VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.5% → 99.2%

Time: 44.8s

Alternatives: 22

Speedup: 1.4×

• could not determine a ground truth

  1. F = -1.2413343466367262e-33
  2. B = 3.26693031549645e-266
  3. x = -8.421441521497844e+124

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.5% 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: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;F \leq -8 \cdot 10^{+83}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 110000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{x}{-\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ 1.0 (tan B)))))
   (if (<= F -8e+83)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 110000000.0)
       (fma
        (/ F (sin B))
        (pow (fma x 2.0 (fma F F 2.0)) -0.5)
        (/ x (- (tan B))))
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x * (1.0 / tan(B));
	double tmp;
	if (F <= -8e+83) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 110000000.0) {
		tmp = fma((F / sin(B)), pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5), (x / -tan(B)));
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(1.0 / tan(B)))
	tmp = 0.0
	if (F <= -8e+83)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 110000000.0)
		tmp = fma(Float64(F / sin(B)), (fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5), Float64(x / Float64(-tan(B))));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -8e+83], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 110000000.0], N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] + N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -8.00000000000000025e83

   1. Initial program 48.4%

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

   3. Taylor expanded in F around -inf 99.7%

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

   if -8.00000000000000025e83 < F < 1.1e8

   1. Initial program 99.5%

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

   if 1.1e8 < F

   1. Initial program 52.4%

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

   3. Taylor expanded in F around inf 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -8 \cdot 10^{+83}:\\ \;\;\;\;\frac{-1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq 110000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{F}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{x}{-\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;F \leq -4.45 \cdot 10^{+151}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 125000000:\\ \;\;\;\;F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ 1.0 (tan B)))))
   (if (<= F -4.45e+151)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 125000000.0)
       (- (* F (/ (pow (fma x 2.0 (fma F F 2.0)) -0.5) (sin B))) (/ x (tan B)))
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x * (1.0 / tan(B));
	double tmp;
	if (F <= -4.45e+151) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 125000000.0) {
		tmp = (F * (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) / sin(B))) - (x / tan(B));
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(1.0 / tan(B)))
	tmp = 0.0
	if (F <= -4.45e+151)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 125000000.0)
		tmp = Float64(Float64(F * Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) / sin(B))) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -4.45e+151], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 125000000.0], N[(N[(F * N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.4500000000000003e151

   1. Initial program 37.7%

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

   3. Taylor expanded in F around -inf 99.7%

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

   if -4.4500000000000003e151 < F < 1.25e8

   1. Initial program 97.4%

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

      1. distribute-lft-neg-in 97.4%

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

      2. +-commutative 97.4%

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

      3. associate-*l/ 99.6%

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

      4. associate-/l* 99.5%

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

      5. fma-define 99.5%

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

      6. /-rgt-identity 99.5%

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

      7. remove-double-neg 99.5%

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

      8. fma-neg 99.5%

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

   3. Simplified 99.6%

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

   if 1.25e8 < F

   1. Initial program 52.4%

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

   3. Taylor expanded in F around inf 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.45 \cdot 10^{+151}:\\ \;\;\;\;\frac{-1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq 125000000:\\ \;\;\;\;F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;F \leq -8 \cdot 10^{+97}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 110000000:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ 1.0 (tan B)))))
   (if (<= F -8e+97)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 110000000.0)
       (+
        (* x (/ -1.0 (tan B)))
        (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)))
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x * (1.0 / tan(B));
	double tmp;
	if (F <= -8e+97) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 110000000.0) {
		tmp = (x * (-1.0 / tan(B))) + ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5));
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (1.0d0 / tan(b))
    if (f <= (-8d+97)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 110000000.0d0) then
        tmp = (x * ((-1.0d0) / tan(b))) + ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)))
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x * (1.0 / Math.tan(B));
	double tmp;
	if (F <= -8e+97) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 110000000.0) {
		tmp = (x * (-1.0 / Math.tan(B))) + ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5));
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * (1.0 / math.tan(B))
	tmp = 0
	if F <= -8e+97:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 110000000.0:
		tmp = (x * (-1.0 / math.tan(B))) + ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5))
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(1.0 / tan(B)))
	tmp = 0.0
	if (F <= -8e+97)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 110000000.0)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x * (1.0 / tan(B));
	tmp = 0.0;
	if (F <= -8e+97)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 110000000.0)
		tmp = (x * (-1.0 / tan(B))) + ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5));
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -8e+97], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 110000000.0], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -8.0000000000000006e97

   1. Initial program 46.8%

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

   3. Taylor expanded in F around -inf 99.7%

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

   if -8.0000000000000006e97 < F < 1.1e8

   1. Initial program 99.5%

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

   if 1.1e8 < F

   1. Initial program 52.4%

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

   3. Taylor expanded in F around inf 99.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -8 \cdot 10^{+97}:\\ \;\;\;\;\frac{-1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq 110000000:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;F \leq -7.6 \cdot 10^{+83}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 15000000:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ 1.0 (tan B)))))
   (if (<= F -7.6e+83)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 15000000.0)
       (+
        (/ -1.0 (/ (tan B) x))
        (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)))
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x * (1.0 / tan(B));
	double tmp;
	if (F <= -7.6e+83) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 15000000.0) {
		tmp = (-1.0 / (tan(B) / x)) + ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5));
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (1.0d0 / tan(b))
    if (f <= (-7.6d+83)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 15000000.0d0) then
        tmp = ((-1.0d0) / (tan(b) / x)) + ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)))
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x * (1.0 / Math.tan(B));
	double tmp;
	if (F <= -7.6e+83) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 15000000.0) {
		tmp = (-1.0 / (Math.tan(B) / x)) + ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5));
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * (1.0 / math.tan(B))
	tmp = 0
	if F <= -7.6e+83:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 15000000.0:
		tmp = (-1.0 / (math.tan(B) / x)) + ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5))
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(1.0 / tan(B)))
	tmp = 0.0
	if (F <= -7.6e+83)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 15000000.0)
		tmp = Float64(Float64(-1.0 / Float64(tan(B) / x)) + Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x * (1.0 / tan(B));
	tmp = 0.0;
	if (F <= -7.6e+83)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 15000000.0)
		tmp = (-1.0 / (tan(B) / x)) + ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5));
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -7.6e+83], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 15000000.0], N[(N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -7.6000000000000004e83

   1. Initial program 48.4%

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

   3. Taylor expanded in F around -inf 99.7%

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

   if -7.6000000000000004e83 < F < 1.5e7

   1. Initial program 99.5%

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

      1. div-inv 99.7%

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

      2. clear-num 99.6%

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

   4. Applied egg-rr 99.6%

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

   if 1.5e7 < F

   1. Initial program 52.4%

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

   3. Taylor expanded in F around inf 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7.6 \cdot 10^{+83}:\\ \;\;\;\;\frac{-1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq 15000000:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;F \leq -1.55:\\ \;\;\;\;\frac{-1}{\sin B} - t\_1\\ \mathbf{elif}\;F \leq 1.46:\\ \;\;\;\;F \cdot \left(t\_0 \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\right) - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;t\_0 - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))) (t_1 (* x (/ 1.0 (tan B)))))
   (if (<= F -1.55)
     (- (/ -1.0 (sin B)) t_1)
     (if (<= F 1.46)
       (- (* F (* t_0 (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))))) (/ x (tan B)))
       (- t_0 t_1)))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / sin(B);
	double t_1 = x * (1.0 / tan(B));
	double tmp;
	if (F <= -1.55) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= 1.46) {
		tmp = (F * (t_0 * sqrt((1.0 / (2.0 + (x * 2.0)))))) - (x / tan(B));
	} else {
		tmp = t_0 - t_1;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 / sin(b)
    t_1 = x * (1.0d0 / tan(b))
    if (f <= (-1.55d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_1
    else if (f <= 1.46d0) then
        tmp = (f * (t_0 * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))))) - (x / tan(b))
    else
        tmp = t_0 - t_1
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = 1.0 / Math.sin(B);
	double t_1 = x * (1.0 / Math.tan(B));
	double tmp;
	if (F <= -1.55) {
		tmp = (-1.0 / Math.sin(B)) - t_1;
	} else if (F <= 1.46) {
		tmp = (F * (t_0 * Math.sqrt((1.0 / (2.0 + (x * 2.0)))))) - (x / Math.tan(B));
	} else {
		tmp = t_0 - t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = 1.0 / math.sin(B)
	t_1 = x * (1.0 / math.tan(B))
	tmp = 0
	if F <= -1.55:
		tmp = (-1.0 / math.sin(B)) - t_1
	elif F <= 1.46:
		tmp = (F * (t_0 * math.sqrt((1.0 / (2.0 + (x * 2.0)))))) - (x / math.tan(B))
	else:
		tmp = t_0 - t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / sin(B))
	t_1 = Float64(x * Float64(1.0 / tan(B)))
	tmp = 0.0
	if (F <= -1.55)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= 1.46)
		tmp = Float64(Float64(F * Float64(t_0 * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0)))))) - Float64(x / tan(B)));
	else
		tmp = Float64(t_0 - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = 1.0 / sin(B);
	t_1 = x * (1.0 / tan(B));
	tmp = 0.0;
	if (F <= -1.55)
		tmp = (-1.0 / sin(B)) - t_1;
	elseif (F <= 1.46)
		tmp = (F * (t_0 * sqrt((1.0 / (2.0 + (x * 2.0)))))) - (x / tan(B));
	else
		tmp = t_0 - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.55], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, 1.46], N[(N[(F * N[(t$95$0 * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.55000000000000004

   1. Initial program 58.1%

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

   3. Taylor expanded in F around -inf 99.7%

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

   if -1.55000000000000004 < F < 1.46

   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. associate-*l/ 99.5%

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

      4. associate-/l* 99.4%

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

      5. fma-define 99.4%

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

      6. /-rgt-identity 99.4%

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

      7. remove-double-neg 99.4%

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

      8. fma-neg 99.4%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in F around 0 99.1%

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

   if 1.46 < F

   1. Initial program 53.7%

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

   3. Taylor expanded in F around inf 99.3%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.55:\\ \;\;\;\;\frac{-1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq 1.46:\\ \;\;\;\;F \cdot \left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\right) - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1.45:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B \cdot \sqrt{2 + x \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ 1.0 (tan B)))))
   (if (<= F -1.4)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.45)
       (+ (* x (/ -1.0 (tan B))) (/ F (* (sin B) (sqrt (+ 2.0 (* x 2.0))))))
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x * (1.0 / tan(B));
	double tmp;
	if (F <= -1.4) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.45) {
		tmp = (x * (-1.0 / tan(B))) + (F / (sin(B) * sqrt((2.0 + (x * 2.0)))));
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (1.0d0 / tan(b))
    if (f <= (-1.4d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.45d0) then
        tmp = (x * ((-1.0d0) / tan(b))) + (f / (sin(b) * sqrt((2.0d0 + (x * 2.0d0)))))
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x * (1.0 / Math.tan(B));
	double tmp;
	if (F <= -1.4) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.45) {
		tmp = (x * (-1.0 / Math.tan(B))) + (F / (Math.sin(B) * Math.sqrt((2.0 + (x * 2.0)))));
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * (1.0 / math.tan(B))
	tmp = 0
	if F <= -1.4:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.45:
		tmp = (x * (-1.0 / math.tan(B))) + (F / (math.sin(B) * math.sqrt((2.0 + (x * 2.0)))))
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(1.0 / tan(B)))
	tmp = 0.0
	if (F <= -1.4)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.45)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(F / Float64(sin(B) * sqrt(Float64(2.0 + Float64(x * 2.0))))));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x * (1.0 / tan(B));
	tmp = 0.0;
	if (F <= -1.4)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.45)
		tmp = (x * (-1.0 / tan(B))) + (F / (sin(B) * sqrt((2.0 + (x * 2.0)))));
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.4], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.45], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.3999999999999999

   1. Initial program 58.1%

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

   3. Taylor expanded in F around -inf 99.7%

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

   if -1.3999999999999999 < F < 1.44999999999999996

   1. Initial program 99.5%

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

      1. associate-*l/ 99.5%

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

      2. +-commutative 99.5%

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

      3. *-commutative 99.5%

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

      4. fma-undefine 99.5%

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

      5. fma-define 99.5%

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

      6. metadata-eval 99.5%

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

      7. metadata-eval 99.5%

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

      8. associate-*r/ 99.4%

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

      9. clear-num 99.4%

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

      10. un-div-inv 99.5%

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

      11. fma-define 99.5%

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

      12. fma-undefine 99.5%

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

      13. *-commutative 99.5%

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

      14. fma-define 99.5%

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

      15. fma-define 99.5%

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in F around 0 99.0%

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

   if 1.44999999999999996 < F

   1. Initial program 53.7%

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

   3. Taylor expanded in F around inf 99.3%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq 1.45:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B \cdot \sqrt{2 + x \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;F \leq -0.205:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 9.6 \cdot 10^{-116}:\\ \;\;\;\;F \cdot \left(\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{1}{B}\right) - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 155000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ 1.0 (tan B)))))
   (if (<= F -0.205)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 9.6e-116)
       (- (* F (* (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))) (/ 1.0 B))) (/ x (tan B)))
       (if (<= F 155000.0)
         (- (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)) (/ x B))
         (- (/ 1.0 (sin B)) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (1.0 / tan(B));
	double tmp;
	if (F <= -0.205) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 9.6e-116) {
		tmp = (F * (sqrt((1.0 / (2.0 + (x * 2.0)))) * (1.0 / B))) - (x / tan(B));
	} else if (F <= 155000.0) {
		tmp = ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (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 * (1.0d0 / tan(b))
    if (f <= (-0.205d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 9.6d-116) then
        tmp = (f * (sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))) * (1.0d0 / b))) - (x / tan(b))
    else if (f <= 155000.0d0) then
        tmp = ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0))) - (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 * (1.0 / Math.tan(B));
	double tmp;
	if (F <= -0.205) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 9.6e-116) {
		tmp = (F * (Math.sqrt((1.0 / (2.0 + (x * 2.0)))) * (1.0 / B))) - (x / Math.tan(B));
	} else if (F <= 155000.0) {
		tmp = ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B);
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * (1.0 / math.tan(B))
	tmp = 0
	if F <= -0.205:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 9.6e-116:
		tmp = (F * (math.sqrt((1.0 / (2.0 + (x * 2.0)))) * (1.0 / B))) - (x / math.tan(B))
	elif F <= 155000.0:
		tmp = ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B)
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(1.0 / tan(B)))
	tmp = 0.0
	if (F <= -0.205)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 9.6e-116)
		tmp = Float64(Float64(F * Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0)))) * Float64(1.0 / B))) - Float64(x / tan(B)));
	elseif (F <= 155000.0)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)) - 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 * (1.0 / tan(B));
	tmp = 0.0;
	if (F <= -0.205)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 9.6e-116)
		tmp = (F * (sqrt((1.0 / (2.0 + (x * 2.0)))) * (1.0 / B))) - (x / tan(B));
	elseif (F <= 155000.0)
		tmp = ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5)) - (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[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.205], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 9.6e-116], N[(N[(F * N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 155000.0], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - N[(x / 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 < -0.204999999999999988

   1. Initial program 58.1%

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

   3. Taylor expanded in F around -inf 99.7%

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

   if -0.204999999999999988 < F < 9.59999999999999972e-116

   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. associate-*l/ 99.6%

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

      4. associate-/l* 99.5%

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

      5. fma-define 99.5%

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

      6. /-rgt-identity 99.5%

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

      7. remove-double-neg 99.5%

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

      8. fma-neg 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in B around 0 87.1%

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

   6. Taylor expanded in F around 0 87.1%

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

   if 9.59999999999999972e-116 < F < 155000

   1. Initial program 99.3%

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

   3. Taylor expanded in B around 0 88.5%

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

   if 155000 < F

   1. Initial program 52.4%

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

   3. Taylor expanded in F around inf 99.7%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.205:\\ \;\;\;\;\frac{-1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq 9.6 \cdot 10^{-116}:\\ \;\;\;\;F \cdot \left(\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{1}{B}\right) - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 155000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;F \leq -0.37:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 0.34:\\ \;\;\;\;F \cdot \left(\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{1}{B}\right) - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ 1.0 (tan B)))))
   (if (<= F -0.37)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 0.34)
       (- (* F (* (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))) (/ 1.0 B))) (/ x (tan B)))
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x * (1.0 / tan(B));
	double tmp;
	if (F <= -0.37) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 0.34) {
		tmp = (F * (sqrt((1.0 / (2.0 + (x * 2.0)))) * (1.0 / B))) - (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 * (1.0d0 / tan(b))
    if (f <= (-0.37d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 0.34d0) then
        tmp = (f * (sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))) * (1.0d0 / b))) - (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 * (1.0 / Math.tan(B));
	double tmp;
	if (F <= -0.37) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 0.34) {
		tmp = (F * (Math.sqrt((1.0 / (2.0 + (x * 2.0)))) * (1.0 / B))) - (x / Math.tan(B));
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * (1.0 / math.tan(B))
	tmp = 0
	if F <= -0.37:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 0.34:
		tmp = (F * (math.sqrt((1.0 / (2.0 + (x * 2.0)))) * (1.0 / B))) - (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 * Float64(1.0 / tan(B)))
	tmp = 0.0
	if (F <= -0.37)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 0.34)
		tmp = Float64(Float64(F * Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0)))) * Float64(1.0 / B))) - 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 * (1.0 / tan(B));
	tmp = 0.0;
	if (F <= -0.37)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 0.34)
		tmp = (F * (sqrt((1.0 / (2.0 + (x * 2.0)))) * (1.0 / B))) - (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[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.37], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.34], N[(N[(F * N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.37

   1. Initial program 58.1%

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

   3. Taylor expanded in F around -inf 99.7%

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

   if -0.37 < F < 0.340000000000000024

   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. associate-*l/ 99.5%

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

      4. associate-/l* 99.4%

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

      5. fma-define 99.4%

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

      6. /-rgt-identity 99.4%

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

      7. remove-double-neg 99.4%

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

      8. fma-neg 99.4%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in B around 0 82.7%

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

   6. Taylor expanded in F around 0 82.2%

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

   if 0.340000000000000024 < F

   1. Initial program 53.7%

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

   3. Taylor expanded in F around inf 99.3%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.37:\\ \;\;\;\;\frac{-1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq 0.34:\\ \;\;\;\;F \cdot \left(\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{1}{B}\right) - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;F \leq -3.4 \cdot 10^{-10}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq -4.5 \cdot 10^{-100}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{elif}\;F \leq 1.3 \cdot 10^{-33}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ 1.0 (tan B)))))
   (if (<= F -3.4e-10)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F -4.5e-100)
       (/ (- (* F (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))) x) B)
       (if (<= F 1.3e-33) (/ x (- (tan B))) (- (/ 1.0 (sin B)) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (1.0 / tan(B));
	double tmp;
	if (F <= -3.4e-10) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= -4.5e-100) {
		tmp = ((F * sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B;
	} else if (F <= 1.3e-33) {
		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 * (1.0d0 / tan(b))
    if (f <= (-3.4d-10)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= (-4.5d-100)) then
        tmp = ((f * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))) - x) / b
    else if (f <= 1.3d-33) 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 * (1.0 / Math.tan(B));
	double tmp;
	if (F <= -3.4e-10) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= -4.5e-100) {
		tmp = ((F * Math.sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B;
	} else if (F <= 1.3e-33) {
		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 * (1.0 / math.tan(B))
	tmp = 0
	if F <= -3.4e-10:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= -4.5e-100:
		tmp = ((F * math.sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B
	elif F <= 1.3e-33:
		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 * Float64(1.0 / tan(B)))
	tmp = 0.0
	if (F <= -3.4e-10)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= -4.5e-100)
		tmp = Float64(Float64(Float64(F * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))) - x) / B);
	elseif (F <= 1.3e-33)
		tmp = Float64(x / Float64(-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 * (1.0 / tan(B));
	tmp = 0.0;
	if (F <= -3.4e-10)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= -4.5e-100)
		tmp = ((F * sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B;
	elseif (F <= 1.3e-33)
		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[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -3.4e-10], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -4.5e-100], N[(N[(N[(F * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.3e-33], 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 < -3.40000000000000015e-10

   1. Initial program 59.1%

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

   3. Taylor expanded in F around -inf 99.8%

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

   if -3.40000000000000015e-10 < F < -4.5000000000000001e-100

   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-define 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-define 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-define 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/ 99.4%

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

      11. *-rgt-identity 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in B around 0 70.4%

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

   6. Taylor expanded in F around 0 70.4%

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

   if -4.5000000000000001e-100 < F < 1.29999999999999997e-33

   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-define 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-define 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-define 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/ 99.8%

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

      11. *-rgt-identity 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in F around 0 75.4%

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

      1. mul-1-neg 75.4%

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

      2. associate-/l* 75.2%

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

   7. Simplified 75.2%

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

      1. clear-num 75.1%

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

      2. un-div-inv 75.3%

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

      3. quot-tan 75.5%

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

   9. Applied egg-rr 75.5%

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

   if 1.29999999999999997e-33 < F

   1. Initial program 59.3%

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

   3. Taylor expanded in F around inf 90.5%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.4 \cdot 10^{-10}:\\ \;\;\;\;\frac{-1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq -4.5 \cdot 10^{-100}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{elif}\;F \leq 1.3 \cdot 10^{-33}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{2 + x \cdot 2}}\\ \mathbf{if}\;F \leq -1.9 \cdot 10^{-9}:\\ \;\;\;\;\frac{-1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq -5 \cdot 10^{-88}:\\ \;\;\;\;\frac{F \cdot t\_0 - x}{B}\\ \mathbf{elif}\;F \leq 2.8 \cdot 10^{-101}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 32500:\\ \;\;\;\;t\_0 \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.1 \cdot 10^{+213} \lor \neg \left(F \leq 8.4 \cdot 10^{+273}\right):\\ \;\;\;\;F \cdot \frac{1}{F \cdot B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \frac{1}{F} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))))
   (if (<= F -1.9e-9)
     (- (/ -1.0 (sin B)) (* x (/ 1.0 (tan B))))
     (if (<= F -5e-88)
       (/ (- (* F t_0) x) B)
       (if (<= F 2.8e-101)
         (/ x (- (tan B)))
         (if (<= F 32500.0)
           (- (* t_0 (/ F B)) (/ x B))
           (if (or (<= F 3.1e+213) (not (<= F 8.4e+273)))
             (- (* F (/ 1.0 (* F B))) (/ x (tan B)))
             (- (* (/ F (sin B)) (/ 1.0 F)) (/ x B)))))))))

C

double code(double F, double B, double x) {
	double t_0 = sqrt((1.0 / (2.0 + (x * 2.0))));
	double tmp;
	if (F <= -1.9e-9) {
		tmp = (-1.0 / sin(B)) - (x * (1.0 / tan(B)));
	} else if (F <= -5e-88) {
		tmp = ((F * t_0) - x) / B;
	} else if (F <= 2.8e-101) {
		tmp = x / -tan(B);
	} else if (F <= 32500.0) {
		tmp = (t_0 * (F / B)) - (x / B);
	} else if ((F <= 3.1e+213) || !(F <= 8.4e+273)) {
		tmp = (F * (1.0 / (F * B))) - (x / tan(B));
	} else {
		tmp = ((F / sin(B)) * (1.0 / F)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))
    if (f <= (-1.9d-9)) then
        tmp = ((-1.0d0) / sin(b)) - (x * (1.0d0 / tan(b)))
    else if (f <= (-5d-88)) then
        tmp = ((f * t_0) - x) / b
    else if (f <= 2.8d-101) then
        tmp = x / -tan(b)
    else if (f <= 32500.0d0) then
        tmp = (t_0 * (f / b)) - (x / b)
    else if ((f <= 3.1d+213) .or. (.not. (f <= 8.4d+273))) then
        tmp = (f * (1.0d0 / (f * b))) - (x / tan(b))
    else
        tmp = ((f / sin(b)) * (1.0d0 / f)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = Math.sqrt((1.0 / (2.0 + (x * 2.0))));
	double tmp;
	if (F <= -1.9e-9) {
		tmp = (-1.0 / Math.sin(B)) - (x * (1.0 / Math.tan(B)));
	} else if (F <= -5e-88) {
		tmp = ((F * t_0) - x) / B;
	} else if (F <= 2.8e-101) {
		tmp = x / -Math.tan(B);
	} else if (F <= 32500.0) {
		tmp = (t_0 * (F / B)) - (x / B);
	} else if ((F <= 3.1e+213) || !(F <= 8.4e+273)) {
		tmp = (F * (1.0 / (F * B))) - (x / Math.tan(B));
	} else {
		tmp = ((F / Math.sin(B)) * (1.0 / F)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = math.sqrt((1.0 / (2.0 + (x * 2.0))))
	tmp = 0
	if F <= -1.9e-9:
		tmp = (-1.0 / math.sin(B)) - (x * (1.0 / math.tan(B)))
	elif F <= -5e-88:
		tmp = ((F * t_0) - x) / B
	elif F <= 2.8e-101:
		tmp = x / -math.tan(B)
	elif F <= 32500.0:
		tmp = (t_0 * (F / B)) - (x / B)
	elif (F <= 3.1e+213) or not (F <= 8.4e+273):
		tmp = (F * (1.0 / (F * B))) - (x / math.tan(B))
	else:
		tmp = ((F / math.sin(B)) * (1.0 / F)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))
	tmp = 0.0
	if (F <= -1.9e-9)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x * Float64(1.0 / tan(B))));
	elseif (F <= -5e-88)
		tmp = Float64(Float64(Float64(F * t_0) - x) / B);
	elseif (F <= 2.8e-101)
		tmp = Float64(x / Float64(-tan(B)));
	elseif (F <= 32500.0)
		tmp = Float64(Float64(t_0 * Float64(F / B)) - Float64(x / B));
	elseif ((F <= 3.1e+213) || !(F <= 8.4e+273))
		tmp = Float64(Float64(F * Float64(1.0 / Float64(F * B))) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(Float64(F / sin(B)) * Float64(1.0 / F)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = sqrt((1.0 / (2.0 + (x * 2.0))));
	tmp = 0.0;
	if (F <= -1.9e-9)
		tmp = (-1.0 / sin(B)) - (x * (1.0 / tan(B)));
	elseif (F <= -5e-88)
		tmp = ((F * t_0) - x) / B;
	elseif (F <= 2.8e-101)
		tmp = x / -tan(B);
	elseif (F <= 32500.0)
		tmp = (t_0 * (F / B)) - (x / B);
	elseif ((F <= 3.1e+213) || ~((F <= 8.4e+273)))
		tmp = (F * (1.0 / (F * B))) - (x / tan(B));
	else
		tmp = ((F / sin(B)) * (1.0 / F)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[F, -1.9e-9], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -5e-88], N[(N[(N[(F * t$95$0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 2.8e-101], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 32500.0], N[(N[(t$95$0 * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[F, 3.1e+213], N[Not[LessEqual[F, 8.4e+273]], $MachinePrecision]], N[(N[(F * N[(1.0 / N[(F * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(1.0 / F), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if F < -1.90000000000000006e-9

   1. Initial program 59.1%

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

   3. Taylor expanded in F around -inf 99.8%

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

   if -1.90000000000000006e-9 < F < -5.00000000000000009e-88

   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-define 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-define 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-define 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/ 99.4%

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

      11. *-rgt-identity 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in B around 0 70.4%

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

   6. Taylor expanded in F around 0 70.4%

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

   if -5.00000000000000009e-88 < F < 2.79999999999999989e-101

   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-define 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-define 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-define 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/ 99.8%

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

      11. *-rgt-identity 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in F around 0 78.8%

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

      1. mul-1-neg 78.8%

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

      2. associate-/l* 78.7%

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

   7. Simplified 78.7%

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

      1. clear-num 78.6%

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

      2. un-div-inv 78.7%

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

      3. quot-tan 78.9%

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

   9. Applied egg-rr 78.9%

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

   if 2.79999999999999989e-101 < F < 32500

   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. Step-by-step derivation

      1. distribute-lft-neg-in 99.3%

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

      2. +-commutative 99.3%

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

      3. fma-define 99.3%

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

      4. +-commutative 99.3%

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

      5. *-commutative 99.3%

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

      6. fma-define 99.3%

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

      7. fma-define 99.3%

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

      8. metadata-eval 99.3%

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

      9. metadata-eval 99.3%

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

      10. associate-*r/ 99.4%

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

      11. *-rgt-identity 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in B around 0 60.2%

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

   6. Taylor expanded in F around 0 57.9%

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

   if 32500 < F < 3.09999999999999991e213 or 8.40000000000000007e273 < F

   1. Initial program 61.4%

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

      1. distribute-lft-neg-in 61.4%

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

      2. +-commutative 61.4%

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

      3. associate-*l/ 82.1%

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

      4. associate-/l* 82.0%

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

      5. fma-define 82.0%

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

      6. /-rgt-identity 82.0%

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

      7. remove-double-neg 82.0%

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

      8. fma-neg 82.0%

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

   3. Simplified 82.1%

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

   5. Taylor expanded in B around 0 74.1%

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

   6. Taylor expanded in F around inf 82.9%

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

   if 3.09999999999999991e213 < F < 8.40000000000000007e273

   1. Initial program 16.6%

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

   3. Taylor expanded in F around inf 92.8%

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

   4. Taylor expanded in B around 0 85.4%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot \frac{1}{F} \]
3. Recombined 6 regimes into one program.

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.9 \cdot 10^{-9}:\\ \;\;\;\;\frac{-1}{\sin B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq -5 \cdot 10^{-88}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{elif}\;F \leq 2.8 \cdot 10^{-101}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 32500:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.1 \cdot 10^{+213} \lor \neg \left(F \leq 8.4 \cdot 10^{+273}\right):\\ \;\;\;\;F \cdot \frac{1}{F \cdot B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \frac{1}{F} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 68.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2.9 \cdot 10^{-8}:\\ \;\;\;\;\frac{-1}{B} - t\_1\\ \mathbf{elif}\;F \leq -5.8 \cdot 10^{-86}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{-99}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 32500:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 3.3 \cdot 10^{+213} \lor \neg \left(F \leq 8.4 \cdot 10^{+273}\right):\\ \;\;\;\;F \cdot \frac{1}{F \cdot B} - t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \frac{1}{F} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- (* F (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))) x) B))
        (t_1 (/ x (tan B))))
   (if (<= F -2.9e-8)
     (- (/ -1.0 B) t_1)
     (if (<= F -5.8e-86)
       t_0
       (if (<= F 4.5e-99)
         (/ x (- (tan B)))
         (if (<= F 32500.0)
           t_0
           (if (or (<= F 3.3e+213) (not (<= F 8.4e+273)))
             (- (* F (/ 1.0 (* F B))) t_1)
             (- (* (/ F (sin B)) (/ 1.0 F)) (/ x B)))))))))

C

double code(double F, double B, double x) {
	double t_0 = ((F * sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B;
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -2.9e-8) {
		tmp = (-1.0 / B) - t_1;
	} else if (F <= -5.8e-86) {
		tmp = t_0;
	} else if (F <= 4.5e-99) {
		tmp = x / -tan(B);
	} else if (F <= 32500.0) {
		tmp = t_0;
	} else if ((F <= 3.3e+213) || !(F <= 8.4e+273)) {
		tmp = (F * (1.0 / (F * B))) - t_1;
	} else {
		tmp = ((F / sin(B)) * (1.0 / F)) - (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 = ((f * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))) - x) / b
    t_1 = x / tan(b)
    if (f <= (-2.9d-8)) then
        tmp = ((-1.0d0) / b) - t_1
    else if (f <= (-5.8d-86)) then
        tmp = t_0
    else if (f <= 4.5d-99) then
        tmp = x / -tan(b)
    else if (f <= 32500.0d0) then
        tmp = t_0
    else if ((f <= 3.3d+213) .or. (.not. (f <= 8.4d+273))) then
        tmp = (f * (1.0d0 / (f * b))) - t_1
    else
        tmp = ((f / sin(b)) * (1.0d0 / f)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = ((F * Math.sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B;
	double t_1 = x / Math.tan(B);
	double tmp;
	if (F <= -2.9e-8) {
		tmp = (-1.0 / B) - t_1;
	} else if (F <= -5.8e-86) {
		tmp = t_0;
	} else if (F <= 4.5e-99) {
		tmp = x / -Math.tan(B);
	} else if (F <= 32500.0) {
		tmp = t_0;
	} else if ((F <= 3.3e+213) || !(F <= 8.4e+273)) {
		tmp = (F * (1.0 / (F * B))) - t_1;
	} else {
		tmp = ((F / Math.sin(B)) * (1.0 / F)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = ((F * math.sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B
	t_1 = x / math.tan(B)
	tmp = 0
	if F <= -2.9e-8:
		tmp = (-1.0 / B) - t_1
	elif F <= -5.8e-86:
		tmp = t_0
	elif F <= 4.5e-99:
		tmp = x / -math.tan(B)
	elif F <= 32500.0:
		tmp = t_0
	elif (F <= 3.3e+213) or not (F <= 8.4e+273):
		tmp = (F * (1.0 / (F * B))) - t_1
	else:
		tmp = ((F / math.sin(B)) * (1.0 / F)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(Float64(F * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))) - x) / B)
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -2.9e-8)
		tmp = Float64(Float64(-1.0 / B) - t_1);
	elseif (F <= -5.8e-86)
		tmp = t_0;
	elseif (F <= 4.5e-99)
		tmp = Float64(x / Float64(-tan(B)));
	elseif (F <= 32500.0)
		tmp = t_0;
	elseif ((F <= 3.3e+213) || !(F <= 8.4e+273))
		tmp = Float64(Float64(F * Float64(1.0 / Float64(F * B))) - t_1);
	else
		tmp = Float64(Float64(Float64(F / sin(B)) * Float64(1.0 / F)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = ((F * sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B;
	t_1 = x / tan(B);
	tmp = 0.0;
	if (F <= -2.9e-8)
		tmp = (-1.0 / B) - t_1;
	elseif (F <= -5.8e-86)
		tmp = t_0;
	elseif (F <= 4.5e-99)
		tmp = x / -tan(B);
	elseif (F <= 32500.0)
		tmp = t_0;
	elseif ((F <= 3.3e+213) || ~((F <= 8.4e+273)))
		tmp = (F * (1.0 / (F * B))) - t_1;
	else
		tmp = ((F / sin(B)) * (1.0 / F)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(N[(F * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.9e-8], N[(N[(-1.0 / B), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, -5.8e-86], t$95$0, If[LessEqual[F, 4.5e-99], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 32500.0], t$95$0, If[Or[LessEqual[F, 3.3e+213], N[Not[LessEqual[F, 8.4e+273]], $MachinePrecision]], N[(N[(F * N[(1.0 / N[(F * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(1.0 / F), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -2.9000000000000002e-8

   1. Initial program 59.1%

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

      1. distribute-lft-neg-in 59.1%

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

      2. +-commutative 59.1%

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

      3. associate-*l/ 74.0%

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

      4. associate-/l* 73.9%

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

      5. fma-define 73.9%

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

      6. /-rgt-identity 73.9%

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

      7. remove-double-neg 73.9%

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

      8. fma-neg 73.9%

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

   3. Simplified 74.0%

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

   5. Taylor expanded in F around -inf 99.6%

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

   6. Taylor expanded in B around 0 74.0%

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

   if -2.9000000000000002e-8 < F < -5.7999999999999998e-86 or 4.5000000000000003e-99 < F < 32500

   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. Step-by-step derivation

      1. distribute-lft-neg-in 99.3%

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

      2. +-commutative 99.3%

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

      3. fma-define 99.3%

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

      4. +-commutative 99.3%

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

      5. *-commutative 99.3%

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

      6. fma-define 99.3%

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

      7. fma-define 99.3%

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

      8. metadata-eval 99.3%

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

      9. metadata-eval 99.3%

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

      10. associate-*r/ 99.4%

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

      11. *-rgt-identity 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in B around 0 64.0%

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

   6. Taylor expanded in F around 0 62.5%

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

   if -5.7999999999999998e-86 < F < 4.5000000000000003e-99

   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-define 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-define 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-define 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/ 99.8%

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

      11. *-rgt-identity 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in F around 0 78.8%

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

      1. mul-1-neg 78.8%

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

      2. associate-/l* 78.7%

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

   7. Simplified 78.7%

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

      1. clear-num 78.6%

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

      2. un-div-inv 78.7%

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

      3. quot-tan 78.9%

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

   9. Applied egg-rr 78.9%

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

   if 32500 < F < 3.3000000000000001e213 or 8.40000000000000007e273 < F

   1. Initial program 61.4%

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

      1. distribute-lft-neg-in 61.4%

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

      2. +-commutative 61.4%

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

      3. associate-*l/ 82.1%

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

      4. associate-/l* 82.0%

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

      5. fma-define 82.0%

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

      6. /-rgt-identity 82.0%

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

      7. remove-double-neg 82.0%

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

      8. fma-neg 82.0%

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

   3. Simplified 82.1%

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

   5. Taylor expanded in B around 0 74.1%

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

   6. Taylor expanded in F around inf 82.9%

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

   if 3.3000000000000001e213 < F < 8.40000000000000007e273

   1. Initial program 16.6%

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

   3. Taylor expanded in F around inf 92.8%

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

   4. Taylor expanded in B around 0 85.4%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.9 \cdot 10^{-8}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -5.8 \cdot 10^{-86}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{-99}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 32500:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{elif}\;F \leq 3.3 \cdot 10^{+213} \lor \neg \left(F \leq 8.4 \cdot 10^{+273}\right):\\ \;\;\;\;F \cdot \frac{1}{F \cdot B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \frac{1}{F} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 68.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{2 + x \cdot 2}}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3.9 \cdot 10^{-11}:\\ \;\;\;\;\frac{-1}{B} - t\_1\\ \mathbf{elif}\;F \leq -1.7 \cdot 10^{-100}:\\ \;\;\;\;\frac{F \cdot t\_0 - x}{B}\\ \mathbf{elif}\;F \leq 2.8 \cdot 10^{-101}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 32500:\\ \;\;\;\;t\_0 \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.5 \cdot 10^{+211} \lor \neg \left(F \leq 8.4 \cdot 10^{+273}\right):\\ \;\;\;\;F \cdot \frac{1}{F \cdot B} - t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \frac{1}{F} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))) (t_1 (/ x (tan B))))
   (if (<= F -3.9e-11)
     (- (/ -1.0 B) t_1)
     (if (<= F -1.7e-100)
       (/ (- (* F t_0) x) B)
       (if (<= F 2.8e-101)
         (/ x (- (tan B)))
         (if (<= F 32500.0)
           (- (* t_0 (/ F B)) (/ x B))
           (if (or (<= F 1.5e+211) (not (<= F 8.4e+273)))
             (- (* F (/ 1.0 (* F B))) t_1)
             (- (* (/ F (sin B)) (/ 1.0 F)) (/ x B)))))))))

C

double code(double F, double B, double x) {
	double t_0 = sqrt((1.0 / (2.0 + (x * 2.0))));
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -3.9e-11) {
		tmp = (-1.0 / B) - t_1;
	} else if (F <= -1.7e-100) {
		tmp = ((F * t_0) - x) / B;
	} else if (F <= 2.8e-101) {
		tmp = x / -tan(B);
	} else if (F <= 32500.0) {
		tmp = (t_0 * (F / B)) - (x / B);
	} else if ((F <= 1.5e+211) || !(F <= 8.4e+273)) {
		tmp = (F * (1.0 / (F * B))) - t_1;
	} else {
		tmp = ((F / sin(B)) * (1.0 / F)) - (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 = sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))
    t_1 = x / tan(b)
    if (f <= (-3.9d-11)) then
        tmp = ((-1.0d0) / b) - t_1
    else if (f <= (-1.7d-100)) then
        tmp = ((f * t_0) - x) / b
    else if (f <= 2.8d-101) then
        tmp = x / -tan(b)
    else if (f <= 32500.0d0) then
        tmp = (t_0 * (f / b)) - (x / b)
    else if ((f <= 1.5d+211) .or. (.not. (f <= 8.4d+273))) then
        tmp = (f * (1.0d0 / (f * b))) - t_1
    else
        tmp = ((f / sin(b)) * (1.0d0 / f)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = Math.sqrt((1.0 / (2.0 + (x * 2.0))));
	double t_1 = x / Math.tan(B);
	double tmp;
	if (F <= -3.9e-11) {
		tmp = (-1.0 / B) - t_1;
	} else if (F <= -1.7e-100) {
		tmp = ((F * t_0) - x) / B;
	} else if (F <= 2.8e-101) {
		tmp = x / -Math.tan(B);
	} else if (F <= 32500.0) {
		tmp = (t_0 * (F / B)) - (x / B);
	} else if ((F <= 1.5e+211) || !(F <= 8.4e+273)) {
		tmp = (F * (1.0 / (F * B))) - t_1;
	} else {
		tmp = ((F / Math.sin(B)) * (1.0 / F)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = math.sqrt((1.0 / (2.0 + (x * 2.0))))
	t_1 = x / math.tan(B)
	tmp = 0
	if F <= -3.9e-11:
		tmp = (-1.0 / B) - t_1
	elif F <= -1.7e-100:
		tmp = ((F * t_0) - x) / B
	elif F <= 2.8e-101:
		tmp = x / -math.tan(B)
	elif F <= 32500.0:
		tmp = (t_0 * (F / B)) - (x / B)
	elif (F <= 1.5e+211) or not (F <= 8.4e+273):
		tmp = (F * (1.0 / (F * B))) - t_1
	else:
		tmp = ((F / math.sin(B)) * (1.0 / F)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -3.9e-11)
		tmp = Float64(Float64(-1.0 / B) - t_1);
	elseif (F <= -1.7e-100)
		tmp = Float64(Float64(Float64(F * t_0) - x) / B);
	elseif (F <= 2.8e-101)
		tmp = Float64(x / Float64(-tan(B)));
	elseif (F <= 32500.0)
		tmp = Float64(Float64(t_0 * Float64(F / B)) - Float64(x / B));
	elseif ((F <= 1.5e+211) || !(F <= 8.4e+273))
		tmp = Float64(Float64(F * Float64(1.0 / Float64(F * B))) - t_1);
	else
		tmp = Float64(Float64(Float64(F / sin(B)) * Float64(1.0 / F)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = sqrt((1.0 / (2.0 + (x * 2.0))));
	t_1 = x / tan(B);
	tmp = 0.0;
	if (F <= -3.9e-11)
		tmp = (-1.0 / B) - t_1;
	elseif (F <= -1.7e-100)
		tmp = ((F * t_0) - x) / B;
	elseif (F <= 2.8e-101)
		tmp = x / -tan(B);
	elseif (F <= 32500.0)
		tmp = (t_0 * (F / B)) - (x / B);
	elseif ((F <= 1.5e+211) || ~((F <= 8.4e+273)))
		tmp = (F * (1.0 / (F * B))) - t_1;
	else
		tmp = ((F / sin(B)) * (1.0 / F)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -3.9e-11], N[(N[(-1.0 / B), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, -1.7e-100], N[(N[(N[(F * t$95$0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 2.8e-101], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 32500.0], N[(N[(t$95$0 * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[F, 1.5e+211], N[Not[LessEqual[F, 8.4e+273]], $MachinePrecision]], N[(N[(F * N[(1.0 / N[(F * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(1.0 / F), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if F < -3.9000000000000001e-11

   1. Initial program 59.1%

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

      1. distribute-lft-neg-in 59.1%

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

      2. +-commutative 59.1%

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

      3. associate-*l/ 74.0%

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

      4. associate-/l* 73.9%

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

      5. fma-define 73.9%

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

      6. /-rgt-identity 73.9%

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

      7. remove-double-neg 73.9%

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

      8. fma-neg 73.9%

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

   3. Simplified 74.0%

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

   5. Taylor expanded in F around -inf 99.6%

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

   6. Taylor expanded in B around 0 74.0%

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

   if -3.9000000000000001e-11 < F < -1.69999999999999988e-100

   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-define 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-define 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-define 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/ 99.4%

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

      11. *-rgt-identity 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in B around 0 70.4%

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

   6. Taylor expanded in F around 0 70.4%

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

   if -1.69999999999999988e-100 < F < 2.79999999999999989e-101

   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-define 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-define 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-define 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/ 99.8%

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

      11. *-rgt-identity 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in F around 0 78.8%

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

      1. mul-1-neg 78.8%

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

      2. associate-/l* 78.7%

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

   7. Simplified 78.7%

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

      1. clear-num 78.6%

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

      2. un-div-inv 78.7%

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

      3. quot-tan 78.9%

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

   9. Applied egg-rr 78.9%

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

   if 2.79999999999999989e-101 < F < 32500

   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. Step-by-step derivation

      1. distribute-lft-neg-in 99.3%

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

      2. +-commutative 99.3%

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

      3. fma-define 99.3%

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

      4. +-commutative 99.3%

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

      5. *-commutative 99.3%

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

      6. fma-define 99.3%

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

      7. fma-define 99.3%

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

      8. metadata-eval 99.3%

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

      9. metadata-eval 99.3%

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

      10. associate-*r/ 99.4%

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

      11. *-rgt-identity 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in B around 0 60.2%

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

   6. Taylor expanded in F around 0 57.9%

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

   if 32500 < F < 1.5e211 or 8.40000000000000007e273 < F

   1. Initial program 61.4%

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

      1. distribute-lft-neg-in 61.4%

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

      2. +-commutative 61.4%

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

      3. associate-*l/ 82.1%

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

      4. associate-/l* 82.0%

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

      5. fma-define 82.0%

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

      6. /-rgt-identity 82.0%

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

      7. remove-double-neg 82.0%

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

      8. fma-neg 82.0%

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

   3. Simplified 82.1%

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

   5. Taylor expanded in B around 0 74.1%

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

   6. Taylor expanded in F around inf 82.9%

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

   if 1.5e211 < F < 8.40000000000000007e273

   1. Initial program 16.6%

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

   3. Taylor expanded in F around inf 92.8%

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

   4. Taylor expanded in B around 0 85.4%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot \frac{1}{F} \]
3. Recombined 6 regimes into one program.

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.9 \cdot 10^{-11}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.7 \cdot 10^{-100}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{elif}\;F \leq 2.8 \cdot 10^{-101}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 32500:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.5 \cdot 10^{+211} \lor \neg \left(F \leq 8.4 \cdot 10^{+273}\right):\\ \;\;\;\;F \cdot \frac{1}{F \cdot B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \frac{1}{F} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 67.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -4.2 \cdot 10^{-50}:\\ \;\;\;\;\frac{-1}{B} - t\_0\\ \mathbf{elif}\;F \leq 2.25 \cdot 10^{-32}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 3.3 \cdot 10^{+213} \lor \neg \left(F \leq 8.4 \cdot 10^{+273}\right):\\ \;\;\;\;F \cdot \frac{1}{F \cdot B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \frac{1}{F} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -4.2e-50)
     (- (/ -1.0 B) t_0)
     (if (<= F 2.25e-32)
       (/ x (- (tan B)))
       (if (or (<= F 3.3e+213) (not (<= F 8.4e+273)))
         (- (* F (/ 1.0 (* F B))) t_0)
         (- (* (/ F (sin B)) (/ 1.0 F)) (/ x B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -4.2e-50) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= 2.25e-32) {
		tmp = x / -tan(B);
	} else if ((F <= 3.3e+213) || !(F <= 8.4e+273)) {
		tmp = (F * (1.0 / (F * B))) - t_0;
	} else {
		tmp = ((F / sin(B)) * (1.0 / F)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-4.2d-50)) then
        tmp = ((-1.0d0) / b) - t_0
    else if (f <= 2.25d-32) then
        tmp = x / -tan(b)
    else if ((f <= 3.3d+213) .or. (.not. (f <= 8.4d+273))) then
        tmp = (f * (1.0d0 / (f * b))) - t_0
    else
        tmp = ((f / sin(b)) * (1.0d0 / f)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -4.2e-50) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= 2.25e-32) {
		tmp = x / -Math.tan(B);
	} else if ((F <= 3.3e+213) || !(F <= 8.4e+273)) {
		tmp = (F * (1.0 / (F * B))) - t_0;
	} else {
		tmp = ((F / Math.sin(B)) * (1.0 / F)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -4.2e-50:
		tmp = (-1.0 / B) - t_0
	elif F <= 2.25e-32:
		tmp = x / -math.tan(B)
	elif (F <= 3.3e+213) or not (F <= 8.4e+273):
		tmp = (F * (1.0 / (F * B))) - t_0
	else:
		tmp = ((F / math.sin(B)) * (1.0 / F)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -4.2e-50)
		tmp = Float64(Float64(-1.0 / B) - t_0);
	elseif (F <= 2.25e-32)
		tmp = Float64(x / Float64(-tan(B)));
	elseif ((F <= 3.3e+213) || !(F <= 8.4e+273))
		tmp = Float64(Float64(F * Float64(1.0 / Float64(F * B))) - t_0);
	else
		tmp = Float64(Float64(Float64(F / sin(B)) * Float64(1.0 / F)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -4.2e-50)
		tmp = (-1.0 / B) - t_0;
	elseif (F <= 2.25e-32)
		tmp = x / -tan(B);
	elseif ((F <= 3.3e+213) || ~((F <= 8.4e+273)))
		tmp = (F * (1.0 / (F * B))) - t_0;
	else
		tmp = ((F / sin(B)) * (1.0 / F)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -4.2e-50], N[(N[(-1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 2.25e-32], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], If[Or[LessEqual[F, 3.3e+213], N[Not[LessEqual[F, 8.4e+273]], $MachinePrecision]], N[(N[(F * N[(1.0 / N[(F * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(1.0 / F), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -4.2000000000000002e-50

   1. Initial program 63.1%

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

      1. distribute-lft-neg-in 63.1%

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

      2. +-commutative 63.1%

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

      3. associate-*l/ 76.5%

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

      4. associate-/l* 76.4%

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

      5. fma-define 76.4%

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

      6. /-rgt-identity 76.4%

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

      7. remove-double-neg 76.4%

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

      8. fma-neg 76.4%

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

   3. Simplified 76.5%

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

   5. Taylor expanded in F around -inf 92.6%

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

   6. Taylor expanded in B around 0 69.4%

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

   if -4.2000000000000002e-50 < F < 2.25000000000000002e-32

   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-define 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-define 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-define 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. Add Preprocessing

   5. Taylor expanded in F around 0 72.7%

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

      1. mul-1-neg 72.7%

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

      2. associate-/l* 72.6%

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

   7. Simplified 72.6%

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

      1. clear-num 72.4%

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

      2. un-div-inv 72.7%

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

      3. quot-tan 72.8%

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

   9. Applied egg-rr 72.8%

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

   if 2.25000000000000002e-32 < F < 3.3000000000000001e213 or 8.40000000000000007e273 < F

   1. Initial program 66.1%

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

      1. distribute-lft-neg-in 66.1%

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

      2. +-commutative 66.1%

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

      3. associate-*l/ 84.3%

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

      4. associate-/l* 84.1%

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

      5. fma-define 84.1%

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

      6. /-rgt-identity 84.1%

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

      7. remove-double-neg 84.1%

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

      8. fma-neg 84.1%

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

   3. Simplified 84.2%

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

   5. Taylor expanded in B around 0 71.2%

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

   6. Taylor expanded in F around inf 74.8%

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

   if 3.3000000000000001e213 < F < 8.40000000000000007e273

   1. Initial program 16.6%

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

   3. Taylor expanded in F around inf 92.8%

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

   4. Taylor expanded in B around 0 85.4%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.2 \cdot 10^{-50}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.25 \cdot 10^{-32}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 3.3 \cdot 10^{+213} \lor \neg \left(F \leq 8.4 \cdot 10^{+273}\right):\\ \;\;\;\;F \cdot \frac{1}{F \cdot B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \frac{1}{F} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 69.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -6.2 \cdot 10^{-50}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.25 \cdot 10^{-32}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -6.2e-50)
   (- (/ -1.0 B) (/ x (tan B)))
   (if (<= F 2.25e-32) (/ x (- (tan B))) (- (/ 1.0 B) (* x (/ 1.0 (tan B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -6.2e-50) {
		tmp = (-1.0 / B) - (x / tan(B));
	} else if (F <= 2.25e-32) {
		tmp = x / -tan(B);
	} else {
		tmp = (1.0 / B) - (x * (1.0 / tan(B)));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-6.2d-50)) then
        tmp = ((-1.0d0) / b) - (x / tan(b))
    else if (f <= 2.25d-32) then
        tmp = x / -tan(b)
    else
        tmp = (1.0d0 / b) - (x * (1.0d0 / tan(b)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -6.2e-50) {
		tmp = (-1.0 / B) - (x / Math.tan(B));
	} else if (F <= 2.25e-32) {
		tmp = x / -Math.tan(B);
	} else {
		tmp = (1.0 / B) - (x * (1.0 / Math.tan(B)));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -6.2e-50:
		tmp = (-1.0 / B) - (x / math.tan(B))
	elif F <= 2.25e-32:
		tmp = x / -math.tan(B)
	else:
		tmp = (1.0 / B) - (x * (1.0 / math.tan(B)))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -6.2e-50)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / tan(B)));
	elseif (F <= 2.25e-32)
		tmp = Float64(x / Float64(-tan(B)));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x * Float64(1.0 / tan(B))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -6.2e-50)
		tmp = (-1.0 / B) - (x / tan(B));
	elseif (F <= 2.25e-32)
		tmp = x / -tan(B);
	else
		tmp = (1.0 / B) - (x * (1.0 / tan(B)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -6.2e-50], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.25e-32], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -6.2000000000000004e-50

   1. Initial program 63.1%

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

      1. distribute-lft-neg-in 63.1%

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

      2. +-commutative 63.1%

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

      3. associate-*l/ 76.5%

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

      4. associate-/l* 76.4%

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

      5. fma-define 76.4%

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

      6. /-rgt-identity 76.4%

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

      7. remove-double-neg 76.4%

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

      8. fma-neg 76.4%

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

   3. Simplified 76.5%

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

   5. Taylor expanded in F around -inf 92.6%

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

   6. Taylor expanded in B around 0 69.4%

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

   if -6.2000000000000004e-50 < F < 2.25000000000000002e-32

   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-define 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-define 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-define 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. Add Preprocessing

   5. Taylor expanded in F around 0 72.7%

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

      1. mul-1-neg 72.7%

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

      2. associate-/l* 72.6%

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

   7. Simplified 72.6%

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

      1. clear-num 72.4%

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

      2. un-div-inv 72.7%

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

      3. quot-tan 72.8%

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

   9. Applied egg-rr 72.8%

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

   if 2.25000000000000002e-32 < F

   1. Initial program 57.2%

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

   3. Taylor expanded in F around inf 73.8%

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

   4. Taylor expanded in B around 0 69.4%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6.2 \cdot 10^{-50}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.25 \cdot 10^{-32}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 56.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.4 \cdot 10^{+74}:\\ \;\;\;\;0.3333333333333333 \cdot \left(x \cdot B\right) + \frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.6 \cdot 10^{-32}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.4e+74)
   (+ (* 0.3333333333333333 (* x B)) (/ (- -1.0 x) B))
   (if (<= F 2.6e-32) (/ x (- (tan B))) (- (/ 1.0 B) (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.4e+74) {
		tmp = (0.3333333333333333 * (x * B)) + ((-1.0 - x) / B);
	} else if (F <= 2.6e-32) {
		tmp = x / -tan(B);
	} else {
		tmp = (1.0 / 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 <= (-3.4d+74)) then
        tmp = (0.3333333333333333d0 * (x * b)) + (((-1.0d0) - x) / b)
    else if (f <= 2.6d-32) then
        tmp = x / -tan(b)
    else
        tmp = (1.0d0 / b) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.4e+74) {
		tmp = (0.3333333333333333 * (x * B)) + ((-1.0 - x) / B);
	} else if (F <= 2.6e-32) {
		tmp = x / -Math.tan(B);
	} else {
		tmp = (1.0 / B) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.4e+74:
		tmp = (0.3333333333333333 * (x * B)) + ((-1.0 - x) / B)
	elif F <= 2.6e-32:
		tmp = x / -math.tan(B)
	else:
		tmp = (1.0 / B) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.4e+74)
		tmp = Float64(Float64(0.3333333333333333 * Float64(x * B)) + Float64(Float64(-1.0 - x) / B));
	elseif (F <= 2.6e-32)
		tmp = Float64(x / Float64(-tan(B)));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -3.4e+74)
		tmp = (0.3333333333333333 * (x * B)) + ((-1.0 - x) / B);
	elseif (F <= 2.6e-32)
		tmp = x / -tan(B);
	else
		tmp = (1.0 / B) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.4e+74], N[(N[(0.3333333333333333 * N[(x * B), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.6e-32], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.3999999999999999e74

   1. Initial program 50.7%

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

      1. distribute-lft-neg-in 50.7%

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

      2. +-commutative 50.7%

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

      3. associate-*l/ 68.6%

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

      4. associate-/l* 68.6%

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

      5. fma-define 68.6%

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

      6. /-rgt-identity 68.6%

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

      7. remove-double-neg 68.6%

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

      8. fma-neg 68.6%

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

   3. Simplified 68.7%

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

   5. Taylor expanded in F around -inf 99.6%

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

   6. Taylor expanded in B around 0 78.6%

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

      1. *-commutative 78.6%

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

   8. Simplified 78.6%

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

   9. Taylor expanded in B around 0 59.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B} + 0.3333333333333333 \cdot \left(B \cdot x\right)} \]
   10. Step-by-step derivation

       1. +-commutative 59.3%

          \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(B \cdot x\right) + -1 \cdot \frac{1 + x}{B}} \]

       2. mul-1-neg 59.3%

          \[\leadsto 0.3333333333333333 \cdot \left(B \cdot x\right) + \color{blue}{\left(-\frac{1 + x}{B}\right)} \]

       3. unsub-neg 59.3%

          \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(B \cdot x\right) - \frac{1 + x}{B}} \]

       4. +-commutative 59.3%

          \[\leadsto 0.3333333333333333 \cdot \left(B \cdot x\right) - \frac{\color{blue}{x + 1}}{B} \]

   11. Simplified 59.3%

       \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(B \cdot x\right) - \frac{x + 1}{B}} \]

   if -3.3999999999999999e74 < F < 2.5999999999999997e-32

   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-define 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-define 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-define 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. Add Preprocessing

   5. Taylor expanded in F around 0 66.0%

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

      1. mul-1-neg 66.0%

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

      2. associate-/l* 66.0%

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

   7. Simplified 66.0%

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

      1. clear-num 65.8%

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

      2. un-div-inv 66.0%

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

      3. quot-tan 66.1%

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

   9. Applied egg-rr 66.1%

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

   if 2.5999999999999997e-32 < 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. fma-define 57.2%

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

      4. +-commutative 57.2%

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

      5. *-commutative 57.2%

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

      6. fma-define 57.2%

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

      7. fma-define 57.2%

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

      8. metadata-eval 57.2%

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

      9. metadata-eval 57.2%

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

      10. associate-*r/ 57.3%

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

      11. *-rgt-identity 57.3%

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

   3. Simplified 57.3%

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

   5. Taylor expanded in B around 0 43.8%

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

   6. Taylor expanded in F around inf 50.4%

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

      1. mul-1-neg 50.4%

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

      2. unsub-neg 50.4%

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

   8. Simplified 50.4%

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

   9. Taylor expanded in x around 0 50.5%

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

       1. +-commutative 50.5%

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

       2. neg-mul-1 50.5%

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

       3. sub-neg 50.5%

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

   11. Simplified 50.5%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.4 \cdot 10^{+74}:\\ \;\;\;\;0.3333333333333333 \cdot \left(x \cdot B\right) + \frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.6 \cdot 10^{-32}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 63.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.6 \cdot 10^{-50}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.6 \cdot 10^{-32}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.6e-50)
   (- (/ -1.0 B) (/ x (tan B)))
   (if (<= F 2.6e-32) (/ x (- (tan B))) (- (/ 1.0 B) (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.6e-50) {
		tmp = (-1.0 / B) - (x / tan(B));
	} else if (F <= 2.6e-32) {
		tmp = x / -tan(B);
	} else {
		tmp = (1.0 / 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 <= (-3.6d-50)) then
        tmp = ((-1.0d0) / b) - (x / tan(b))
    else if (f <= 2.6d-32) then
        tmp = x / -tan(b)
    else
        tmp = (1.0d0 / b) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.6e-50) {
		tmp = (-1.0 / B) - (x / Math.tan(B));
	} else if (F <= 2.6e-32) {
		tmp = x / -Math.tan(B);
	} else {
		tmp = (1.0 / B) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.6e-50:
		tmp = (-1.0 / B) - (x / math.tan(B))
	elif F <= 2.6e-32:
		tmp = x / -math.tan(B)
	else:
		tmp = (1.0 / B) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.6e-50)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / tan(B)));
	elseif (F <= 2.6e-32)
		tmp = Float64(x / Float64(-tan(B)));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -3.6e-50)
		tmp = (-1.0 / B) - (x / tan(B));
	elseif (F <= 2.6e-32)
		tmp = x / -tan(B);
	else
		tmp = (1.0 / B) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.6e-50], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.6e-32], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.59999999999999979e-50

   1. Initial program 63.1%

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

      1. distribute-lft-neg-in 63.1%

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

      2. +-commutative 63.1%

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

      3. associate-*l/ 76.5%

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

      4. associate-/l* 76.4%

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

      5. fma-define 76.4%

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

      6. /-rgt-identity 76.4%

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

      7. remove-double-neg 76.4%

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

      8. fma-neg 76.4%

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

   3. Simplified 76.5%

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

   5. Taylor expanded in F around -inf 92.6%

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

   6. Taylor expanded in B around 0 69.4%

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

   if -3.59999999999999979e-50 < F < 2.5999999999999997e-32

   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-define 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-define 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-define 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. Add Preprocessing

   5. Taylor expanded in F around 0 72.7%

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

      1. mul-1-neg 72.7%

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

      2. associate-/l* 72.6%

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

   7. Simplified 72.6%

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

      1. clear-num 72.4%

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

      2. un-div-inv 72.7%

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

      3. quot-tan 72.8%

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

   9. Applied egg-rr 72.8%

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

   if 2.5999999999999997e-32 < 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. fma-define 57.2%

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

      4. +-commutative 57.2%

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

      5. *-commutative 57.2%

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

      6. fma-define 57.2%

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

      7. fma-define 57.2%

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

      8. metadata-eval 57.2%

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

      9. metadata-eval 57.2%

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

      10. associate-*r/ 57.3%

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

      11. *-rgt-identity 57.3%

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

   3. Simplified 57.3%

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

   5. Taylor expanded in B around 0 43.8%

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

   6. Taylor expanded in F around inf 50.4%

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

      1. mul-1-neg 50.4%

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

      2. unsub-neg 50.4%

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

   8. Simplified 50.4%

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

   9. Taylor expanded in x around 0 50.5%

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

       1. +-commutative 50.5%

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

       2. neg-mul-1 50.5%

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

       3. sub-neg 50.5%

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

   11. Simplified 50.5%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.6 \cdot 10^{-50}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.6 \cdot 10^{-32}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 43.8% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.3 \cdot 10^{-59}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.5 \cdot 10^{-32}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.3e-59)
   (/ (- -1.0 x) B)
   (if (<= F 2.5e-32) (/ (- x) B) (- (/ 1.0 B) (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.3e-59) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 2.5e-32) {
		tmp = -x / B;
	} else {
		tmp = (1.0 / 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 <= (-1.3d-59)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 2.5d-32) then
        tmp = -x / b
    else
        tmp = (1.0d0 / b) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.3e-59) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 2.5e-32) {
		tmp = -x / B;
	} else {
		tmp = (1.0 / B) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.3e-59:
		tmp = (-1.0 - x) / B
	elif F <= 2.5e-32:
		tmp = -x / B
	else:
		tmp = (1.0 / B) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.3e-59)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 2.5e-32)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.3e-59)
		tmp = (-1.0 - x) / B;
	elseif (F <= 2.5e-32)
		tmp = -x / B;
	else
		tmp = (1.0 / B) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.3e-59], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 2.5e-32], N[((-x) / B), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.29999999999999999e-59

   1. Initial program 63.1%

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

      1. distribute-lft-neg-in 63.1%

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

      2. +-commutative 63.1%

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

      3. associate-*l/ 76.5%

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

      4. associate-/l* 76.4%

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

      5. fma-define 76.4%

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

      6. /-rgt-identity 76.4%

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

      7. remove-double-neg 76.4%

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

      8. fma-neg 76.4%

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

   3. Simplified 76.5%

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

   5. Taylor expanded in F around -inf 92.6%

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

   6. Taylor expanded in B around 0 69.4%

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

      1. *-commutative 69.4%

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

   8. Simplified 69.4%

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

   9. Taylor expanded in B around 0 50.3%

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

       1. mul-1-neg 50.3%

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

       2. distribute-neg-frac2 50.3%

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

       3. +-commutative 50.3%

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

   11. Simplified 50.3%

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

   if -1.29999999999999999e-59 < F < 2.5e-32

   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-define 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-define 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-define 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. Add Preprocessing

   5. Taylor expanded in B around 0 47.1%

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

   6. Taylor expanded in x around inf 34.8%

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

      1. associate-*r/ 34.8%

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

      2. mul-1-neg 34.8%

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

   8. Simplified 34.8%

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

   if 2.5e-32 < 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. fma-define 57.2%

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

      4. +-commutative 57.2%

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

      5. *-commutative 57.2%

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

      6. fma-define 57.2%

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

      7. fma-define 57.2%

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

      8. metadata-eval 57.2%

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

      9. metadata-eval 57.2%

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

      10. associate-*r/ 57.3%

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

      11. *-rgt-identity 57.3%

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

   3. Simplified 57.3%

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

   5. Taylor expanded in B around 0 43.8%

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

   6. Taylor expanded in F around inf 50.4%

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

      1. mul-1-neg 50.4%

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

      2. unsub-neg 50.4%

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

   8. Simplified 50.4%

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

   9. Taylor expanded in x around 0 50.5%

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

       1. +-commutative 50.5%

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

       2. neg-mul-1 50.5%

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

       3. sub-neg 50.5%

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

   11. Simplified 50.5%

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

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.3 \cdot 10^{-59}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.5 \cdot 10^{-32}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 43.9% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -6.2 \cdot 10^{-59}:\\ \;\;\;\;0.3333333333333333 \cdot \left(x \cdot B\right) + \frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.85 \cdot 10^{-32}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -6.2e-59)
   (+ (* 0.3333333333333333 (* x B)) (/ (- -1.0 x) B))
   (if (<= F 1.85e-32) (/ (- x) B) (- (/ 1.0 B) (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -6.2e-59) {
		tmp = (0.3333333333333333 * (x * B)) + ((-1.0 - x) / B);
	} else if (F <= 1.85e-32) {
		tmp = -x / B;
	} else {
		tmp = (1.0 / 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 <= (-6.2d-59)) then
        tmp = (0.3333333333333333d0 * (x * b)) + (((-1.0d0) - x) / b)
    else if (f <= 1.85d-32) then
        tmp = -x / b
    else
        tmp = (1.0d0 / b) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -6.2e-59) {
		tmp = (0.3333333333333333 * (x * B)) + ((-1.0 - x) / B);
	} else if (F <= 1.85e-32) {
		tmp = -x / B;
	} else {
		tmp = (1.0 / B) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -6.2e-59:
		tmp = (0.3333333333333333 * (x * B)) + ((-1.0 - x) / B)
	elif F <= 1.85e-32:
		tmp = -x / B
	else:
		tmp = (1.0 / B) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -6.2e-59)
		tmp = Float64(Float64(0.3333333333333333 * Float64(x * B)) + Float64(Float64(-1.0 - x) / B));
	elseif (F <= 1.85e-32)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -6.2e-59)
		tmp = (0.3333333333333333 * (x * B)) + ((-1.0 - x) / B);
	elseif (F <= 1.85e-32)
		tmp = -x / B;
	else
		tmp = (1.0 / B) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -6.2e-59], N[(N[(0.3333333333333333 * N[(x * B), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.85e-32], N[((-x) / B), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -6.19999999999999998e-59

   1. Initial program 63.1%

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

      1. distribute-lft-neg-in 63.1%

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

      2. +-commutative 63.1%

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

      3. associate-*l/ 76.5%

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

      4. associate-/l* 76.4%

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

      5. fma-define 76.4%

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

      6. /-rgt-identity 76.4%

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

      7. remove-double-neg 76.4%

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

      8. fma-neg 76.4%

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

   3. Simplified 76.5%

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

   5. Taylor expanded in F around -inf 92.6%

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

   6. Taylor expanded in B around 0 69.4%

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

      1. *-commutative 69.4%

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

   8. Simplified 69.4%

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

   9. Taylor expanded in B around 0 50.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B} + 0.3333333333333333 \cdot \left(B \cdot x\right)} \]
   10. Step-by-step derivation

       1. +-commutative 50.9%

          \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(B \cdot x\right) + -1 \cdot \frac{1 + x}{B}} \]

       2. mul-1-neg 50.9%

          \[\leadsto 0.3333333333333333 \cdot \left(B \cdot x\right) + \color{blue}{\left(-\frac{1 + x}{B}\right)} \]

       3. unsub-neg 50.9%

          \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(B \cdot x\right) - \frac{1 + x}{B}} \]

       4. +-commutative 50.9%

          \[\leadsto 0.3333333333333333 \cdot \left(B \cdot x\right) - \frac{\color{blue}{x + 1}}{B} \]

   11. Simplified 50.9%

       \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(B \cdot x\right) - \frac{x + 1}{B}} \]

   if -6.19999999999999998e-59 < F < 1.85e-32

   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-define 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-define 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-define 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. Add Preprocessing

   5. Taylor expanded in B around 0 47.1%

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

   6. Taylor expanded in x around inf 34.8%

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

      1. associate-*r/ 34.8%

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

      2. mul-1-neg 34.8%

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

   8. Simplified 34.8%

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

   if 1.85e-32 < 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. fma-define 57.2%

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

      4. +-commutative 57.2%

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

      5. *-commutative 57.2%

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

      6. fma-define 57.2%

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

      7. fma-define 57.2%

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

      8. metadata-eval 57.2%

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

      9. metadata-eval 57.2%

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

      10. associate-*r/ 57.3%

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

      11. *-rgt-identity 57.3%

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

   3. Simplified 57.3%

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

   5. Taylor expanded in B around 0 43.8%

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

   6. Taylor expanded in F around inf 50.4%

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

      1. mul-1-neg 50.4%

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

      2. unsub-neg 50.4%

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

   8. Simplified 50.4%

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

   9. Taylor expanded in x around 0 50.5%

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

       1. +-commutative 50.5%

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

       2. neg-mul-1 50.5%

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

       3. sub-neg 50.5%

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

   11. Simplified 50.5%

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

4. Final simplification 45.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6.2 \cdot 10^{-59}:\\ \;\;\;\;0.3333333333333333 \cdot \left(x \cdot B\right) + \frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.85 \cdot 10^{-32}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 43.8% accurate, 21.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.9 \cdot 10^{-56}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.1 \cdot 10^{-32}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.9e-56)
   (/ (- -1.0 x) B)
   (if (<= F 2.1e-32) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.9e-56) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 2.1e-32) {
		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 <= (-2.9d-56)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 2.1d-32) 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 <= -2.9e-56) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 2.1e-32) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.9e-56:
		tmp = (-1.0 - x) / B
	elif F <= 2.1e-32:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.9e-56)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 2.1e-32)
		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 <= -2.9e-56)
		tmp = (-1.0 - x) / B;
	elseif (F <= 2.1e-32)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.9e-56], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 2.1e-32], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.89999999999999991e-56

   1. Initial program 63.1%

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

      1. distribute-lft-neg-in 63.1%

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

      2. +-commutative 63.1%

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

      3. associate-*l/ 76.5%

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

      4. associate-/l* 76.4%

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

      5. fma-define 76.4%

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

      6. /-rgt-identity 76.4%

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

      7. remove-double-neg 76.4%

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

      8. fma-neg 76.4%

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

   3. Simplified 76.5%

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

   5. Taylor expanded in F around -inf 92.6%

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

   6. Taylor expanded in B around 0 69.4%

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

      1. *-commutative 69.4%

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

   8. Simplified 69.4%

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

   9. Taylor expanded in B around 0 50.3%

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

       1. mul-1-neg 50.3%

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

       2. distribute-neg-frac2 50.3%

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

       3. +-commutative 50.3%

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

   11. Simplified 50.3%

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

   if -2.89999999999999991e-56 < F < 2.0999999999999999e-32

   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-define 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-define 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-define 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. Add Preprocessing

   5. Taylor expanded in B around 0 47.1%

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

   6. Taylor expanded in x around inf 34.8%

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

      1. associate-*r/ 34.8%

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

      2. mul-1-neg 34.8%

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

   8. Simplified 34.8%

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

   if 2.0999999999999999e-32 < 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. fma-define 57.2%

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

      4. +-commutative 57.2%

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

      5. *-commutative 57.2%

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

      6. fma-define 57.2%

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

      7. fma-define 57.2%

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

      8. metadata-eval 57.2%

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

      9. metadata-eval 57.2%

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

      10. associate-*r/ 57.3%

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

      11. *-rgt-identity 57.3%

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

   3. Simplified 57.3%

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

   5. Taylor expanded in B around 0 43.8%

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

   6. Taylor expanded in F around inf 50.4%

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

      1. mul-1-neg 50.4%

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

      2. unsub-neg 50.4%

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

   8. Simplified 50.4%

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

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.9 \cdot 10^{-56}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.1 \cdot 10^{-32}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 30.0% accurate, 23.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-101} \lor \neg \left(x \leq 6.5 \cdot 10^{-57}\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (or (<= x -4e-101) (not (<= x 6.5e-57))) (/ (- x) B) (/ 1.0 B)))

C

double code(double F, double B, double x) {
	double tmp;
	if ((x <= -4e-101) || !(x <= 6.5e-57)) {
		tmp = -x / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-4d-101)) .or. (.not. (x <= 6.5d-57))) then
        tmp = -x / b
    else
        tmp = 1.0d0 / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if ((x <= -4e-101) || !(x <= 6.5e-57)) {
		tmp = -x / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if (x <= -4e-101) or not (x <= 6.5e-57):
		tmp = -x / B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if ((x <= -4e-101) || !(x <= 6.5e-57))
		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 ((x <= -4e-101) || ~((x <= 6.5e-57)))
		tmp = -x / B;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[Or[LessEqual[x, -4e-101], N[Not[LessEqual[x, 6.5e-57]], $MachinePrecision]], N[((-x) / B), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.00000000000000021e-101 or 6.49999999999999992e-57 < x

   1. Initial program 78.7%

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

      1. distribute-lft-neg-in 78.7%

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

      2. +-commutative 78.7%

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

      3. fma-define 78.7%

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

      4. +-commutative 78.7%

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

      5. *-commutative 78.7%

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

      6. fma-define 78.7%

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

      7. fma-define 78.7%

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

      8. metadata-eval 78.7%

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

      9. metadata-eval 78.7%

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

      10. associate-*r/ 78.9%

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

      11. *-rgt-identity 78.9%

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

   3. Simplified 78.9%

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

   5. Taylor expanded in B around 0 50.1%

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

   6. Taylor expanded in x around inf 45.4%

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

      1. associate-*r/ 45.4%

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

      2. mul-1-neg 45.4%

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

   8. Simplified 45.4%

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

   if -4.00000000000000021e-101 < x < 6.49999999999999992e-57

   1. Initial program 66.4%

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

      1. distribute-lft-neg-in 66.4%

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

      2. +-commutative 66.4%

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

      3. fma-define 66.4%

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

      4. +-commutative 66.4%

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

      5. *-commutative 66.4%

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

      6. fma-define 66.4%

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

      7. fma-define 66.4%

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

      8. metadata-eval 66.4%

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

      9. metadata-eval 66.4%

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

      10. associate-*r/ 66.4%

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

      11. *-rgt-identity 66.4%

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

   3. Simplified 66.4%

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

   5. Taylor expanded in B around 0 35.4%

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

   6. Taylor expanded in F around inf 16.4%

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

      1. mul-1-neg 16.4%

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

      2. unsub-neg 16.4%

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

   8. Simplified 16.4%

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

   9. Taylor expanded in x around 0 16.4%

      \[\leadsto \color{blue}{\frac{1}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-101} \lor \neg \left(x \leq 6.5 \cdot 10^{-57}\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 36.3% accurate, 32.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 2.3 \cdot 10^{-32}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F 2.3e-32) (/ (- x) B) (/ (- 1.0 x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= 2.3e-32) {
		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 <= 2.3d-32) 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 <= 2.3e-32) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= 2.3e-32:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= 2.3e-32)
		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 <= 2.3e-32)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, 2.3e-32], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 2.3000000000000001e-32

   1. Initial program 80.9%

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

      1. distribute-lft-neg-in 80.9%

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

      2. +-commutative 80.9%

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

      3. fma-define 80.9%

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

      4. +-commutative 80.9%

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

      5. *-commutative 80.9%

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

      6. fma-define 80.9%

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

      7. fma-define 80.9%

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

      8. metadata-eval 80.9%

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

      9. metadata-eval 80.9%

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

      10. associate-*r/ 81.0%

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

      11. *-rgt-identity 81.0%

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

   3. Simplified 81.0%

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

   5. Taylor expanded in B around 0 44.2%

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

   6. Taylor expanded in x around inf 32.1%

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

      1. associate-*r/ 32.1%

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

      2. mul-1-neg 32.1%

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

   8. Simplified 32.1%

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

   if 2.3000000000000001e-32 < 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. fma-define 57.2%

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

      4. +-commutative 57.2%

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

      5. *-commutative 57.2%

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

      6. fma-define 57.2%

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

      7. fma-define 57.2%

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

      8. metadata-eval 57.2%

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

      9. metadata-eval 57.2%

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

      10. associate-*r/ 57.3%

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

      11. *-rgt-identity 57.3%

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

   3. Simplified 57.3%

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

   5. Taylor expanded in B around 0 43.8%

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

   6. Taylor expanded in F around inf 50.4%

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

      1. mul-1-neg 50.4%

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

      2. unsub-neg 50.4%

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

   8. Simplified 50.4%

      \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 2.3 \cdot 10^{-32}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 9.9% 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 73.7%

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

   1. distribute-lft-neg-in 73.7%

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

   2. +-commutative 73.7%

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

   3. fma-define 73.7%

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

   4. +-commutative 73.7%

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

   5. *-commutative 73.7%

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

   6. fma-define 73.7%

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

   7. fma-define 73.7%

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

   8. metadata-eval 73.7%

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

   9. metadata-eval 73.7%

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

   10. associate-*r/ 73.8%

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

   11. *-rgt-identity 73.8%

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

3. Simplified 73.8%

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

5. Taylor expanded in B around 0 44.1%

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

6. Taylor expanded in F around inf 31.6%

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

   1. mul-1-neg 31.6%

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

   2. unsub-neg 31.6%

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

8. Simplified 31.6%

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

9. Taylor expanded in x around 0 10.1%

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

10. Final simplification 10.1%

    \[\leadsto \frac{1}{B} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024052 
(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))))))