VandenBroeck and Keller, Equation (23)

Percentage Accurate: 77.4% → 99.7%

Time: 19.4s

Alternatives: 24

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.5 \cdot 10^{+20}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 300000000:\\ \;\;\;\;F \cdot \frac{1}{\frac{\sin B}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.5e+20)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 300000000.0)
       (- (* F (/ 1.0 (/ (sin B) (pow (fma x 2.0 (fma F F 2.0)) -0.5)))) t_0)
       (- (/ 1.0 (sin B)) (/ (* x (cos B)) (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.5e+20) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 300000000.0) {
		tmp = (F * (1.0 / (sin(B) / pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5)))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - ((x * cos(B)) / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.5e+20)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 300000000.0)
		tmp = Float64(Float64(F * Float64(1.0 / Float64(sin(B) / (fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5)))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(Float64(x * cos(B)) / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.5e+20], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 300000000.0], N[(N[(F * N[(1.0 / N[(N[Sin[B], $MachinePrecision] / N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.5e20

   1. Initial program 54.0%

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

   3. Simplified 68.9%

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

   5. Taylor expanded in F around -inf 99.6%

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

   if -1.5e20 < F < 3e8

   1. Initial program 99.5%

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

   3. Simplified 99.7%

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

      1. clear-num 99.7%

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

      2. inv-pow 99.7%

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

      3. fma-undefine 99.7%

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

      4. *-commutative 99.7%

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

      5. fma-define 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. unpow-1 99.7%

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

      2. fma-define 99.7%

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

      3. *-commutative 99.7%

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

      4. fma-define 99.7%

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

   8. Simplified 99.7%

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

   if 3e8 < F

   1. Initial program 64.4%

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

   3. Simplified 77.0%

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

      1. associate-*r/ 77.0%

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

      2. fma-undefine 77.0%

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

      3. +-commutative 77.0%

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

      4. *-commutative 77.0%

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

      5. fma-define 77.0%

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

      6. metadata-eval 77.0%

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

      7. metadata-eval 77.0%

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

      8. fma-define 77.0%

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

      9. +-commutative 77.0%

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

      10. fma-define 77.0%

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

      11. metadata-eval 77.0%

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

      12. metadata-eval 77.0%

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

   6. Applied egg-rr 77.0%

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

   7. Taylor expanded in F around inf 99.7%

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

Alternative 2: 99.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -9 \cdot 10^{+74}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 300000000:\\ \;\;\;\;F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -9e+74)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 300000000.0)
       (- (* F (/ (pow (fma x 2.0 (fma F F 2.0)) -0.5) (sin B))) t_0)
       (- (/ 1.0 (sin B)) (/ (* x (cos B)) (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -9e+74) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 300000000.0) {
		tmp = (F * (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) / sin(B))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - ((x * cos(B)) / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -9e+74)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 300000000.0)
		tmp = Float64(Float64(F * Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) / sin(B))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(Float64(x * cos(B)) / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -9e+74], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 300000000.0], N[(N[(F * N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -8.9999999999999999e74

   1. Initial program 47.2%

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

   3. Simplified 64.2%

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

   5. Taylor expanded in F around -inf 99.6%

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

   if -8.9999999999999999e74 < F < 3e8

   1. Initial program 99.5%

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

   3. Simplified 99.7%

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

   if 3e8 < F

   1. Initial program 64.4%

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

   3. Simplified 77.0%

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

      1. associate-*r/ 77.0%

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

      2. fma-undefine 77.0%

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

      3. +-commutative 77.0%

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

      4. *-commutative 77.0%

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

      5. fma-define 77.0%

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

      6. metadata-eval 77.0%

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

      7. metadata-eval 77.0%

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

      8. fma-define 77.0%

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

      9. +-commutative 77.0%

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

      10. fma-define 77.0%

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

      11. metadata-eval 77.0%

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

      12. metadata-eval 77.0%

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

   6. Applied egg-rr 77.0%

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

   7. Taylor expanded in F around inf 99.7%

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

Alternative 3: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{+17}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 100000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - x \cdot \frac{\cos B}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2e+17)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 100000000.0)
     (-
      (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5))
      (* x (/ (cos B) (sin B))))
     (- (/ 1.0 (sin B)) (/ (* x (cos B)) (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2e+17) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 100000000.0) {
		tmp = ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x * (cos(B) / sin(B)));
	} else {
		tmp = (1.0 / sin(B)) - ((x * cos(B)) / sin(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-2d+17)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 100000000.0d0) then
        tmp = ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0))) - (x * (cos(b) / sin(b)))
    else
        tmp = (1.0d0 / sin(b)) - ((x * cos(b)) / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2e+17) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 100000000.0) {
		tmp = ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x * (Math.cos(B) / Math.sin(B)));
	} else {
		tmp = (1.0 / Math.sin(B)) - ((x * Math.cos(B)) / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2e+17:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 100000000.0:
		tmp = ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x * (math.cos(B) / math.sin(B)))
	else:
		tmp = (1.0 / math.sin(B)) - ((x * math.cos(B)) / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2e+17)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 100000000.0)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)) - Float64(x * Float64(cos(B) / sin(B))));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(Float64(x * cos(B)) / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2e+17)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 100000000.0)
		tmp = ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5)) - (x * (cos(B) / sin(B)));
	else
		tmp = (1.0 / sin(B)) - ((x * cos(B)) / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2e+17], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 100000000.0], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - N[(x * N[(N[Cos[B], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2e17

   1. Initial program 55.3%

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

   3. Simplified 69.7%

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

   5. Taylor expanded in F around -inf 99.6%

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

   if -2e17 < F < 1e8

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 99.5%

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

      1. mul-1-neg 99.5%

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

      2. associate-/l* 99.5%

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

   5. Simplified 99.5%

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

   if 1e8 < F

   1. Initial program 64.9%

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

   3. Simplified 77.3%

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

      1. associate-*r/ 77.3%

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

      2. fma-undefine 77.3%

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

      3. +-commutative 77.3%

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

      4. *-commutative 77.3%

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

      5. fma-define 77.3%

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

      6. metadata-eval 77.3%

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

      7. metadata-eval 77.3%

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

      8. fma-define 77.3%

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

      9. +-commutative 77.3%

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

      10. fma-define 77.3%

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

      11. metadata-eval 77.3%

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

      12. metadata-eval 77.3%

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

   6. Applied egg-rr 77.3%

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

   7. Taylor expanded in F around inf 99.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{+17}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 100000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - x \cdot \frac{\cos B}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.45 \cdot 10^{+17}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 105000000:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{1}{\frac{\sin B}{F}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.45e+17)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 105000000.0)
     (+
      (* x (/ -1.0 (tan B)))
      (* (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5) (/ 1.0 (/ (sin B) F))))
     (- (/ 1.0 (sin B)) (/ (* x (cos B)) (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.45e+17) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 105000000.0) {
		tmp = (x * (-1.0 / tan(B))) + (pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (1.0 / (sin(B) / F)));
	} else {
		tmp = (1.0 / sin(B)) - ((x * cos(B)) / sin(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-2.45d+17)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 105000000.0d0) then
        tmp = (x * ((-1.0d0) / tan(b))) + ((((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)) * (1.0d0 / (sin(b) / f)))
    else
        tmp = (1.0d0 / sin(b)) - ((x * cos(b)) / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.45e+17) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 105000000.0) {
		tmp = (x * (-1.0 / Math.tan(B))) + (Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (1.0 / (Math.sin(B) / F)));
	} else {
		tmp = (1.0 / Math.sin(B)) - ((x * Math.cos(B)) / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.45e+17:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 105000000.0:
		tmp = (x * (-1.0 / math.tan(B))) + (math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (1.0 / (math.sin(B) / F)))
	else:
		tmp = (1.0 / math.sin(B)) - ((x * math.cos(B)) / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.45e+17)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 105000000.0)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64((Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5) * Float64(1.0 / Float64(sin(B) / F))));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(Float64(x * cos(B)) / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.45e+17)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 105000000.0)
		tmp = (x * (-1.0 / tan(B))) + ((((2.0 + (F * F)) + (x * 2.0)) ^ -0.5) * (1.0 / (sin(B) / F)));
	else
		tmp = (1.0 / sin(B)) - ((x * cos(B)) / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.45e+17], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 105000000.0], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(1.0 / N[(N[Sin[B], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.45e17

   1. Initial program 54.7%

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

   3. Simplified 69.3%

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

   5. Taylor expanded in F around -inf 99.6%

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

   if -2.45e17 < F < 1.05e8

   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. clear-num 99.5%

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

      2. inv-pow 99.5%

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

   4. Applied egg-rr 99.5%

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

      1. unpow-1 99.5%

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

   6. Simplified 99.5%

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

   if 1.05e8 < F

   1. Initial program 64.9%

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

   3. Simplified 77.3%

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

      1. associate-*r/ 77.3%

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

      2. fma-undefine 77.3%

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

      3. +-commutative 77.3%

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

      4. *-commutative 77.3%

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

      5. fma-define 77.3%

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

      6. metadata-eval 77.3%

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

      7. metadata-eval 77.3%

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

      8. fma-define 77.3%

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

      9. +-commutative 77.3%

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

      10. fma-define 77.3%

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

      11. metadata-eval 77.3%

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

      12. metadata-eval 77.3%

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

   6. Applied egg-rr 77.3%

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

   7. Taylor expanded in F around inf 99.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.45 \cdot 10^{+17}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 105000000:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{1}{\frac{\sin B}{F}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -6.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 125000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} + x \cdot \frac{-1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -6.2e+15)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 125000000.0)
     (+
      (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5))
      (* x (/ -1.0 (tan B))))
     (- (/ 1.0 (sin B)) (/ (* x (cos B)) (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -6.2e+15) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 125000000.0) {
		tmp = ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) + (x * (-1.0 / tan(B)));
	} else {
		tmp = (1.0 / sin(B)) - ((x * cos(B)) / sin(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-6.2d+15)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 125000000.0d0) then
        tmp = ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0))) + (x * ((-1.0d0) / tan(b)))
    else
        tmp = (1.0d0 / sin(b)) - ((x * cos(b)) / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -6.2e+15) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 125000000.0) {
		tmp = ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) + (x * (-1.0 / Math.tan(B)));
	} else {
		tmp = (1.0 / Math.sin(B)) - ((x * Math.cos(B)) / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -6.2e+15:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 125000000.0:
		tmp = ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) + (x * (-1.0 / math.tan(B)))
	else:
		tmp = (1.0 / math.sin(B)) - ((x * math.cos(B)) / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -6.2e+15)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 125000000.0)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)) + Float64(x * Float64(-1.0 / tan(B))));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(Float64(x * cos(B)) / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -6.2e+15)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 125000000.0)
		tmp = ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5)) + (x * (-1.0 / tan(B)));
	else
		tmp = (1.0 / sin(B)) - ((x * cos(B)) / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -6.2e+15], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 125000000.0], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] + N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -6.2e15

   1. Initial program 55.3%

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

   3. Simplified 69.7%

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

   5. Taylor expanded in F around -inf 99.6%

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

   if -6.2e15 < F < 1.25e8

   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 1.25e8 < F

   1. Initial program 64.9%

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

   3. Simplified 77.3%

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

      1. associate-*r/ 77.3%

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

      2. fma-undefine 77.3%

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

      3. +-commutative 77.3%

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

      4. *-commutative 77.3%

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

      5. fma-define 77.3%

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

      6. metadata-eval 77.3%

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

      7. metadata-eval 77.3%

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

      8. fma-define 77.3%

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

      9. +-commutative 77.3%

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

      10. fma-define 77.3%

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

      11. metadata-eval 77.3%

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

      12. metadata-eval 77.3%

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

   6. Applied egg-rr 77.3%

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

   7. Taylor expanded in F around inf 99.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 125000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} + x \cdot \frac{-1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{+17}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 108000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} + \frac{-1}{\frac{\tan B}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2e+17)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 108000000.0)
     (+
      (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5))
      (/ -1.0 (/ (tan B) x)))
     (- (/ 1.0 (sin B)) (/ (* x (cos B)) (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2e+17) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 108000000.0) {
		tmp = ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) + (-1.0 / (tan(B) / x));
	} else {
		tmp = (1.0 / sin(B)) - ((x * cos(B)) / sin(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-2d+17)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 108000000.0d0) then
        tmp = ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0))) + ((-1.0d0) / (tan(b) / x))
    else
        tmp = (1.0d0 / sin(b)) - ((x * cos(b)) / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2e+17) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 108000000.0) {
		tmp = ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) + (-1.0 / (Math.tan(B) / x));
	} else {
		tmp = (1.0 / Math.sin(B)) - ((x * Math.cos(B)) / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2e+17:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 108000000.0:
		tmp = ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) + (-1.0 / (math.tan(B) / x))
	else:
		tmp = (1.0 / math.sin(B)) - ((x * math.cos(B)) / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2e+17)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 108000000.0)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)) + Float64(-1.0 / Float64(tan(B) / x)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(Float64(x * cos(B)) / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2e+17)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 108000000.0)
		tmp = ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5)) + (-1.0 / (tan(B) / x));
	else
		tmp = (1.0 / sin(B)) - ((x * cos(B)) / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2e+17], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 108000000.0], 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[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2e17

   1. Initial program 55.3%

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

   3. Simplified 69.7%

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

   5. Taylor expanded in F around -inf 99.6%

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

   if -2e17 < F < 1.08e8

   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. neg-mul-1 99.5%

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

      2. div-inv 99.6%

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

      3. clear-num 99.4%

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

      4. un-div-inv 99.4%

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

   4. Applied egg-rr 99.4%

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

   if 1.08e8 < F

   1. Initial program 64.9%

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

   3. Simplified 77.3%

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

      1. associate-*r/ 77.3%

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

      2. fma-undefine 77.3%

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

      3. +-commutative 77.3%

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

      4. *-commutative 77.3%

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

      5. fma-define 77.3%

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

      6. metadata-eval 77.3%

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

      7. metadata-eval 77.3%

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

      8. fma-define 77.3%

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

      9. +-commutative 77.3%

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

      10. fma-define 77.3%

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

      11. metadata-eval 77.3%

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

      12. metadata-eval 77.3%

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

   6. Applied egg-rr 77.3%

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

   7. Taylor expanded in F around inf 99.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{+17}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 108000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} + \frac{-1}{\frac{\tan B}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -310000:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1.45:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{2 + x \cdot 2}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -310000.0)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.45)
       (- (/ F (* (sin B) (sqrt (+ 2.0 (* x 2.0))))) t_0)
       (- (/ 1.0 (sin B)) (/ (* x (cos B)) (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -310000.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.45) {
		tmp = (F / (sin(B) * sqrt((2.0 + (x * 2.0))))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - ((x * cos(B)) / sin(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-310000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.45d0) then
        tmp = (f / (sin(b) * sqrt((2.0d0 + (x * 2.0d0))))) - t_0
    else
        tmp = (1.0d0 / sin(b)) - ((x * cos(b)) / sin(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 <= -310000.0) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.45) {
		tmp = (F / (Math.sin(B) * Math.sqrt((2.0 + (x * 2.0))))) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - ((x * Math.cos(B)) / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -310000.0:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.45:
		tmp = (F / (math.sin(B) * math.sqrt((2.0 + (x * 2.0))))) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - ((x * math.cos(B)) / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -310000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.45)
		tmp = Float64(Float64(F / Float64(sin(B) * sqrt(Float64(2.0 + Float64(x * 2.0))))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(Float64(x * cos(B)) / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -310000.0)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.45)
		tmp = (F / (sin(B) * sqrt((2.0 + (x * 2.0))))) - t_0;
	else
		tmp = (1.0 / sin(B)) - ((x * cos(B)) / sin(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, -310000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.45], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.1e5

   1. Initial program 55.9%

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

   3. Simplified 70.2%

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

   5. Taylor expanded in F around -inf 99.7%

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

   if -3.1e5 < F < 1.44999999999999996

   1. Initial program 99.5%

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

   3. Simplified 99.7%

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

      1. clear-num 99.7%

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

      2. un-div-inv 99.6%

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

      3. fma-undefine 99.6%

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

      4. *-commutative 99.6%

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

      5. fma-define 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in F around 0 99.3%

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

   if 1.44999999999999996 < F

   1. Initial program 65.4%

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

   3. Simplified 77.6%

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

      1. associate-*r/ 77.6%

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

      2. fma-undefine 77.6%

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

      3. +-commutative 77.6%

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

      4. *-commutative 77.6%

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

      5. fma-define 77.5%

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

      6. metadata-eval 77.5%

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

      7. metadata-eval 77.5%

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

      8. fma-define 77.6%

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

      9. +-commutative 77.6%

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

      10. fma-define 77.6%

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

      11. metadata-eval 77.6%

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

      12. metadata-eval 77.6%

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

   6. Applied egg-rr 77.6%

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

   7. Taylor expanded in F around inf 99.5%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -310000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.45:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{2 + x \cdot 2}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -6.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1850000:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -6.2e+15)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1850000.0)
       (- (* (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5) (/ F B)) t_0)
       (- (/ 1.0 (sin B)) (/ (* x (cos B)) (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -6.2e+15) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1850000.0) {
		tmp = (pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - ((x * cos(B)) / sin(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-6.2d+15)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1850000.0d0) then
        tmp = ((((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)) * (f / b)) - t_0
    else
        tmp = (1.0d0 / sin(b)) - ((x * cos(b)) / sin(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 <= -6.2e+15) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1850000.0) {
		tmp = (Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - ((x * Math.cos(B)) / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -6.2e+15:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1850000.0:
		tmp = (math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - ((x * math.cos(B)) / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -6.2e+15)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1850000.0)
		tmp = Float64(Float64((Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5) * Float64(F / B)) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(Float64(x * cos(B)) / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -6.2e+15)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1850000.0)
		tmp = ((((2.0 + (F * F)) + (x * 2.0)) ^ -0.5) * (F / B)) - t_0;
	else
		tmp = (1.0 / sin(B)) - ((x * cos(B)) / sin(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, -6.2e+15], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1850000.0], N[(N[(N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -6.2e15

   1. Initial program 55.3%

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

   3. Simplified 69.7%

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

   5. Taylor expanded in F around -inf 99.6%

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

   if -6.2e15 < F < 1.85e6

   1. Initial program 99.5%

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

   3. Taylor expanded in B around 0 87.4%

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

      1. neg-sub0 87.4%

         \[\leadsto \color{blue}{\left(0 - x \cdot \frac{1}{\tan B}\right)} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. div-inv 87.5%

         \[\leadsto \left(0 - \color{blue}{\frac{x}{\tan B}}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   5. Applied egg-rr 87.5%

      \[\leadsto \color{blue}{\left(0 - \frac{x}{\tan B}\right)} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   6. Step-by-step derivation

      1. neg-sub0 87.5%

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

      2. distribute-neg-frac 87.5%

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

   7. Simplified 87.5%

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

   if 1.85e6 < F

   1. Initial program 64.9%

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

   3. Simplified 77.3%

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

      1. associate-*r/ 77.3%

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

      2. fma-undefine 77.3%

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

      3. +-commutative 77.3%

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

      4. *-commutative 77.3%

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

      5. fma-define 77.3%

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

      6. metadata-eval 77.3%

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

      7. metadata-eval 77.3%

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

      8. fma-define 77.3%

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

      9. +-commutative 77.3%

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

      10. fma-define 77.3%

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

      11. metadata-eval 77.3%

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

      12. metadata-eval 77.3%

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

   6. Applied egg-rr 77.3%

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

   7. Taylor expanded in F around inf 99.7%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1850000:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -6.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1850000:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -6.2e+15)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1850000.0)
       (- (* (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5) (/ F B)) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -6.2e+15) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1850000.0) {
		tmp = (pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-6.2d+15)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1850000.0d0) then
        tmp = ((((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)) * (f / b)) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -6.2e+15) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1850000.0) {
		tmp = (Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -6.2e+15:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1850000.0:
		tmp = (math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -6.2e+15)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1850000.0)
		tmp = Float64(Float64((Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5) * Float64(F / B)) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -6.2e+15)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1850000.0)
		tmp = ((((2.0 + (F * F)) + (x * 2.0)) ^ -0.5) * (F / B)) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -6.2e+15], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1850000.0], N[(N[(N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -6.2e15

   1. Initial program 55.3%

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

   3. Simplified 69.7%

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

   5. Taylor expanded in F around -inf 99.6%

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

   if -6.2e15 < F < 1.85e6

   1. Initial program 99.5%

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

   3. Taylor expanded in B around 0 87.4%

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

      1. neg-sub0 87.4%

         \[\leadsto \color{blue}{\left(0 - x \cdot \frac{1}{\tan B}\right)} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. div-inv 87.5%

         \[\leadsto \left(0 - \color{blue}{\frac{x}{\tan B}}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   5. Applied egg-rr 87.5%

      \[\leadsto \color{blue}{\left(0 - \frac{x}{\tan B}\right)} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   6. Step-by-step derivation

      1. neg-sub0 87.5%

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

      2. distribute-neg-frac 87.5%

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

   7. Simplified 87.5%

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

   if 1.85e6 < F

   1. Initial program 64.9%

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

   3. Simplified 77.3%

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

   5. Taylor expanded in F around inf 99.6%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1850000:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 85.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.2 \cdot 10^{-16}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq -3.35 \cdot 10^{-77}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 1.55 \cdot 10^{-118}:\\ \;\;\;\;x \cdot \frac{\cos B}{-\sin B}\\ \mathbf{elif}\;F \leq 1850000:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.2e-16)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F -3.35e-77)
       (/ (* F (sqrt 0.5)) (sin B))
       (if (<= F 1.55e-118)
         (* x (/ (cos B) (- (sin B))))
         (if (<= F 1850000.0)
           (- (* (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5) (/ F B)) (/ x B))
           (- (/ 1.0 (sin B)) t_0)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.2e-16) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= -3.35e-77) {
		tmp = (F * sqrt(0.5)) / sin(B);
	} else if (F <= 1.55e-118) {
		tmp = x * (cos(B) / -sin(B));
	} else if (F <= 1850000.0) {
		tmp = (pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.2d-16)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= (-3.35d-77)) then
        tmp = (f * sqrt(0.5d0)) / sin(b)
    else if (f <= 1.55d-118) then
        tmp = x * (cos(b) / -sin(b))
    else if (f <= 1850000.0d0) then
        tmp = ((((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)) * (f / b)) - (x / b)
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.2e-16) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= -3.35e-77) {
		tmp = (F * Math.sqrt(0.5)) / Math.sin(B);
	} else if (F <= 1.55e-118) {
		tmp = x * (Math.cos(B) / -Math.sin(B));
	} else if (F <= 1850000.0) {
		tmp = (Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.2e-16:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= -3.35e-77:
		tmp = (F * math.sqrt(0.5)) / math.sin(B)
	elif F <= 1.55e-118:
		tmp = x * (math.cos(B) / -math.sin(B))
	elif F <= 1850000.0:
		tmp = (math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B)
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.2e-16)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= -3.35e-77)
		tmp = Float64(Float64(F * sqrt(0.5)) / sin(B));
	elseif (F <= 1.55e-118)
		tmp = Float64(x * Float64(cos(B) / Float64(-sin(B))));
	elseif (F <= 1850000.0)
		tmp = Float64(Float64((Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5) * Float64(F / B)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.2e-16)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= -3.35e-77)
		tmp = (F * sqrt(0.5)) / sin(B);
	elseif (F <= 1.55e-118)
		tmp = x * (cos(B) / -sin(B));
	elseif (F <= 1850000.0)
		tmp = ((((2.0 + (F * F)) + (x * 2.0)) ^ -0.5) * (F / B)) - (x / B);
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.2e-16], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -3.35e-77], N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.55e-118], N[(x * N[(N[Cos[B], $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1850000.0], N[(N[(N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -1.20000000000000002e-16

   1. Initial program 58.2%

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

   3. Simplified 71.7%

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

   5. Taylor expanded in F around -inf 98.5%

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

   if -1.20000000000000002e-16 < F < -3.3499999999999999e-77

   1. Initial program 99.1%

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

   3. Simplified 99.1%

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

      1. associate-*r/ 99.5%

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

      2. fma-undefine 99.5%

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

      3. +-commutative 99.5%

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

      4. *-commutative 99.5%

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

      5. fma-define 99.5%

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

      6. metadata-eval 99.5%

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

      7. metadata-eval 99.5%

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

      8. fma-define 99.5%

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

      9. +-commutative 99.5%

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

      10. fma-define 99.5%

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

      11. metadata-eval 99.5%

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

      12. metadata-eval 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in x around 0 77.5%

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

      1. associate-*l/ 77.9%

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

      2. *-lft-identity 77.9%

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

      3. times-frac 77.5%

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

      4. /-rgt-identity 77.5%

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

      5. +-commutative 77.5%

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

      6. unpow2 77.5%

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

      7. fma-undefine 77.5%

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

   9. Simplified 77.5%

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

   10. Taylor expanded in F around 0 77.9%

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

   if -3.3499999999999999e-77 < F < 1.5500000000000001e-118

   1. Initial program 99.5%

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

   3. Simplified 99.7%

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

      1. associate-*r/ 99.6%

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

      2. fma-undefine 99.6%

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

      3. +-commutative 99.6%

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

      4. *-commutative 99.6%

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

      5. fma-define 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. fma-define 99.6%

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

      9. +-commutative 99.6%

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

      10. fma-define 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in F around 0 83.0%

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

      1. mul-1-neg 83.0%

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

      2. associate-/l* 83.2%

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

      3. distribute-lft-neg-in 83.2%

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

   9. Simplified 83.2%

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

   if 1.5500000000000001e-118 < F < 1.85e6

   1. Initial program 99.5%

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

   3. Taylor expanded in B around 0 86.4%

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

   4. Taylor expanded in B around 0 77.7%

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

      1. associate-*r/ 77.7%

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

      2. mul-1-neg 77.7%

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

   6. Simplified 77.7%

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

   if 1.85e6 < F

   1. Initial program 64.9%

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

   3. Simplified 77.3%

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

   5. Taylor expanded in F around inf 99.6%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.2 \cdot 10^{-16}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -3.35 \cdot 10^{-77}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 1.55 \cdot 10^{-118}:\\ \;\;\;\;x \cdot \frac{\cos B}{-\sin B}\\ \mathbf{elif}\;F \leq 1850000:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 70.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -3 \cdot 10^{+138}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -3.1 \cdot 10^{-19}:\\ \;\;\;\;\frac{-1}{B} - t\_0\\ \mathbf{elif}\;F \leq -5.2 \cdot 10^{-76}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 9.6 \cdot 10^{-119}:\\ \;\;\;\;x \cdot \frac{\cos B}{-\sin B}\\ \mathbf{elif}\;F \leq 1850000:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -3e+138)
     (- (/ -1.0 (sin B)) (/ x B))
     (if (<= F -3.1e-19)
       (- (/ -1.0 B) t_0)
       (if (<= F -5.2e-76)
         (/ (* F (sqrt 0.5)) (sin B))
         (if (<= F 9.6e-119)
           (* x (/ (cos B) (- (sin B))))
           (if (<= F 1850000.0)
             (- (* (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5) (/ F B)) (/ x B))
             (- (/ 1.0 B) t_0))))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -3e+138) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -3.1e-19) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= -5.2e-76) {
		tmp = (F * sqrt(0.5)) / sin(B);
	} else if (F <= 9.6e-119) {
		tmp = x * (cos(B) / -sin(B));
	} else if (F <= 1850000.0) {
		tmp = (pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else {
		tmp = (1.0 / B) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-3d+138)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-3.1d-19)) then
        tmp = ((-1.0d0) / b) - t_0
    else if (f <= (-5.2d-76)) then
        tmp = (f * sqrt(0.5d0)) / sin(b)
    else if (f <= 9.6d-119) then
        tmp = x * (cos(b) / -sin(b))
    else if (f <= 1850000.0d0) then
        tmp = ((((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)) * (f / b)) - (x / b)
    else
        tmp = (1.0d0 / b) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -3e+138) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -3.1e-19) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= -5.2e-76) {
		tmp = (F * Math.sqrt(0.5)) / Math.sin(B);
	} else if (F <= 9.6e-119) {
		tmp = x * (Math.cos(B) / -Math.sin(B));
	} else if (F <= 1850000.0) {
		tmp = (Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else {
		tmp = (1.0 / B) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -3e+138:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -3.1e-19:
		tmp = (-1.0 / B) - t_0
	elif F <= -5.2e-76:
		tmp = (F * math.sqrt(0.5)) / math.sin(B)
	elif F <= 9.6e-119:
		tmp = x * (math.cos(B) / -math.sin(B))
	elif F <= 1850000.0:
		tmp = (math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B)
	else:
		tmp = (1.0 / B) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -3e+138)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -3.1e-19)
		tmp = Float64(Float64(-1.0 / B) - t_0);
	elseif (F <= -5.2e-76)
		tmp = Float64(Float64(F * sqrt(0.5)) / sin(B));
	elseif (F <= 9.6e-119)
		tmp = Float64(x * Float64(cos(B) / Float64(-sin(B))));
	elseif (F <= 1850000.0)
		tmp = Float64(Float64((Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5) * Float64(F / B)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / B) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -3e+138)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -3.1e-19)
		tmp = (-1.0 / B) - t_0;
	elseif (F <= -5.2e-76)
		tmp = (F * sqrt(0.5)) / sin(B);
	elseif (F <= 9.6e-119)
		tmp = x * (cos(B) / -sin(B));
	elseif (F <= 1850000.0)
		tmp = ((((2.0 + (F * F)) + (x * 2.0)) ^ -0.5) * (F / B)) - (x / B);
	else
		tmp = (1.0 / B) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -3e+138], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -3.1e-19], N[(N[(-1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -5.2e-76], N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 9.6e-119], N[(x * N[(N[Cos[B], $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1850000.0], N[(N[(N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if F < -3.0000000000000001e138

   1. Initial program 37.4%

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

   3. Simplified 53.6%

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

   5. Taylor expanded in F around -inf 99.6%

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

   6. Taylor expanded in B around 0 78.7%

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

   if -3.0000000000000001e138 < F < -3.0999999999999999e-19

   1. Initial program 90.1%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in F around -inf 96.7%

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

   6. Taylor expanded in B around 0 82.0%

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

   if -3.0999999999999999e-19 < F < -5.1999999999999999e-76

   1. Initial program 99.1%

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

   3. Simplified 99.1%

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

      1. associate-*r/ 99.5%

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

      2. fma-undefine 99.5%

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

      3. +-commutative 99.5%

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

      4. *-commutative 99.5%

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

      5. fma-define 99.5%

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

      6. metadata-eval 99.5%

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

      7. metadata-eval 99.5%

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

      8. fma-define 99.5%

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

      9. +-commutative 99.5%

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

      10. fma-define 99.5%

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

      11. metadata-eval 99.5%

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

      12. metadata-eval 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in x around 0 77.5%

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

      1. associate-*l/ 77.9%

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

      2. *-lft-identity 77.9%

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

      3. times-frac 77.5%

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

      4. /-rgt-identity 77.5%

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

      5. +-commutative 77.5%

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

      6. unpow2 77.5%

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

      7. fma-undefine 77.5%

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

   9. Simplified 77.5%

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

   10. Taylor expanded in F around 0 77.9%

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

   if -5.1999999999999999e-76 < F < 9.60000000000000034e-119

   1. Initial program 99.5%

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

   3. Simplified 99.7%

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

      1. associate-*r/ 99.6%

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

      2. fma-undefine 99.6%

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

      3. +-commutative 99.6%

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

      4. *-commutative 99.6%

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

      5. fma-define 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. fma-define 99.6%

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

      9. +-commutative 99.6%

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

      10. fma-define 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in F around 0 83.0%

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

      1. mul-1-neg 83.0%

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

      2. associate-/l* 83.2%

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

      3. distribute-lft-neg-in 83.2%

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

   9. Simplified 83.2%

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

   if 9.60000000000000034e-119 < F < 1.85e6

   1. Initial program 99.5%

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

   3. Taylor expanded in B around 0 86.4%

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

   4. Taylor expanded in B around 0 77.7%

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

      1. associate-*r/ 77.7%

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

      2. mul-1-neg 77.7%

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

   6. Simplified 77.7%

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

   if 1.85e6 < F

   1. Initial program 64.9%

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

   3. Taylor expanded in B around 0 42.8%

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

      1. neg-sub0 42.8%

         \[\leadsto \color{blue}{\left(0 - x \cdot \frac{1}{\tan B}\right)} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. div-inv 42.8%

         \[\leadsto \left(0 - \color{blue}{\frac{x}{\tan B}}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   5. Applied egg-rr 42.8%

      \[\leadsto \color{blue}{\left(0 - \frac{x}{\tan B}\right)} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   6. Step-by-step derivation

      1. neg-sub0 42.8%

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

      2. distribute-neg-frac 42.8%

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

   7. Simplified 42.8%

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

   8. Taylor expanded in F around inf 64.8%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3 \cdot 10^{+138}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -3.1 \cdot 10^{-19}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -5.2 \cdot 10^{-76}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 9.6 \cdot 10^{-119}:\\ \;\;\;\;x \cdot \frac{\cos B}{-\sin B}\\ \mathbf{elif}\;F \leq 1850000:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 91.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -310000:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 0.027:\\ \;\;\;\;\frac{F}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -310000.0)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 0.027)
       (- (* (/ F B) (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -310000.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 0.027) {
		tmp = ((F / B) * sqrt((1.0 / (2.0 + (x * 2.0))))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-310000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 0.027d0) then
        tmp = ((f / b) * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -310000.0) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 0.027) {
		tmp = ((F / B) * Math.sqrt((1.0 / (2.0 + (x * 2.0))))) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -310000.0:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 0.027:
		tmp = ((F / B) * math.sqrt((1.0 / (2.0 + (x * 2.0))))) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -310000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 0.027)
		tmp = Float64(Float64(Float64(F / B) * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -310000.0)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 0.027)
		tmp = ((F / B) * sqrt((1.0 / (2.0 + (x * 2.0))))) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -310000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.027], N[(N[(N[(F / B), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.1e5

   1. Initial program 55.9%

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

   3. Simplified 70.2%

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

   5. Taylor expanded in F around -inf 99.7%

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

   if -3.1e5 < F < 0.0269999999999999997

   1. Initial program 99.5%

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

   3. Taylor expanded in B around 0 87.2%

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

      1. neg-sub0 87.2%

         \[\leadsto \color{blue}{\left(0 - x \cdot \frac{1}{\tan B}\right)} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. div-inv 87.3%

         \[\leadsto \left(0 - \color{blue}{\frac{x}{\tan B}}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   5. Applied egg-rr 87.3%

      \[\leadsto \color{blue}{\left(0 - \frac{x}{\tan B}\right)} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   6. Step-by-step derivation

      1. neg-sub0 87.3%

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

      2. distribute-neg-frac 87.3%

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

   7. Simplified 87.3%

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

   8. Taylor expanded in F around 0 87.0%

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

   if 0.0269999999999999997 < F

   1. Initial program 65.4%

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

   3. Simplified 77.6%

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

   5. Taylor expanded in F around inf 99.4%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -310000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.027:\\ \;\;\;\;\frac{F}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 77.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.5 \cdot 10^{-16}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq -2.35 \cdot 10^{-76}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{-122}:\\ \;\;\;\;x \cdot \frac{\cos B}{-\sin B}\\ \mathbf{elif}\;F \leq 1850000:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.5e-16)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F -2.35e-76)
       (/ (* F (sqrt 0.5)) (sin B))
       (if (<= F 5e-122)
         (* x (/ (cos B) (- (sin B))))
         (if (<= F 1850000.0)
           (- (* (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5) (/ F B)) (/ x B))
           (- (/ 1.0 B) t_0)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.5e-16) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= -2.35e-76) {
		tmp = (F * sqrt(0.5)) / sin(B);
	} else if (F <= 5e-122) {
		tmp = x * (cos(B) / -sin(B));
	} else if (F <= 1850000.0) {
		tmp = (pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else {
		tmp = (1.0 / B) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.5d-16)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= (-2.35d-76)) then
        tmp = (f * sqrt(0.5d0)) / sin(b)
    else if (f <= 5d-122) then
        tmp = x * (cos(b) / -sin(b))
    else if (f <= 1850000.0d0) then
        tmp = ((((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)) * (f / b)) - (x / b)
    else
        tmp = (1.0d0 / b) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.5e-16) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= -2.35e-76) {
		tmp = (F * Math.sqrt(0.5)) / Math.sin(B);
	} else if (F <= 5e-122) {
		tmp = x * (Math.cos(B) / -Math.sin(B));
	} else if (F <= 1850000.0) {
		tmp = (Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else {
		tmp = (1.0 / B) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.5e-16:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= -2.35e-76:
		tmp = (F * math.sqrt(0.5)) / math.sin(B)
	elif F <= 5e-122:
		tmp = x * (math.cos(B) / -math.sin(B))
	elif F <= 1850000.0:
		tmp = (math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B)
	else:
		tmp = (1.0 / B) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.5e-16)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= -2.35e-76)
		tmp = Float64(Float64(F * sqrt(0.5)) / sin(B));
	elseif (F <= 5e-122)
		tmp = Float64(x * Float64(cos(B) / Float64(-sin(B))));
	elseif (F <= 1850000.0)
		tmp = Float64(Float64((Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5) * Float64(F / B)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / B) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.5e-16)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= -2.35e-76)
		tmp = (F * sqrt(0.5)) / sin(B);
	elseif (F <= 5e-122)
		tmp = x * (cos(B) / -sin(B));
	elseif (F <= 1850000.0)
		tmp = ((((2.0 + (F * F)) + (x * 2.0)) ^ -0.5) * (F / B)) - (x / B);
	else
		tmp = (1.0 / B) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.5e-16], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -2.35e-76], N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5e-122], N[(x * N[(N[Cos[B], $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1850000.0], N[(N[(N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -1.49999999999999997e-16

   1. Initial program 58.2%

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

   3. Simplified 71.7%

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

   5. Taylor expanded in F around -inf 98.5%

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

   if -1.49999999999999997e-16 < F < -2.3500000000000001e-76

   1. Initial program 99.1%

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

   3. Simplified 99.1%

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

      1. associate-*r/ 99.5%

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

      2. fma-undefine 99.5%

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

      3. +-commutative 99.5%

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

      4. *-commutative 99.5%

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

      5. fma-define 99.5%

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

      6. metadata-eval 99.5%

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

      7. metadata-eval 99.5%

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

      8. fma-define 99.5%

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

      9. +-commutative 99.5%

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

      10. fma-define 99.5%

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

      11. metadata-eval 99.5%

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

      12. metadata-eval 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in x around 0 77.5%

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

      1. associate-*l/ 77.9%

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

      2. *-lft-identity 77.9%

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

      3. times-frac 77.5%

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

      4. /-rgt-identity 77.5%

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

      5. +-commutative 77.5%

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

      6. unpow2 77.5%

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

      7. fma-undefine 77.5%

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

   9. Simplified 77.5%

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

   10. Taylor expanded in F around 0 77.9%

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

   if -2.3500000000000001e-76 < F < 4.9999999999999999e-122

   1. Initial program 99.5%

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

   3. Simplified 99.7%

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

      1. associate-*r/ 99.6%

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

      2. fma-undefine 99.6%

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

      3. +-commutative 99.6%

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

      4. *-commutative 99.6%

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

      5. fma-define 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. fma-define 99.6%

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

      9. +-commutative 99.6%

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

      10. fma-define 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in F around 0 83.0%

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

      1. mul-1-neg 83.0%

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

      2. associate-/l* 83.2%

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

      3. distribute-lft-neg-in 83.2%

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

   9. Simplified 83.2%

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

   if 4.9999999999999999e-122 < F < 1.85e6

   1. Initial program 99.5%

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

   3. Taylor expanded in B around 0 86.4%

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

   4. Taylor expanded in B around 0 77.7%

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

      1. associate-*r/ 77.7%

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

      2. mul-1-neg 77.7%

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

   6. Simplified 77.7%

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

   if 1.85e6 < F

   1. Initial program 64.9%

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

   3. Taylor expanded in B around 0 42.8%

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

      1. neg-sub0 42.8%

         \[\leadsto \color{blue}{\left(0 - x \cdot \frac{1}{\tan B}\right)} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. div-inv 42.8%

         \[\leadsto \left(0 - \color{blue}{\frac{x}{\tan B}}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   5. Applied egg-rr 42.8%

      \[\leadsto \color{blue}{\left(0 - \frac{x}{\tan B}\right)} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   6. Step-by-step derivation

      1. neg-sub0 42.8%

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

      2. distribute-neg-frac 42.8%

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

   7. Simplified 42.8%

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

   8. Taylor expanded in F around inf 64.8%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.5 \cdot 10^{-16}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -2.35 \cdot 10^{-76}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{-122}:\\ \;\;\;\;x \cdot \frac{\cos B}{-\sin B}\\ \mathbf{elif}\;F \leq 1850000:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -4.1 \cdot 10^{+138}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3 \cdot 10^{-178}:\\ \;\;\;\;\frac{-1}{B} - t\_0\\ \mathbf{elif}\;F \leq 2500000:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -4.1e+138)
     (- (/ -1.0 (sin B)) (/ x B))
     (if (<= F 3e-178)
       (- (/ -1.0 B) t_0)
       (if (<= F 2500000.0)
         (- (* (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5) (/ F B)) (/ x B))
         (- (/ 1.0 B) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -4.1e+138) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 3e-178) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= 2500000.0) {
		tmp = (pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else {
		tmp = (1.0 / B) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-4.1d+138)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 3d-178) then
        tmp = ((-1.0d0) / b) - t_0
    else if (f <= 2500000.0d0) then
        tmp = ((((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)) * (f / b)) - (x / b)
    else
        tmp = (1.0d0 / b) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -4.1e+138) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 3e-178) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= 2500000.0) {
		tmp = (Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else {
		tmp = (1.0 / B) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -4.1e+138:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 3e-178:
		tmp = (-1.0 / B) - t_0
	elif F <= 2500000.0:
		tmp = (math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B)
	else:
		tmp = (1.0 / B) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -4.1e+138)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 3e-178)
		tmp = Float64(Float64(-1.0 / B) - t_0);
	elseif (F <= 2500000.0)
		tmp = Float64(Float64((Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5) * Float64(F / B)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / B) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -4.1e+138)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 3e-178)
		tmp = (-1.0 / B) - t_0;
	elseif (F <= 2500000.0)
		tmp = ((((2.0 + (F * F)) + (x * 2.0)) ^ -0.5) * (F / B)) - (x / B);
	else
		tmp = (1.0 / B) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -4.1e+138], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3e-178], N[(N[(-1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 2500000.0], N[(N[(N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -4.0999999999999998e138

   1. Initial program 37.4%

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

   3. Simplified 53.6%

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

   5. Taylor expanded in F around -inf 99.6%

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

   6. Taylor expanded in B around 0 78.7%

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

   if -4.0999999999999998e138 < F < 2.9999999999999999e-178

   1. Initial program 96.8%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in F around -inf 57.2%

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

   6. Taylor expanded in B around 0 67.5%

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

   if 2.9999999999999999e-178 < F < 2.5e6

   1. Initial program 99.6%

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

   3. Taylor expanded in B around 0 84.7%

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

   4. Taylor expanded in B around 0 69.5%

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

      1. associate-*r/ 69.5%

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

      2. mul-1-neg 69.5%

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

   6. Simplified 69.5%

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

   if 2.5e6 < F

   1. Initial program 64.9%

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

   3. Taylor expanded in B around 0 42.8%

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

      1. neg-sub0 42.8%

         \[\leadsto \color{blue}{\left(0 - x \cdot \frac{1}{\tan B}\right)} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. div-inv 42.8%

         \[\leadsto \left(0 - \color{blue}{\frac{x}{\tan B}}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   5. Applied egg-rr 42.8%

      \[\leadsto \color{blue}{\left(0 - \frac{x}{\tan B}\right)} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   6. Step-by-step derivation

      1. neg-sub0 42.8%

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

      2. distribute-neg-frac 42.8%

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

   7. Simplified 42.8%

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

   8. Taylor expanded in F around inf 64.8%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.1 \cdot 10^{+138}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3 \cdot 10^{-178}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2500000:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 54.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -4.1 \cdot 10^{+138}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 210000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 2.1 \cdot 10^{+105}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{+254}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{F \cdot \sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ -1.0 B) (/ x (tan B)))))
   (if (<= F -4.1e+138)
     (- (/ -1.0 (sin B)) (/ x B))
     (if (<= F 210000.0)
       t_0
       (if (<= F 2.1e+105)
         (/ 1.0 (sin B))
         (if (<= F 7.5e+254) t_0 (* F (/ 1.0 (* F (sin B))))))))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 / B) - (x / tan(B));
	double tmp;
	if (F <= -4.1e+138) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 210000.0) {
		tmp = t_0;
	} else if (F <= 2.1e+105) {
		tmp = 1.0 / sin(B);
	} else if (F <= 7.5e+254) {
		tmp = t_0;
	} else {
		tmp = F * (1.0 / (F * sin(B)));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-1.0d0) / b) - (x / tan(b))
    if (f <= (-4.1d+138)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 210000.0d0) then
        tmp = t_0
    else if (f <= 2.1d+105) then
        tmp = 1.0d0 / sin(b)
    else if (f <= 7.5d+254) then
        tmp = t_0
    else
        tmp = f * (1.0d0 / (f * sin(b)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (-1.0 / B) - (x / Math.tan(B));
	double tmp;
	if (F <= -4.1e+138) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 210000.0) {
		tmp = t_0;
	} else if (F <= 2.1e+105) {
		tmp = 1.0 / Math.sin(B);
	} else if (F <= 7.5e+254) {
		tmp = t_0;
	} else {
		tmp = F * (1.0 / (F * Math.sin(B)));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (-1.0 / B) - (x / math.tan(B))
	tmp = 0
	if F <= -4.1e+138:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 210000.0:
		tmp = t_0
	elif F <= 2.1e+105:
		tmp = 1.0 / math.sin(B)
	elif F <= 7.5e+254:
		tmp = t_0
	else:
		tmp = F * (1.0 / (F * math.sin(B)))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 / B) - Float64(x / tan(B)))
	tmp = 0.0
	if (F <= -4.1e+138)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 210000.0)
		tmp = t_0;
	elseif (F <= 2.1e+105)
		tmp = Float64(1.0 / sin(B));
	elseif (F <= 7.5e+254)
		tmp = t_0;
	else
		tmp = Float64(F * Float64(1.0 / Float64(F * sin(B))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (-1.0 / B) - (x / tan(B));
	tmp = 0.0;
	if (F <= -4.1e+138)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 210000.0)
		tmp = t_0;
	elseif (F <= 2.1e+105)
		tmp = 1.0 / sin(B);
	elseif (F <= 7.5e+254)
		tmp = t_0;
	else
		tmp = F * (1.0 / (F * sin(B)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -4.1e+138], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 210000.0], t$95$0, If[LessEqual[F, 2.1e+105], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7.5e+254], t$95$0, N[(F * N[(1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -4.0999999999999998e138

   1. Initial program 37.4%

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

   3. Simplified 53.6%

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

   5. Taylor expanded in F around -inf 99.6%

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

   6. Taylor expanded in B around 0 78.7%

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

   if -4.0999999999999998e138 < F < 2.1e5 or 2.1000000000000001e105 < F < 7.50000000000000043e254

   1. Initial program 90.7%

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

   3. Simplified 95.3%

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

   5. Taylor expanded in F around -inf 54.8%

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

   6. Taylor expanded in B around 0 61.3%

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

   if 2.1e5 < F < 2.1000000000000001e105

   1. Initial program 95.7%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in F around inf 98.9%

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

   6. Taylor expanded in x around 0 66.7%

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

   if 7.50000000000000043e254 < F

   1. Initial program 2.4%

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

   3. Simplified 27.9%

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

      1. associate-*r/ 27.9%

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

      2. fma-undefine 27.9%

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

      3. +-commutative 27.9%

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

      4. *-commutative 27.9%

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

      5. fma-define 27.9%

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

      6. metadata-eval 27.9%

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

      7. metadata-eval 27.9%

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

      8. fma-define 27.9%

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

      9. +-commutative 27.9%

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

      10. fma-define 27.9%

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

      11. metadata-eval 27.9%

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

      12. metadata-eval 27.9%

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

   6. Applied egg-rr 27.9%

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

   7. Taylor expanded in x around 0 1.3%

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

      1. associate-*l/ 2.3%

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

      2. *-lft-identity 2.3%

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

      3. times-frac 2.3%

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

      4. /-rgt-identity 2.3%

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

      5. +-commutative 2.3%

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

      6. unpow2 2.3%

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

      7. fma-undefine 2.3%

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

   9. Simplified 2.3%

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

   10. Taylor expanded in F around inf 72.7%

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

       1. *-commutative 72.7%

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

   12. Simplified 72.7%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.1 \cdot 10^{+138}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 210000:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.1 \cdot 10^{+105}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{+254}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{F \cdot \sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 44.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -8.2 \cdot 10^{+244}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq -9.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 0.55:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{+168}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -8.2e+244)
   (/ (- -1.0 x) B)
   (if (<= F -9.2e-15)
     (/ -1.0 (sin B))
     (if (<= F 0.55)
       (/ x (- B))
       (if (<= F 1.4e+168) (/ 1.0 (sin B)) (/ (- 1.0 x) B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -8.2e+244) {
		tmp = (-1.0 - x) / B;
	} else if (F <= -9.2e-15) {
		tmp = -1.0 / sin(B);
	} else if (F <= 0.55) {
		tmp = x / -B;
	} else if (F <= 1.4e+168) {
		tmp = 1.0 / sin(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 <= (-8.2d+244)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= (-9.2d-15)) then
        tmp = (-1.0d0) / sin(b)
    else if (f <= 0.55d0) then
        tmp = x / -b
    else if (f <= 1.4d+168) then
        tmp = 1.0d0 / sin(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 <= -8.2e+244) {
		tmp = (-1.0 - x) / B;
	} else if (F <= -9.2e-15) {
		tmp = -1.0 / Math.sin(B);
	} else if (F <= 0.55) {
		tmp = x / -B;
	} else if (F <= 1.4e+168) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -8.2e+244:
		tmp = (-1.0 - x) / B
	elif F <= -9.2e-15:
		tmp = -1.0 / math.sin(B)
	elif F <= 0.55:
		tmp = x / -B
	elif F <= 1.4e+168:
		tmp = 1.0 / math.sin(B)
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -8.2e+244)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= -9.2e-15)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 0.55)
		tmp = Float64(x / Float64(-B));
	elseif (F <= 1.4e+168)
		tmp = Float64(1.0 / sin(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 <= -8.2e+244)
		tmp = (-1.0 - x) / B;
	elseif (F <= -9.2e-15)
		tmp = -1.0 / sin(B);
	elseif (F <= 0.55)
		tmp = x / -B;
	elseif (F <= 1.4e+168)
		tmp = 1.0 / sin(B);
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -8.2e+244], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, -9.2e-15], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.55], N[(x / (-B)), $MachinePrecision], If[LessEqual[F, 1.4e+168], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -8.19999999999999985e244

   1. Initial program 27.7%

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

   3. Simplified 53.8%

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

   5. Taylor expanded in F around -inf 99.8%

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

   6. Taylor expanded in B around 0 67.6%

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

      1. associate-*r/ 67.6%

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

      2. neg-mul-1 67.6%

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

      3. +-commutative 67.6%

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

   8. Simplified 67.6%

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

   9. Taylor expanded in x around 0 67.5%

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

       1. associate-*r/ 67.5%

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

       2. neg-mul-1 67.5%

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

       3. div-sub 67.6%

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

       4. sub-neg 67.6%

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

       5. metadata-eval 67.6%

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

       6. +-commutative 67.6%

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

       7. unsub-neg 67.6%

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

   11. Simplified 67.6%

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

   if -8.19999999999999985e244 < F < -9.19999999999999961e-15

   1. Initial program 68.7%

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

   3. Simplified 77.8%

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

      1. associate-*r/ 77.9%

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

      2. fma-undefine 77.9%

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

      3. +-commutative 77.9%

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

      4. *-commutative 77.9%

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

      5. fma-define 77.9%

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

      6. metadata-eval 77.9%

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

      7. metadata-eval 77.9%

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

      8. fma-define 77.9%

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

      9. +-commutative 77.9%

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

      10. fma-define 77.9%

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

      11. metadata-eval 77.9%

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

      12. metadata-eval 77.9%

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

   6. Applied egg-rr 77.9%

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

   7. Taylor expanded in x around 0 32.7%

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

      1. associate-*l/ 38.1%

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

      2. *-lft-identity 38.1%

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

      3. times-frac 38.0%

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

      4. /-rgt-identity 38.0%

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

      5. +-commutative 38.0%

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

      6. unpow2 38.0%

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

      7. fma-undefine 38.0%

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

   9. Simplified 38.0%

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

   10. Taylor expanded in F around -inf 58.4%

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

   if -9.19999999999999961e-15 < F < 0.55000000000000004

   1. Initial program 99.5%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in F around -inf 41.5%

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

   6. Taylor expanded in B around 0 18.7%

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

      1. associate-*r/ 18.7%

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

      2. neg-mul-1 18.7%

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

      3. +-commutative 18.7%

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

   8. Simplified 18.7%

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

   9. Taylor expanded in x around inf 30.0%

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

       1. associate-*r/ 30.0%

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

       2. neg-mul-1 30.0%

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

   11. Simplified 30.0%

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

   if 0.55000000000000004 < F < 1.39999999999999995e168

   1. Initial program 90.2%

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

   3. Simplified 94.9%

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

   5. Taylor expanded in F around inf 99.2%

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

   6. Taylor expanded in x around 0 60.3%

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

   if 1.39999999999999995e168 < F

   1. Initial program 31.8%

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

   3. Simplified 54.2%

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

   5. Taylor expanded in F around inf 99.6%

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

   6. Taylor expanded in B around 0 59.1%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -8.2 \cdot 10^{+244}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq -9.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 0.55:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{+168}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 59.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.14 \cdot 10^{+139}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.7 \cdot 10^{-194}:\\ \;\;\;\;\frac{-1}{B} - t\_0\\ \mathbf{elif}\;F \leq -1.75 \cdot 10^{-290}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.14e+139)
     (- (/ -1.0 (sin B)) (/ x B))
     (if (<= F -1.7e-194)
       (- (/ -1.0 B) t_0)
       (if (<= F -1.75e-290) (/ x (- B)) (- (/ 1.0 B) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.14e+139) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -1.7e-194) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= -1.75e-290) {
		tmp = x / -B;
	} else {
		tmp = (1.0 / B) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.14d+139)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-1.7d-194)) then
        tmp = ((-1.0d0) / b) - t_0
    else if (f <= (-1.75d-290)) then
        tmp = x / -b
    else
        tmp = (1.0d0 / b) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.14e+139) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -1.7e-194) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= -1.75e-290) {
		tmp = x / -B;
	} else {
		tmp = (1.0 / B) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.14e+139:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -1.7e-194:
		tmp = (-1.0 / B) - t_0
	elif F <= -1.75e-290:
		tmp = x / -B
	else:
		tmp = (1.0 / B) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.14e+139)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -1.7e-194)
		tmp = Float64(Float64(-1.0 / B) - t_0);
	elseif (F <= -1.75e-290)
		tmp = Float64(x / Float64(-B));
	else
		tmp = Float64(Float64(1.0 / B) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.14e+139)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -1.7e-194)
		tmp = (-1.0 / B) - t_0;
	elseif (F <= -1.75e-290)
		tmp = x / -B;
	else
		tmp = (1.0 / B) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.14e+139], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.7e-194], N[(N[(-1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -1.75e-290], N[(x / (-B)), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.13999999999999997e139

   1. Initial program 37.4%

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

   3. Simplified 53.6%

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

   5. Taylor expanded in F around -inf 99.6%

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

   6. Taylor expanded in B around 0 78.7%

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

   if -1.13999999999999997e139 < F < -1.70000000000000005e-194

   1. Initial program 94.7%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in F around -inf 71.1%

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

   6. Taylor expanded in B around 0 71.7%

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

   if -1.70000000000000005e-194 < F < -1.74999999999999991e-290

   1. Initial program 99.9%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in F around -inf 30.9%

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

   6. Taylor expanded in B around 0 25.8%

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

      1. associate-*r/ 25.8%

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

      2. neg-mul-1 25.8%

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

      3. +-commutative 25.8%

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

   8. Simplified 25.8%

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

   9. Taylor expanded in x around inf 60.9%

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

       1. associate-*r/ 60.9%

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

       2. neg-mul-1 60.9%

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

   11. Simplified 60.9%

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

   if -1.74999999999999991e-290 < F

   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. Add Preprocessing

   3. Taylor expanded in B around 0 63.4%

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

      1. neg-sub0 63.4%

         \[\leadsto \color{blue}{\left(0 - x \cdot \frac{1}{\tan B}\right)} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. div-inv 63.5%

         \[\leadsto \left(0 - \color{blue}{\frac{x}{\tan B}}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   5. Applied egg-rr 63.5%

      \[\leadsto \color{blue}{\left(0 - \frac{x}{\tan B}\right)} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   6. Step-by-step derivation

      1. neg-sub0 63.5%

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

      2. distribute-neg-frac 63.5%

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

   7. Simplified 63.5%

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

   8. Taylor expanded in F around inf 61.9%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.14 \cdot 10^{+139}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.7 \cdot 10^{-194}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.75 \cdot 10^{-290}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 56.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;x \leq -8.6 \cdot 10^{-10}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -1.95 \cdot 10^{-175}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-81}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ -1.0 B) (/ x (tan B)))))
   (if (<= x -8.6e-10)
     t_0
     (if (<= x -1.95e-175)
       (/ 1.0 (sin B))
       (if (<= x 2.5e-81) (/ -1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 / B) - (x / tan(B));
	double tmp;
	if (x <= -8.6e-10) {
		tmp = t_0;
	} else if (x <= -1.95e-175) {
		tmp = 1.0 / sin(B);
	} else if (x <= 2.5e-81) {
		tmp = -1.0 / sin(B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-1.0d0) / b) - (x / tan(b))
    if (x <= (-8.6d-10)) then
        tmp = t_0
    else if (x <= (-1.95d-175)) then
        tmp = 1.0d0 / sin(b)
    else if (x <= 2.5d-81) then
        tmp = (-1.0d0) / sin(b)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (-1.0 / B) - (x / Math.tan(B));
	double tmp;
	if (x <= -8.6e-10) {
		tmp = t_0;
	} else if (x <= -1.95e-175) {
		tmp = 1.0 / Math.sin(B);
	} else if (x <= 2.5e-81) {
		tmp = -1.0 / Math.sin(B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (-1.0 / B) - (x / math.tan(B))
	tmp = 0
	if x <= -8.6e-10:
		tmp = t_0
	elif x <= -1.95e-175:
		tmp = 1.0 / math.sin(B)
	elif x <= 2.5e-81:
		tmp = -1.0 / math.sin(B)
	else:
		tmp = t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 / B) - Float64(x / tan(B)))
	tmp = 0.0
	if (x <= -8.6e-10)
		tmp = t_0;
	elseif (x <= -1.95e-175)
		tmp = Float64(1.0 / sin(B));
	elseif (x <= 2.5e-81)
		tmp = Float64(-1.0 / sin(B));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (-1.0 / B) - (x / tan(B));
	tmp = 0.0;
	if (x <= -8.6e-10)
		tmp = t_0;
	elseif (x <= -1.95e-175)
		tmp = 1.0 / sin(B);
	elseif (x <= 2.5e-81)
		tmp = -1.0 / sin(B);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -8.6e-10], t$95$0, If[LessEqual[x, -1.95e-175], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.5e-81], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.60000000000000029e-10 or 2.4999999999999999e-81 < x

   1. Initial program 83.5%

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

   3. Simplified 94.9%

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

   5. Taylor expanded in F around -inf 86.8%

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

   6. Taylor expanded in B around 0 89.9%

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

   if -8.60000000000000029e-10 < x < -1.94999999999999999e-175

   1. Initial program 65.3%

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

   3. Simplified 73.2%

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

   5. Taylor expanded in F around inf 41.0%

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

   6. Taylor expanded in x around 0 40.4%

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

   if -1.94999999999999999e-175 < x < 2.4999999999999999e-81

   1. Initial program 73.3%

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

   3. Simplified 74.7%

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

      1. associate-*r/ 74.7%

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

      2. fma-undefine 74.7%

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

      3. +-commutative 74.7%

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

      4. *-commutative 74.7%

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

      5. fma-define 74.7%

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

      6. metadata-eval 74.7%

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

      7. metadata-eval 74.7%

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

      8. fma-define 74.7%

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

      9. +-commutative 74.7%

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

      10. fma-define 74.7%

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

      11. metadata-eval 74.7%

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

      12. metadata-eval 74.7%

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

   6. Applied egg-rr 74.7%

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

   7. Taylor expanded in x around 0 60.0%

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

      1. associate-*l/ 61.4%

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

      2. *-lft-identity 61.4%

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

      3. times-frac 61.3%

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

      4. /-rgt-identity 61.3%

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

      5. +-commutative 61.3%

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

      6. unpow2 61.3%

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

      7. fma-undefine 61.3%

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

   9. Simplified 61.3%

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

   10. Taylor expanded in F around -inf 30.8%

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

Alternative 19: 43.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -6.2 \cdot 10^{+245}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq -9.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 8.6 \cdot 10^{-107}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -6.2e+245)
   (/ (- -1.0 x) B)
   (if (<= F -9.2e-15)
     (/ -1.0 (sin B))
     (if (<= F 8.6e-107) (/ x (- B)) (/ (- 1.0 x) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -6.2e+245) {
		tmp = (-1.0 - x) / B;
	} else if (F <= -9.2e-15) {
		tmp = -1.0 / sin(B);
	} else if (F <= 8.6e-107) {
		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 <= (-6.2d+245)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= (-9.2d-15)) then
        tmp = (-1.0d0) / sin(b)
    else if (f <= 8.6d-107) 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 <= -6.2e+245) {
		tmp = (-1.0 - x) / B;
	} else if (F <= -9.2e-15) {
		tmp = -1.0 / Math.sin(B);
	} else if (F <= 8.6e-107) {
		tmp = x / -B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -6.2e+245:
		tmp = (-1.0 - x) / B
	elif F <= -9.2e-15:
		tmp = -1.0 / math.sin(B)
	elif F <= 8.6e-107:
		tmp = x / -B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -6.2e+245)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= -9.2e-15)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 8.6e-107)
		tmp = Float64(x / Float64(-B));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -6.2e+245)
		tmp = (-1.0 - x) / B;
	elseif (F <= -9.2e-15)
		tmp = -1.0 / sin(B);
	elseif (F <= 8.6e-107)
		tmp = x / -B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -6.2e+245], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, -9.2e-15], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 8.6e-107], N[(x / (-B)), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -6.1999999999999998e245

   1. Initial program 27.7%

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

   3. Simplified 53.8%

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

   5. Taylor expanded in F around -inf 99.8%

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

   6. Taylor expanded in B around 0 67.6%

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

      1. associate-*r/ 67.6%

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

      2. neg-mul-1 67.6%

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

      3. +-commutative 67.6%

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

   8. Simplified 67.6%

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

   9. Taylor expanded in x around 0 67.5%

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

       1. associate-*r/ 67.5%

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

       2. neg-mul-1 67.5%

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

       3. div-sub 67.6%

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

       4. sub-neg 67.6%

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

       5. metadata-eval 67.6%

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

       6. +-commutative 67.6%

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

       7. unsub-neg 67.6%

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

   11. Simplified 67.6%

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

   if -6.1999999999999998e245 < F < -9.19999999999999961e-15

   1. Initial program 68.7%

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

   3. Simplified 77.8%

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

      1. associate-*r/ 77.9%

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

      2. fma-undefine 77.9%

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

      3. +-commutative 77.9%

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

      4. *-commutative 77.9%

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

      5. fma-define 77.9%

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

      6. metadata-eval 77.9%

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

      7. metadata-eval 77.9%

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

      8. fma-define 77.9%

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

      9. +-commutative 77.9%

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

      10. fma-define 77.9%

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

      11. metadata-eval 77.9%

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

      12. metadata-eval 77.9%

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

   6. Applied egg-rr 77.9%

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

   7. Taylor expanded in x around 0 32.7%

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

      1. associate-*l/ 38.1%

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

      2. *-lft-identity 38.1%

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

      3. times-frac 38.0%

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

      4. /-rgt-identity 38.0%

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

      5. +-commutative 38.0%

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

      6. unpow2 38.0%

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

      7. fma-undefine 38.0%

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

   9. Simplified 38.0%

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

   10. Taylor expanded in F around -inf 58.4%

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

   if -9.19999999999999961e-15 < F < 8.5999999999999995e-107

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in F around -inf 43.0%

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

   6. Taylor expanded in B around 0 17.5%

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

      1. associate-*r/ 17.5%

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

      2. neg-mul-1 17.5%

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

      3. +-commutative 17.5%

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

   8. Simplified 17.5%

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

   9. Taylor expanded in x around inf 30.7%

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

       1. associate-*r/ 30.7%

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

       2. neg-mul-1 30.7%

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

   11. Simplified 30.7%

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

   if 8.5999999999999995e-107 < F

   1. Initial program 72.4%

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

   3. Simplified 82.2%

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

   5. Taylor expanded in F around inf 86.9%

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

   6. Taylor expanded in B around 0 42.5%

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

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6.2 \cdot 10^{+245}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq -9.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 8.6 \cdot 10^{-107}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 42.6% accurate, 21.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.4 \cdot 10^{-76}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 8.6 \cdot 10^{-107}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.4e-76)
   (/ (- -1.0 x) B)
   (if (<= F 8.6e-107) (/ x (- B)) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.4e-76) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 8.6e-107) {
		tmp = x / -B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.4d-76)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 8.6d-107) then
        tmp = x / -b
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.4e-76) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 8.6e-107) {
		tmp = x / -B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.4e-76:
		tmp = (-1.0 - x) / B
	elif F <= 8.6e-107:
		tmp = x / -B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.4e-76)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 8.6e-107)
		tmp = Float64(x / Float64(-B));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.4e-76)
		tmp = (-1.0 - x) / B;
	elseif (F <= 8.6e-107)
		tmp = x / -B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.4e-76], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 8.6e-107], N[(x / (-B)), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.40000000000000005e-76

   1. Initial program 62.5%

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

   3. Simplified 74.6%

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

   5. Taylor expanded in F around -inf 89.9%

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

   6. Taylor expanded in B around 0 42.5%

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

      1. associate-*r/ 42.5%

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

      2. neg-mul-1 42.5%

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

      3. +-commutative 42.5%

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

   8. Simplified 42.5%

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

   9. Taylor expanded in x around 0 42.5%

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

       1. associate-*r/ 42.5%

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

       2. neg-mul-1 42.5%

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

       3. div-sub 42.5%

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

       4. sub-neg 42.5%

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

       5. metadata-eval 42.5%

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

       6. +-commutative 42.5%

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

       7. unsub-neg 42.5%

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

   11. Simplified 42.5%

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

   if -1.40000000000000005e-76 < F < 8.5999999999999995e-107

   1. Initial program 99.5%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in F around -inf 44.5%

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

   6. Taylor expanded in B around 0 17.0%

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

      1. associate-*r/ 17.0%

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

      2. neg-mul-1 17.0%

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

      3. +-commutative 17.0%

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

   8. Simplified 17.0%

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

   9. Taylor expanded in x around inf 32.1%

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

       1. associate-*r/ 32.1%

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

       2. neg-mul-1 32.1%

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

   11. Simplified 32.1%

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

   if 8.5999999999999995e-107 < F

   1. Initial program 72.4%

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

   3. Simplified 82.2%

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

   5. Taylor expanded in F around inf 86.9%

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

   6. Taylor expanded in B around 0 42.5%

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

4. Final simplification 39.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.4 \cdot 10^{-76}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 8.6 \cdot 10^{-107}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 30.2% accurate, 23.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-146} \lor \neg \left(x \leq 1.22 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (or (<= x -6.5e-146) (not (<= x 1.22e-21))) (/ x (- B)) (/ -1.0 B)))

C

double code(double F, double B, double x) {
	double tmp;
	if ((x <= -6.5e-146) || !(x <= 1.22e-21)) {
		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 <= (-6.5d-146)) .or. (.not. (x <= 1.22d-21))) 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 <= -6.5e-146) || !(x <= 1.22e-21)) {
		tmp = x / -B;
	} else {
		tmp = -1.0 / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if (x <= -6.5e-146) or not (x <= 1.22e-21):
		tmp = x / -B
	else:
		tmp = -1.0 / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if ((x <= -6.5e-146) || !(x <= 1.22e-21))
		tmp = Float64(x / Float64(-B));
	else
		tmp = Float64(-1.0 / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if ((x <= -6.5e-146) || ~((x <= 1.22e-21)))
		tmp = x / -B;
	else
		tmp = -1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[Or[LessEqual[x, -6.5e-146], N[Not[LessEqual[x, 1.22e-21]], $MachinePrecision]], N[(x / (-B)), $MachinePrecision], N[(-1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.4999999999999999e-146 or 1.21999999999999991e-21 < x

   1. Initial program 82.4%

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

   3. Simplified 92.8%

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

   5. Taylor expanded in F around -inf 79.4%

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

   6. Taylor expanded in B around 0 37.4%

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

      1. associate-*r/ 37.4%

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

      2. neg-mul-1 37.4%

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

      3. +-commutative 37.4%

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

   8. Simplified 37.4%

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

   9. Taylor expanded in x around inf 40.7%

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

       1. associate-*r/ 40.7%

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

       2. neg-mul-1 40.7%

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

   11. Simplified 40.7%

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

   if -6.4999999999999999e-146 < x < 1.21999999999999991e-21

   1. Initial program 70.9%

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

   3. Simplified 74.7%

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

   5. Taylor expanded in F around -inf 31.0%

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

   6. Taylor expanded in B around 0 17.3%

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

      1. associate-*r/ 17.3%

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

      2. neg-mul-1 17.3%

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

      3. +-commutative 17.3%

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

   8. Simplified 17.3%

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

   9. Taylor expanded in x around 0 17.3%

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

4. Final simplification 30.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-146} \lor \neg \left(x \leq 1.22 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 36.0% accurate, 32.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.22 \cdot 10^{-76}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.22e-76) (/ (- -1.0 x) B) (/ x (- B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.22e-76) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = 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 <= (-1.22d-76)) then
        tmp = ((-1.0d0) - x) / b
    else
        tmp = x / -b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.22e-76) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = x / -B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.22e-76:
		tmp = (-1.0 - x) / B
	else:
		tmp = x / -B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.22e-76)
		tmp = Float64(Float64(-1.0 - x) / B);
	else
		tmp = Float64(x / Float64(-B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.22e-76)
		tmp = (-1.0 - x) / B;
	else
		tmp = x / -B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.22e-76], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[(x / (-B)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -1.22e-76

   1. Initial program 62.5%

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

   3. Simplified 74.6%

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

   5. Taylor expanded in F around -inf 89.9%

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

   6. Taylor expanded in B around 0 42.5%

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

      1. associate-*r/ 42.5%

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

      2. neg-mul-1 42.5%

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

      3. +-commutative 42.5%

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

   8. Simplified 42.5%

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

   9. Taylor expanded in x around 0 42.5%

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

       1. associate-*r/ 42.5%

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

       2. neg-mul-1 42.5%

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

       3. div-sub 42.5%

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

       4. sub-neg 42.5%

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

       5. metadata-eval 42.5%

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

       6. +-commutative 42.5%

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

       7. unsub-neg 42.5%

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

   11. Simplified 42.5%

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

   if -1.22e-76 < F

   1. Initial program 84.9%

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

   3. Simplified 90.3%

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

   5. Taylor expanded in F around -inf 43.3%

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

   6. Taylor expanded in B around 0 22.0%

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

      1. associate-*r/ 22.0%

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

      2. neg-mul-1 22.0%

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

      3. +-commutative 22.0%

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

   8. Simplified 22.0%

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

   9. Taylor expanded in x around inf 29.3%

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

       1. associate-*r/ 29.3%

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

       2. neg-mul-1 29.3%

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

   11. Simplified 29.3%

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

4. Final simplification 33.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.22 \cdot 10^{-76}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 17.2% accurate, 40.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 3.2 \cdot 10^{-167}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (if (<= F 3.2e-167) (/ -1.0 B) (/ 1.0 B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= 3.2e-167) {
		tmp = -1.0 / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= 3.2d-167) then
        tmp = (-1.0d0) / b
    else
        tmp = 1.0d0 / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= 3.2e-167) {
		tmp = -1.0 / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= 3.2e-167:
		tmp = -1.0 / B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= 3.2e-167)
		tmp = Float64(-1.0 / B);
	else
		tmp = Float64(1.0 / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= 3.2e-167)
		tmp = -1.0 / B;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, 3.2e-167], N[(-1.0 / B), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 3.2000000000000002e-167

   1. Initial program 79.1%

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

   3. Simplified 85.8%

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

   5. Taylor expanded in F around -inf 69.2%

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

   6. Taylor expanded in B around 0 30.8%

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

      1. associate-*r/ 30.8%

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

      2. neg-mul-1 30.8%

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

      3. +-commutative 30.8%

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

   8. Simplified 30.8%

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

   9. Taylor expanded in x around 0 15.0%

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

   if 3.2000000000000002e-167 < F

   1. Initial program 75.1%

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

   3. Taylor expanded in B around 0 55.8%

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

   4. Taylor expanded in F around inf 49.4%

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

   5. Taylor expanded in x around 0 17.6%

      \[\leadsto \color{blue}{\frac{1}{B} - \frac{1}{B \cdot {F}^{2}}} \]
   6. Step-by-step derivation

      1. associate-/r* 17.6%

         \[\leadsto \frac{1}{B} - \color{blue}{\frac{\frac{1}{B}}{{F}^{2}}} \]

   7. Simplified 17.6%

      \[\leadsto \color{blue}{\frac{1}{B} - \frac{\frac{1}{B}}{{F}^{2}}} \]

   8. Taylor expanded in F around inf 16.7%

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

Alternative 24: 10.5% 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 77.5%

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

3. Simplified 85.1%

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

5. Taylor expanded in F around -inf 58.8%

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

6. Taylor expanded in B around 0 28.8%

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

   1. associate-*r/ 28.8%

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

   2. neg-mul-1 28.8%

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

   3. +-commutative 28.8%

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

8. Simplified 28.8%

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

9. Taylor expanded in x around 0 10.2%

   \[\leadsto \color{blue}{\frac{-1}{B}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024129 
(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))))))