\right)} - 1} \]

      3. associate-/l* 19.2%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{-x}{\frac{\sin B}{\cos B}}}\right)} - 1 \]

      4. tan-quot 19.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{-x}{\color{blue}{\tan B}}\right)} - 1 \]

   8. Applied egg-rr 19.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-x}{\tan B}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 40.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-x}{\tan B}\right)\right)} \]

      2. expm1-log1p 64.7%

         \[\leadsto \color{blue}{\frac{-x}{\tan B}} \]

   10. Simplified 64.7%

       \[\leadsto \color{blue}{\frac{-x}{\tan B}} \]

   if 2.39999999999999991 < F

   1. Initial program 55.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 34.7%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in F around inf 77.2%

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6.8 \cdot 10^{+78}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.4:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 16: 70.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.7 \cdot 10^{+52}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 30:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.7e+52)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 30.0) (/ (- x) (tan B)) (- (/ 1.0 (sin B)) (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.7e+52) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 30.0) {
		tmp = -x / tan(B);
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-2.7d+52)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 30.0d0) then
        tmp = -x / tan(b)
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.7e+52) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 30.0) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.7e+52:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 30.0:
		tmp = -x / math.tan(B)
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.7e+52)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 30.0)
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.7e+52)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 30.0)
		tmp = -x / tan(B);
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.7e+52], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 30.0], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.7e52

   1. Initial program 47.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 31.8%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in F around -inf 83.4%

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   if -2.7e52 < F < 30

   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/ 99.6%

          \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \color{blue}{\frac{\left(-x\right) \cdot 1}{\tan B}}\right) \]

      11. *-rgt-identity 99.6%

          \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{\color{blue}{-x}}{\tan B}\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{-x}{\tan B}\right)} \]

   4. Taylor expanded in F around 0 65.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{\cos B \cdot x}{\sin B}} \]
   5. Step-by-step derivation

      1. associate-*r/ 65.1%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(\cos B \cdot x\right)}{\sin B}} \]

      2. *-commutative 65.1%

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(x \cdot \cos B\right)}}{\sin B} \]

      3. associate-*r* 65.1%

         \[\leadsto \frac{\color{blue}{\left(-1 \cdot x\right) \cdot \cos B}}{\sin B} \]

      4. mul-1-neg 65.1%

         \[\leadsto \frac{\color{blue}{\left(-x\right)} \cdot \cos B}{\sin B} \]

   6. Simplified 65.1%

      \[\leadsto \color{blue}{\frac{\left(-x\right) \cdot \cos B}{\sin B}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 41.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(-x\right) \cdot \cos B}{\sin B}\right)\right)} \]

      2. expm1-udef 19.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(-x\right) \cdot \cos B}{\sin B}\right)} - 1} \]

      3. associate-/l* 19.8%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{-x}{\frac{\sin B}{\cos B}}}\right)} - 1 \]

      4. tan-quot 19.8%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{-x}{\color{blue}{\tan B}}\right)} - 1 \]

   8. Applied egg-rr 19.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-x}{\tan B}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 41.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-x}{\tan B}\right)\right)} \]

      2. expm1-log1p 65.1%

         \[\leadsto \color{blue}{\frac{-x}{\tan B}} \]

   10. Simplified 65.1%

       \[\leadsto \color{blue}{\frac{-x}{\tan B}} \]

   if 30 < F

   1. Initial program 55.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 34.7%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in F around inf 77.2%

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.7 \cdot 10^{+52}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 30:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 17: 45.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.7 \cdot 10^{-34}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 5.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{-x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.7e-34)
   (- (/ -1.0 B) (/ x B))
   (if (<= F 5.5e-6) (/ (- x) (sin B)) (/ 1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.7e-34) {
		tmp = (-1.0 / B) - (x / B);
	} else if (F <= 5.5e-6) {
		tmp = -x / sin(B);
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.7d-34)) then
        tmp = ((-1.0d0) / b) - (x / b)
    else if (f <= 5.5d-6) then
        tmp = -x / sin(b)
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.7e-34) {
		tmp = (-1.0 / B) - (x / B);
	} else if (F <= 5.5e-6) {
		tmp = -x / Math.sin(B);
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.7e-34:
		tmp = (-1.0 / B) - (x / B)
	elif F <= 5.5e-6:
		tmp = -x / math.sin(B)
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.7e-34)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / B));
	elseif (F <= 5.5e-6)
		tmp = Float64(Float64(-x) / sin(B));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.7e-34)
		tmp = (-1.0 / B) - (x / B);
	elseif (F <= 5.5e-6)
		tmp = -x / sin(B);
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.7e-34], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.5e-6], N[((-x) / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.7e-34

   1. Initial program 56.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 43.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in B around 0 24.6%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Taylor expanded in F around -inf 47.7%

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{-1}{B}} \]

   if -1.7e-34 < F < 5.4999999999999999e-6

   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. fma-def 99.4%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{F}{\sin B}, {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}, \left(-x\right) \cdot \frac{1}{\tan B}\right)} \]

      4. +-commutative 99.4%

         \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\color{blue}{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)}, \left(-x\right) \cdot \frac{1}{\tan B}\right) \]

      5. *-commutative 99.4%

         \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\color{blue}{x \cdot 2} + \left(F \cdot F + 2\right)\right)}^{\left(-\frac{1}{2}\right)}, \left(-x\right) \cdot \frac{1}{\tan B}\right) \]

      6. fma-def 99.4%

         \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\color{blue}{\left(\mathsf{fma}\left(x, 2, F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)}, \left(-x\right) \cdot \frac{1}{\tan B}\right) \]

      7. fma-def 99.4%

         \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)\right)}^{\left(-\frac{1}{2}\right)}, \left(-x\right) \cdot \frac{1}{\tan B}\right) \]

      8. metadata-eval 99.4%

         \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(-\color{blue}{0.5}\right)}, \left(-x\right) \cdot \frac{1}{\tan B}\right) \]

      9. metadata-eval 99.4%

         \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{-0.5}}, \left(-x\right) \cdot \frac{1}{\tan B}\right) \]

      10. associate-*r/ 99.6%

          \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \color{blue}{\frac{\left(-x\right) \cdot 1}{\tan B}}\right) \]

      11. *-rgt-identity 99.6%

          \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{\color{blue}{-x}}{\tan B}\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{-x}{\tan B}\right)} \]

   4. Taylor expanded in F around 0 68.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{\cos B \cdot x}{\sin B}} \]
   5. Step-by-step derivation

      1. associate-*r/ 68.2%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(\cos B \cdot x\right)}{\sin B}} \]

      2. *-commutative 68.2%

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(x \cdot \cos B\right)}}{\sin B} \]

      3. associate-*r* 68.2%

         \[\leadsto \frac{\color{blue}{\left(-1 \cdot x\right) \cdot \cos B}}{\sin B} \]

      4. mul-1-neg 68.2%

         \[\leadsto \frac{\color{blue}{\left(-x\right)} \cdot \cos B}{\sin B} \]

   6. Simplified 68.2%

      \[\leadsto \color{blue}{\frac{\left(-x\right) \cdot \cos B}{\sin B}} \]

   7. Taylor expanded in B around 0 33.0%

      \[\leadsto \frac{\color{blue}{-1 \cdot x}}{\sin B} \]
   8. Step-by-step derivation

      1. neg-mul-1 33.0%

         \[\leadsto \frac{\color{blue}{-x}}{\sin B} \]

   9. Simplified 33.0%

      \[\leadsto \frac{\color{blue}{-x}}{\sin B} \]

   if 5.4999999999999999e-6 < F

   1. Initial program 57.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 35.3%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in F around inf 73.2%

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{B}} \]

   4. Taylor expanded in B around inf 54.8%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.7 \cdot 10^{-34}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 5.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{-x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 18: 57.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.35 \cdot 10^{+79}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 43:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.35e+79)
   (- (/ -1.0 B) (/ x B))
   (if (<= F 43.0) (/ (- x) (tan B)) (/ 1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.35e+79) {
		tmp = (-1.0 / B) - (x / B);
	} else if (F <= 43.0) {
		tmp = -x / tan(B);
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-2.35d+79)) then
        tmp = ((-1.0d0) / b) - (x / b)
    else if (f <= 43.0d0) then
        tmp = -x / tan(b)
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.35e+79) {
		tmp = (-1.0 / B) - (x / B);
	} else if (F <= 43.0) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.35e+79:
		tmp = (-1.0 / B) - (x / B)
	elif F <= 43.0:
		tmp = -x / math.tan(B)
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.35e+79)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / B));
	elseif (F <= 43.0)
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.35e+79)
		tmp = (-1.0 / B) - (x / B);
	elseif (F <= 43.0)
		tmp = -x / tan(B);
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.35e+79], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 43.0], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.35000000000000011e79

   1. Initial program 43.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 37.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in B around 0 22.7%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Taylor expanded in F around -inf 54.1%

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{-1}{B}} \]

   if -2.35000000000000011e79 < F < 43

   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/ 99.6%

          \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \color{blue}{\frac{\left(-x\right) \cdot 1}{\tan B}}\right) \]

      11. *-rgt-identity 99.6%

          \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{\color{blue}{-x}}{\tan B}\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{-x}{\tan B}\right)} \]

   4. Taylor expanded in F around 0 64.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{\cos B \cdot x}{\sin B}} \]
   5. Step-by-step derivation

      1. associate-*r/ 64.7%

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(\cos B \cdot x\right)}{\sin B}} \]

      2. *-commutative 64.7%

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(x \cdot \cos B\right)}}{\sin B} \]

      3. associate-*r* 64.7%

         \[\leadsto \frac{\color{blue}{\left(-1 \cdot x\right) \cdot \cos B}}{\sin B} \]

      4. mul-1-neg 64.7%

         \[\leadsto \frac{\color{blue}{\left(-x\right)} \cdot \cos B}{\sin B} \]

   6. Simplified 64.7%

      \[\leadsto \color{blue}{\frac{\left(-x\right) \cdot \cos B}{\sin B}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 40.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(-x\right) \cdot \cos B}{\sin B}\right)\right)} \]

      2. expm1-udef 19.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(-x\right) \cdot \cos B}{\sin B}\right)} - 1} \]

      3. associate-/l* 19.2%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{-x}{\frac{\sin B}{\cos B}}}\right)} - 1 \]

      4. tan-quot 19.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{-x}{\color{blue}{\tan B}}\right)} - 1 \]

   8. Applied egg-rr 19.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-x}{\tan B}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 40.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-x}{\tan B}\right)\right)} \]

      2. expm1-log1p 64.7%

         \[\leadsto \color{blue}{\frac{-x}{\tan B}} \]

   10. Simplified 64.7%

       \[\leadsto \color{blue}{\frac{-x}{\tan B}} \]

   if 43 < F

   1. Initial program 55.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 34.7%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in F around inf 77.2%

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{B}} \]

   4. Taylor expanded in B around inf 57.6%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.35 \cdot 10^{+79}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 43:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 19: 44.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.25 \cdot 10^{-23}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{-x}{B} - B \cdot \left(x \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.25e-23)
   (- (/ -1.0 B) (/ x B))
   (if (<= F 3.4e-5)
     (- (/ (- x) B) (* B (* x -0.3333333333333333)))
     (/ 1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.25e-23) {
		tmp = (-1.0 / B) - (x / B);
	} else if (F <= 3.4e-5) {
		tmp = (-x / B) - (B * (x * -0.3333333333333333));
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.25d-23)) then
        tmp = ((-1.0d0) / b) - (x / b)
    else if (f <= 3.4d-5) then
        tmp = (-x / b) - (b * (x * (-0.3333333333333333d0)))
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.25e-23) {
		tmp = (-1.0 / B) - (x / B);
	} else if (F <= 3.4e-5) {
		tmp = (-x / B) - (B * (x * -0.3333333333333333));
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.25e-23:
		tmp = (-1.0 / B) - (x / B)
	elif F <= 3.4e-5:
		tmp = (-x / B) - (B * (x * -0.3333333333333333))
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.25e-23)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / B));
	elseif (F <= 3.4e-5)
		tmp = Float64(Float64(Float64(-x) / B) - Float64(B * Float64(x * -0.3333333333333333)));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.25e-23)
		tmp = (-1.0 / B) - (x / B);
	elseif (F <= 3.4e-5)
		tmp = (-x / B) - (B * (x * -0.3333333333333333));
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.25e-23], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.4e-5], N[(N[((-x) / B), $MachinePrecision] - N[(B * N[(x * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.2500000000000001e-23

   1. Initial program 55.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 43.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in B around 0 23.9%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Taylor expanded in F around -inf 47.6%

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{-1}{B}} \]

   if -1.2500000000000001e-23 < F < 3.4e-5

   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/ 99.6%

          \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \color{blue}{\frac{\left(-x\right) \cdot 1}{\tan B}}\right) \]

      11. *-rgt-identity 99.6%

          \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{\color{blue}{-x}}{\tan B}\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{-x}{\tan B}\right)} \]

   4. Taylor expanded in F around 0 68.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{\cos B \cdot x}{\sin B}} \]
   5. Step-by-step derivation

      1. mul-1-neg 68.2%

         \[\leadsto \color{blue}{-\frac{\cos B \cdot x}{\sin B}} \]

      2. *-commutative 68.2%

         \[\leadsto -\frac{\color{blue}{x \cdot \cos B}}{\sin B} \]

      3. associate-*l/ 68.3%

         \[\leadsto -\color{blue}{\frac{x}{\sin B} \cdot \cos B} \]

      4. *-commutative 68.3%

         \[\leadsto -\color{blue}{\cos B \cdot \frac{x}{\sin B}} \]

   6. Simplified 68.3%

      \[\leadsto \color{blue}{-\cos B \cdot \frac{x}{\sin B}} \]

   7. Taylor expanded in B around 0 29.8%

      \[\leadsto -\color{blue}{\left(\frac{x}{B} + \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right) \cdot B\right)} \]
   8. Step-by-step derivation

      1. *-commutative 29.8%

         \[\leadsto -\left(\frac{x}{B} + \color{blue}{B \cdot \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right)}\right) \]

      2. distribute-rgt-out-- 29.8%

         \[\leadsto -\left(\frac{x}{B} + B \cdot \color{blue}{\left(x \cdot \left(-0.5 - -0.16666666666666666\right)\right)}\right) \]

      3. metadata-eval 29.8%

         \[\leadsto -\left(\frac{x}{B} + B \cdot \left(x \cdot \color{blue}{-0.3333333333333333}\right)\right) \]

   9. Simplified 29.8%

      \[\leadsto -\color{blue}{\left(\frac{x}{B} + B \cdot \left(x \cdot -0.3333333333333333\right)\right)} \]

   if 3.4e-5 < F

   1. Initial program 57.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 35.7%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in F around inf 74.0%

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{B}} \]

   4. Taylor expanded in B around inf 55.4%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.25 \cdot 10^{-23}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{-x}{B} - B \cdot \left(x \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 20: 43.9% accurate, 21.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.2 \cdot 10^{-36}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 6.3 \cdot 10^{-145}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.2e-36)
   (- (/ -1.0 B) (/ x B))
   (if (<= F 6.3e-145)
     (/ (- x) B)
     (+ (* 0.3333333333333333 (* B x)) (/ (- 1.0 x) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.2e-36) {
		tmp = (-1.0 / B) - (x / B);
	} else if (F <= 6.3e-145) {
		tmp = -x / B;
	} else {
		tmp = (0.3333333333333333 * (B * x)) + ((1.0 - x) / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-2.2d-36)) then
        tmp = ((-1.0d0) / b) - (x / b)
    else if (f <= 6.3d-145) then
        tmp = -x / b
    else
        tmp = (0.3333333333333333d0 * (b * x)) + ((1.0d0 - x) / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.2e-36) {
		tmp = (-1.0 / B) - (x / B);
	} else if (F <= 6.3e-145) {
		tmp = -x / B;
	} else {
		tmp = (0.3333333333333333 * (B * x)) + ((1.0 - x) / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.2e-36:
		tmp = (-1.0 / B) - (x / B)
	elif F <= 6.3e-145:
		tmp = -x / B
	else:
		tmp = (0.3333333333333333 * (B * x)) + ((1.0 - x) / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.2e-36)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / B));
	elseif (F <= 6.3e-145)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(0.3333333333333333 * Float64(B * x)) + Float64(Float64(1.0 - x) / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.2e-36)
		tmp = (-1.0 / B) - (x / B);
	elseif (F <= 6.3e-145)
		tmp = -x / B;
	else
		tmp = (0.3333333333333333 * (B * x)) + ((1.0 - x) / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.2e-36], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 6.3e-145], N[((-x) / B), $MachinePrecision], N[(N[(0.3333333333333333 * N[(B * x), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.1999999999999999e-36

   1. Initial program 56.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 43.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in B around 0 24.6%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Taylor expanded in F around -inf 47.7%

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{-1}{B}} \]

   if -2.1999999999999999e-36 < F < 6.3000000000000001e-145

   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. fma-def 99.4%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{F}{\sin B}, {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}, \left(-x\right) \cdot \frac{1}{\tan B}\right)} \]

      4. +-commutative 99.4%

         \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\color{blue}{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)}, \left(-x\right) \cdot \frac{1}{\tan B}\right) \]

      5. *-commutative 99.4%

         \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\color{blue}{x \cdot 2} + \left(F \cdot F + 2\right)\right)}^{\left(-\frac{1}{2}\right)}, \left(-x\right) \cdot \frac{1}{\tan B}\right) \]

      6. fma-def 99.4%

         \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\color{blue}{\left(\mathsf{fma}\left(x, 2, F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)}, \left(-x\right) \cdot \frac{1}{\tan B}\right) \]

      7. fma-def 99.4%

         \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)\right)}^{\left(-\frac{1}{2}\right)}, \left(-x\right) \cdot \frac{1}{\tan B}\right) \]

      8. metadata-eval 99.4%

         \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(-\color{blue}{0.5}\right)}, \left(-x\right) \cdot \frac{1}{\tan B}\right) \]

      9. metadata-eval 99.4%

         \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{-0.5}}, \left(-x\right) \cdot \frac{1}{\tan B}\right) \]

      10. associate-*r/ 99.6%

          \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \color{blue}{\frac{\left(-x\right) \cdot 1}{\tan B}}\right) \]

      11. *-rgt-identity 99.6%

          \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{\color{blue}{-x}}{\tan B}\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{-x}{\tan B}\right)} \]

   4. Taylor expanded in F around 0 70.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{\cos B \cdot x}{\sin B}} \]
   5. Step-by-step derivation

      1. mul-1-neg 70.5%

         \[\leadsto \color{blue}{-\frac{\cos B \cdot x}{\sin B}} \]

      2. *-commutative 70.5%

         \[\leadsto -\frac{\color{blue}{x \cdot \cos B}}{\sin B} \]

      3. associate-*l/ 70.5%

         \[\leadsto -\color{blue}{\frac{x}{\sin B} \cdot \cos B} \]

      4. *-commutative 70.5%

         \[\leadsto -\color{blue}{\cos B \cdot \frac{x}{\sin B}} \]

   6. Simplified 70.5%

      \[\leadsto \color{blue}{-\cos B \cdot \frac{x}{\sin B}} \]

   7. Taylor expanded in B around 0 32.4%

      \[\leadsto -\color{blue}{\frac{x}{B}} \]

   if 6.3000000000000001e-145 < F

   1. Initial program 65.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 47.8%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in F around inf 54.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \color{blue}{\frac{1}{F}} \]

   4. Taylor expanded in B around 0 46.4%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{1}{B}\right) - \frac{x}{B}} \]
   5. Step-by-step derivation

      1. associate--l+ 46.4%

         \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(B \cdot x\right) + \left(\frac{1}{B} - \frac{x}{B}\right)} \]

      2. *-commutative 46.4%

         \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(x \cdot B\right)} + \left(\frac{1}{B} - \frac{x}{B}\right) \]

      3. div-sub 46.4%

         \[\leadsto 0.3333333333333333 \cdot \left(x \cdot B\right) + \color{blue}{\frac{1 - x}{B}} \]

   6. Simplified 46.4%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(x \cdot B\right) + \frac{1 - x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.2 \cdot 10^{-36}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 6.3 \cdot 10^{-145}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(B \cdot x\right) + \frac{1 - x}{B}\\ \end{array} \]

Alternative 21: 44.4% accurate, 22.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.2 \cdot 10^{-24}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.00018:\\ \;\;\;\;\frac{-x}{B} - B \cdot \left(x \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.2e-24)
   (- (/ -1.0 B) (/ x B))
   (if (<= F 0.00018)
     (- (/ (- x) B) (* B (* x -0.3333333333333333)))
     (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.2e-24) {
		tmp = (-1.0 / B) - (x / B);
	} else if (F <= 0.00018) {
		tmp = (-x / B) - (B * (x * -0.3333333333333333));
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-3.2d-24)) then
        tmp = ((-1.0d0) / b) - (x / b)
    else if (f <= 0.00018d0) then
        tmp = (-x / b) - (b * (x * (-0.3333333333333333d0)))
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.2e-24) {
		tmp = (-1.0 / B) - (x / B);
	} else if (F <= 0.00018) {
		tmp = (-x / B) - (B * (x * -0.3333333333333333));
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.2e-24:
		tmp = (-1.0 / B) - (x / B)
	elif F <= 0.00018:
		tmp = (-x / B) - (B * (x * -0.3333333333333333))
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.2e-24)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / B));
	elseif (F <= 0.00018)
		tmp = Float64(Float64(Float64(-x) / B) - Float64(B * Float64(x * -0.3333333333333333)));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -3.2e-24)
		tmp = (-1.0 / B) - (x / B);
	elseif (F <= 0.00018)
		tmp = (-x / B) - (B * (x * -0.3333333333333333));
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.2e-24], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.00018], N[(N[((-x) / B), $MachinePrecision] - N[(B * N[(x * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.20000000000000012e-24

   1. Initial program 55.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 43.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in B around 0 23.9%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Taylor expanded in F around -inf 47.6%

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{-1}{B}} \]

   if -3.20000000000000012e-24 < F < 1.80000000000000011e-4

   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/ 99.6%

          \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \color{blue}{\frac{\left(-x\right) \cdot 1}{\tan B}}\right) \]

      11. *-rgt-identity 99.6%

          \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{\color{blue}{-x}}{\tan B}\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{-x}{\tan B}\right)} \]

   4. Taylor expanded in F around 0 67.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{\cos B \cdot x}{\sin B}} \]
   5. Step-by-step derivation

      1. mul-1-neg 67.6%

         \[\leadsto \color{blue}{-\frac{\cos B \cdot x}{\sin B}} \]

      2. *-commutative 67.6%

         \[\leadsto -\frac{\color{blue}{x \cdot \cos B}}{\sin B} \]

      3. associate-*l/ 67.6%

         \[\leadsto -\color{blue}{\frac{x}{\sin B} \cdot \cos B} \]

      4. *-commutative 67.6%

         \[\leadsto -\color{blue}{\cos B \cdot \frac{x}{\sin B}} \]

   6. Simplified 67.6%

      \[\leadsto \color{blue}{-\cos B \cdot \frac{x}{\sin B}} \]

   7. Taylor expanded in B around 0 29.6%

      \[\leadsto -\color{blue}{\left(\frac{x}{B} + \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right) \cdot B\right)} \]
   8. Step-by-step derivation

      1. *-commutative 29.6%

         \[\leadsto -\left(\frac{x}{B} + \color{blue}{B \cdot \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right)}\right) \]

      2. distribute-rgt-out-- 29.6%

         \[\leadsto -\left(\frac{x}{B} + B \cdot \color{blue}{\left(x \cdot \left(-0.5 - -0.16666666666666666\right)\right)}\right) \]

      3. metadata-eval 29.6%

         \[\leadsto -\left(\frac{x}{B} + B \cdot \left(x \cdot \color{blue}{-0.3333333333333333}\right)\right) \]

   9. Simplified 29.6%

      \[\leadsto -\color{blue}{\left(\frac{x}{B} + B \cdot \left(x \cdot -0.3333333333333333\right)\right)} \]

   if 1.80000000000000011e-4 < F

   1. Initial program 56.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 41.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in F around inf 53.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \color{blue}{\frac{1}{F}} \]

   4. Taylor expanded in B around 0 52.7%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.2 \cdot 10^{-24}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.00018:\\ \;\;\;\;\frac{-x}{B} - B \cdot \left(x \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 22: 43.8% accurate, 24.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.15 \cdot 10^{-33}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.85 \cdot 10^{-144}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B} + B \cdot 0.16666666666666666\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.15e-33)
   (- (/ -1.0 B) (/ x B))
   (if (<= F 1.85e-144)
     (/ (- x) B)
     (+ (/ (- 1.0 x) B) (* B 0.16666666666666666)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.15e-33) {
		tmp = (-1.0 / B) - (x / B);
	} else if (F <= 1.85e-144) {
		tmp = -x / B;
	} else {
		tmp = ((1.0 - x) / B) + (B * 0.16666666666666666);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.15d-33)) then
        tmp = ((-1.0d0) / b) - (x / b)
    else if (f <= 1.85d-144) then
        tmp = -x / b
    else
        tmp = ((1.0d0 - x) / b) + (b * 0.16666666666666666d0)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.15e-33) {
		tmp = (-1.0 / B) - (x / B);
	} else if (F <= 1.85e-144) {
		tmp = -x / B;
	} else {
		tmp = ((1.0 - x) / B) + (B * 0.16666666666666666);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.15e-33:
		tmp = (-1.0 / B) - (x / B)
	elif F <= 1.85e-144:
		tmp = -x / B
	else:
		tmp = ((1.0 - x) / B) + (B * 0.16666666666666666)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.15e-33)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / B));
	elseif (F <= 1.85e-144)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(Float64(1.0 - x) / B) + Float64(B * 0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.15e-33)
		tmp = (-1.0 / B) - (x / B);
	elseif (F <= 1.85e-144)
		tmp = -x / B;
	else
		tmp = ((1.0 - x) / B) + (B * 0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.15e-33], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.85e-144], N[((-x) / B), $MachinePrecision], N[(N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision] + N[(B * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.14999999999999993e-33

   1. Initial program 56.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 43.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in B around 0 24.6%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Taylor expanded in F around -inf 47.7%

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{-1}{B}} \]

   if -1.14999999999999993e-33 < F < 1.8500000000000001e-144

   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. fma-def 99.4%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{F}{\sin B}, {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}, \left(-x\right) \cdot \frac{1}{\tan B}\right)} \]

      4. +-commutative 99.4%

         \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\color{blue}{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)}, \left(-x\right) \cdot \frac{1}{\tan B}\right) \]

      5. *-commutative 99.4%

         \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\color{blue}{x \cdot 2} + \left(F \cdot F + 2\right)\right)}^{\left(-\frac{1}{2}\right)}, \left(-x\right) \cdot \frac{1}{\tan B}\right) \]

      6. fma-def 99.4%

         \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\color{blue}{\left(\mathsf{fma}\left(x, 2, F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)}, \left(-x\right) \cdot \frac{1}{\tan B}\right) \]

      7. fma-def 99.4%

         \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)\right)}^{\left(-\frac{1}{2}\right)}, \left(-x\right) \cdot \frac{1}{\tan B}\right) \]

      8. metadata-eval 99.4%

         \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(-\color{blue}{0.5}\right)}, \left(-x\right) \cdot \frac{1}{\tan B}\right) \]

      9. metadata-eval 99.4%

         \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{-0.5}}, \left(-x\right) \cdot \frac{1}{\tan B}\right) \]

      10. associate-*r/ 99.6%

          \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \color{blue}{\frac{\left(-x\right) \cdot 1}{\tan B}}\right) \]

      11. *-rgt-identity 99.6%

          \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{\color{blue}{-x}}{\tan B}\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{-x}{\tan B}\right)} \]

   4. Taylor expanded in F around 0 70.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{\cos B \cdot x}{\sin B}} \]
   5. Step-by-step derivation

      1. mul-1-neg 70.9%

         \[\leadsto \color{blue}{-\frac{\cos B \cdot x}{\sin B}} \]

      2. *-commutative 70.9%

         \[\leadsto -\frac{\color{blue}{x \cdot \cos B}}{\sin B} \]

      3. associate-*l/ 70.9%

         \[\leadsto -\color{blue}{\frac{x}{\sin B} \cdot \cos B} \]

      4. *-commutative 70.9%

         \[\leadsto -\color{blue}{\cos B \cdot \frac{x}{\sin B}} \]

   6. Simplified 70.9%

      \[\leadsto \color{blue}{-\cos B \cdot \frac{x}{\sin B}} \]

   7. Taylor expanded in B around 0 32.0%

      \[\leadsto -\color{blue}{\frac{x}{B}} \]

   if 1.8500000000000001e-144 < F

   1. Initial program 64.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 39.9%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in F around inf 64.1%

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{B}} \]

   4. Taylor expanded in B around 0 46.0%

      \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot B + \frac{1}{B}\right) - \frac{x}{B}} \]
   5. Step-by-step derivation

      1. associate--l+ 46.0%

         \[\leadsto \color{blue}{0.16666666666666666 \cdot B + \left(\frac{1}{B} - \frac{x}{B}\right)} \]

      2. *-commutative 46.0%

         \[\leadsto \color{blue}{B \cdot 0.16666666666666666} + \left(\frac{1}{B} - \frac{x}{B}\right) \]

      3. div-sub 46.0%

         \[\leadsto B \cdot 0.16666666666666666 + \color{blue}{\frac{1 - x}{B}} \]

   6. Simplified 46.0%

      \[\leadsto \color{blue}{B \cdot 0.16666666666666666 + \frac{1 - x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.15 \cdot 10^{-33}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.85 \cdot 10^{-144}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B} + B \cdot 0.16666666666666666\\ \end{array} \]

Alternative 23: 43.8% accurate, 35.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -5.5 \cdot 10^{-32}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.85 \cdot 10^{-144}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -5.5e-32)
   (- (/ -1.0 B) (/ x B))
   (if (<= F 1.85e-144) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -5.5e-32) {
		tmp = (-1.0 / B) - (x / B);
	} else if (F <= 1.85e-144) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-5.5d-32)) then
        tmp = ((-1.0d0) / b) - (x / b)
    else if (f <= 1.85d-144) then
        tmp = -x / b
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -5.5e-32) {
		tmp = (-1.0 / B) - (x / B);
	} else if (F <= 1.85e-144) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -5.5e-32:
		tmp = (-1.0 / B) - (x / B)
	elif F <= 1.85e-144:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -5.5e-32)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / B));
	elseif (F <= 1.85e-144)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -5.5e-32)
		tmp = (-1.0 / B) - (x / B);
	elseif (F <= 1.85e-144)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -5.5e-32], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.85e-144], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.50000000000000024e-32

   1. Initial program 56.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 43.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in B around 0 24.6%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Taylor expanded in F around -inf 47.7%

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{-1}{B}} \]

   if -5.50000000000000024e-32 < F < 1.8500000000000001e-144

   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. fma-def 99.4%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{F}{\sin B}, {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}, \left(-x\right) \cdot \frac{1}{\tan B}\right)} \]

      4. +-commutative 99.4%

         \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\color{blue}{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)}, \left(-x\right) \cdot \frac{1}{\tan B}\right) \]

      5. *-commutative 99.4%

         \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\color{blue}{x \cdot 2} + \left(F \cdot F + 2\right)\right)}^{\left(-\frac{1}{2}\right)}, \left(-x\right) \cdot \frac{1}{\tan B}\right) \]

      6. fma-def 99.4%

         \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\color{blue}{\left(\mathsf{fma}\left(x, 2, F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)}, \left(-x\right) \cdot \frac{1}{\tan B}\right) \]

      7. fma-def 99.4%

         \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)\right)}^{\left(-\frac{1}{2}\right)}, \left(-x\right) \cdot \frac{1}{\tan B}\right) \]

      8. metadata-eval 99.4%

         \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(-\color{blue}{0.5}\right)}, \left(-x\right) \cdot \frac{1}{\tan B}\right) \]

      9. metadata-eval 99.4%

         \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{-0.5}}, \left(-x\right) \cdot \frac{1}{\tan B}\right) \]

      10. associate-*r/ 99.6%

          \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \color{blue}{\frac{\left(-x\right) \cdot 1}{\tan B}}\right) \]

      11. *-rgt-identity 99.6%

          \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{\color{blue}{-x}}{\tan B}\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{-x}{\tan B}\right)} \]

   4. Taylor expanded in F around 0 70.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{\cos B \cdot x}{\sin B}} \]
   5. Step-by-step derivation

      1. mul-1-neg 70.9%

         \[\leadsto \color{blue}{-\frac{\cos B \cdot x}{\sin B}} \]

      2. *-commutative 70.9%

         \[\leadsto -\frac{\color{blue}{x \cdot \cos B}}{\sin B} \]

      3. associate-*l/ 70.9%

         \[\leadsto -\color{blue}{\frac{x}{\sin B} \cdot \cos B} \]

      4. *-commutative 70.9%

         \[\leadsto -\color{blue}{\cos B \cdot \frac{x}{\sin B}} \]

   6. Simplified 70.9%

      \[\leadsto \color{blue}{-\cos B \cdot \frac{x}{\sin B}} \]

   7. Taylor expanded in B around 0 32.0%

      \[\leadsto -\color{blue}{\frac{x}{B}} \]

   if 1.8500000000000001e-144 < F

   1. Initial program 64.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 47.3%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in F around inf 53.8%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \color{blue}{\frac{1}{F}} \]

   4. Taylor expanded in B around 0 46.0%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5.5 \cdot 10^{-32}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.85 \cdot 10^{-144}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 24: 36.8% accurate, 45.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 1.85 \cdot 10^{-144}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F 1.85e-144) (/ (- x) B) (/ (- 1.0 x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= 1.85e-144) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= 1.85d-144) then
        tmp = -x / b
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= 1.85e-144) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= 1.85e-144:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= 1.85e-144)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= 1.85e-144)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, 1.85e-144], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 1.8500000000000001e-144

   1. Initial program 77.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. distribute-lft-neg-in 77.5%

         \[\leadsto \color{blue}{\left(-x\right) \cdot \frac{1}{\tan B}} + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. +-commutative 77.5%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x\right) \cdot \frac{1}{\tan B}} \]

      3. fma-def 77.5%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{F}{\sin B}, {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}, \left(-x\right) \cdot \frac{1}{\tan B}\right)} \]

      4. +-commutative 77.5%

         \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\color{blue}{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)}, \left(-x\right) \cdot \frac{1}{\tan B}\right) \]

      5. *-commutative 77.5%

         \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\color{blue}{x \cdot 2} + \left(F \cdot F + 2\right)\right)}^{\left(-\frac{1}{2}\right)}, \left(-x\right) \cdot \frac{1}{\tan B}\right) \]

      6. fma-def 77.5%

         \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\color{blue}{\left(\mathsf{fma}\left(x, 2, F \cdot F + 2\right)\right)}}^{\left(-\frac{1}{2}\right)}, \left(-x\right) \cdot \frac{1}{\tan B}\right) \]

      7. fma-def 77.5%

         \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)\right)}^{\left(-\frac{1}{2}\right)}, \left(-x\right) \cdot \frac{1}{\tan B}\right) \]

      8. metadata-eval 77.5%

         \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(-\color{blue}{0.5}\right)}, \left(-x\right) \cdot \frac{1}{\tan B}\right) \]

      9. metadata-eval 77.5%

         \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{-0.5}}, \left(-x\right) \cdot \frac{1}{\tan B}\right) \]

      10. associate-*r/ 77.6%

          \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \color{blue}{\frac{\left(-x\right) \cdot 1}{\tan B}}\right) \]

      11. *-rgt-identity 77.6%

          \[\leadsto \mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{\color{blue}{-x}}{\tan B}\right) \]

   3. Simplified 77.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{-x}{\tan B}\right)} \]

   4. Taylor expanded in F around 0 56.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{\cos B \cdot x}{\sin B}} \]
   5. Step-by-step derivation

      1. mul-1-neg 56.7%

         \[\leadsto \color{blue}{-\frac{\cos B \cdot x}{\sin B}} \]

      2. *-commutative 56.7%

         \[\leadsto -\frac{\color{blue}{x \cdot \cos B}}{\sin B} \]

      3. associate-*l/ 56.7%

         \[\leadsto -\color{blue}{\frac{x}{\sin B} \cdot \cos B} \]

      4. *-commutative 56.7%

         \[\leadsto -\color{blue}{\cos B \cdot \frac{x}{\sin B}} \]

   6. Simplified 56.7%

      \[\leadsto \color{blue}{-\cos B \cdot \frac{x}{\sin B}} \]

   7. Taylor expanded in B around 0 28.1%

      \[\leadsto -\color{blue}{\frac{x}{B}} \]

   if 1.8500000000000001e-144 < F

   1. Initial program 64.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 47.3%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in F around inf 53.8%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \color{blue}{\frac{1}{F}} \]

   4. Taylor expanded in B around 0 46.0%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.85 \cdot 10^{-144}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 25: 29.6% accurate, 53.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 3.2 \cdot 10^{+110}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (if (<= F 3.2e+110) (/ (- x) B) (/ 1.0 B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= 3.2e+110) {
		tmp = -x / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= 3.2d+110) then
        tmp = -x / b
    else
        tmp = 1.0d0 / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= 3.2e+110) {
		tmp = -x / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= 3.2e+110:
		tmp = -x / B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= 3.2e+110)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(1.0 / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= 3.2e+110)
		tmp = -x / B;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, 3.2e+110], N[((-x) / B), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 3.19999999999999994e110

   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 54.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{\cos B \cdot x}{\sin B}} \]
   5. Step-by-step derivation

      1. mul-1-neg 54.7%

         \[\leadsto \color{blue}{-\frac{\cos B \cdot x}{\sin B}} \]

      2. *-commutative 54.7%

         \[\leadsto -\frac{\color{blue}{x \cdot \cos B}}{\sin B} \]

      3. associate-*l/ 54.7%

         \[\leadsto -\color{blue}{\frac{x}{\sin B} \cdot \cos B} \]

      4. *-commutative 54.7%

         \[\leadsto -\color{blue}{\cos B \cdot \frac{x}{\sin B}} \]

   6. Simplified 54.7%

      \[\leadsto \color{blue}{-\cos B \cdot \frac{x}{\sin B}} \]

   7. Taylor expanded in B around 0 26.5%

      \[\leadsto -\color{blue}{\frac{x}{B}} \]

   if 3.19999999999999994e110 < F

   1. Initial program 29.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0 29.7%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Taylor expanded in F around inf 49.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \color{blue}{\frac{1}{F}} \]

   4. Taylor expanded in B around 0 59.9%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

   5. Taylor expanded in x around 0 44.7%

      \[\leadsto \color{blue}{\frac{1}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 3.2 \cdot 10^{+110}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]

Alternative 26: 10.0% accurate, 108.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (/ 1.0 B))

C

double code(double F, double B, double x) {
	return 1.0 / B;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = 1.0d0 / b
end function

Java

public static double code(double F, double B, double x) {
	return 1.0 / B;
}

Python

def code(F, B, x):
	return 1.0 / B

Julia

function code(F, B, x)
	return Float64(1.0 / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = 1.0 / B;
end

Wolfram

code[F_, B_, x_] := N[(1.0 / B), $MachinePrecision]

Derivation

1. Initial program 72.4%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

2. Taylor expanded in B around 0 56.8%

   \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

3. Taylor expanded in F around inf 46.7%

   \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \color{blue}{\frac{1}{F}} \]

4. Taylor expanded in B around 0 29.0%

   \[\leadsto \color{blue}{\frac{1 - x}{B}} \]

5. Taylor expanded in x around 0 13.1%

   \[\leadsto \color{blue}{\frac{1}{B}} \]

6. Final simplification 13.1%

   \[\leadsto \frac{1}{B} \]

Reproduce

herbie shell --seed 2023264 
(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))))))