VandenBroeck and Keller, Equation (23)

Percentage Accurate: 64.7% → 99.7%

Time: 21.5s

Alternatives: 20

Speedup: 2.8×

• could not determine a ground truth

  1. F = -0.3801168298984202
  2. B = 5.786774557865957e-238
  3. x = -440768.82986555836

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: 64.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1 \cdot 10^{+40}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 400000000:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1e+40)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 400000000.0)
       (- (* F (/ (sqrt (/ 1.0 (fma F F 2.0))) (sin B))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1e+40) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 400000000.0) {
		tmp = (F * (sqrt((1.0 / fma(F, F, 2.0))) / sin(B))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1e+40)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 400000000.0)
		tmp = Float64(Float64(F * Float64(sqrt(Float64(1.0 / fma(F, F, 2.0))) / sin(B))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1e+40], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 400000000.0], N[(N[(F * N[(N[Sqrt[N[(1.0 / N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.00000000000000003e40

   1. Initial program 53.5%

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

   3. Simplified 65.8%

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

   5. Taylor expanded in x around 0 67.7%

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

      1. associate-*l/ 67.7%

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

      2. *-lft-identity 67.7%

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

      3. +-commutative 67.7%

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

      4. unpow2 67.7%

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

      5. fma-undefine 67.7%

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

   7. Simplified 67.7%

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

   8. Taylor expanded in F around -inf 99.8%

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

   if -1.00000000000000003e40 < F < 4e8

   1. Initial program 80.0%

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

   3. Simplified 80.1%

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

   5. Taylor expanded in x around 0 99.4%

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

      1. associate-*l/ 99.4%

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

      2. *-lft-identity 99.4%

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

      3. +-commutative 99.4%

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

      4. unpow2 99.4%

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

      5. fma-undefine 99.4%

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

   7. Simplified 99.4%

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

   if 4e8 < F

   1. Initial program 56.4%

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

   3. Simplified 66.6%

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

   5. Taylor expanded in x around 0 75.3%

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

      1. associate-*l/ 75.3%

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

      2. *-lft-identity 75.3%

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

      3. +-commutative 75.3%

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

      4. unpow2 75.3%

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

      5. fma-undefine 75.3%

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

   7. Simplified 75.3%

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

   8. 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 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{+40}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 400000000:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1300000:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 27:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\frac{\sin B}{\frac{1 + -0.5 \cdot \frac{2 + x \cdot 2}{{F}^{2}}}{F}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1300000.0)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 27.0)
       (- (/ (* F (sqrt 0.5)) (sin B)) t_0)
       (+
        (* x (/ -1.0 (tan B)))
        (/
         F
         (/
          (sin B)
          (/ (+ 1.0 (* -0.5 (/ (+ 2.0 (* x 2.0)) (pow F 2.0)))) F))))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1300000.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 27.0) {
		tmp = ((F * sqrt(0.5)) / sin(B)) - t_0;
	} else {
		tmp = (x * (-1.0 / tan(B))) + (F / (sin(B) / ((1.0 + (-0.5 * ((2.0 + (x * 2.0)) / pow(F, 2.0)))) / F)));
	}
	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 <= (-1300000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 27.0d0) then
        tmp = ((f * sqrt(0.5d0)) / sin(b)) - t_0
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (f / (sin(b) / ((1.0d0 + ((-0.5d0) * ((2.0d0 + (x * 2.0d0)) / (f ** 2.0d0)))) / f)))
    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 <= -1300000.0) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 27.0) {
		tmp = ((F * Math.sqrt(0.5)) / Math.sin(B)) - t_0;
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (F / (Math.sin(B) / ((1.0 + (-0.5 * ((2.0 + (x * 2.0)) / Math.pow(F, 2.0)))) / F)));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1300000.0:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 27.0:
		tmp = ((F * math.sqrt(0.5)) / math.sin(B)) - t_0
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (F / (math.sin(B) / ((1.0 + (-0.5 * ((2.0 + (x * 2.0)) / math.pow(F, 2.0)))) / F)))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1300000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 27.0)
		tmp = Float64(Float64(Float64(F * sqrt(0.5)) / sin(B)) - t_0);
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(F / Float64(sin(B) / Float64(Float64(1.0 + Float64(-0.5 * Float64(Float64(2.0 + Float64(x * 2.0)) / (F ^ 2.0)))) / F))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1300000.0)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 27.0)
		tmp = ((F * sqrt(0.5)) / sin(B)) - t_0;
	else
		tmp = (x * (-1.0 / tan(B))) + (F / (sin(B) / ((1.0 + (-0.5 * ((2.0 + (x * 2.0)) / (F ^ 2.0)))) / F)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1300000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 27.0], N[(N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(F / N[(N[Sin[B], $MachinePrecision] / N[(N[(1.0 + N[(-0.5 * N[(N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision] / N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.3e6

   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 67.1%

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

   5. Taylor expanded in x around 0 70.7%

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

      1. associate-*l/ 70.7%

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

      2. *-lft-identity 70.7%

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

      3. +-commutative 70.7%

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

      4. unpow2 70.7%

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

      5. fma-undefine 70.7%

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

   7. Simplified 70.7%

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

   8. Taylor expanded in F around -inf 99.8%

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

   if -1.3e6 < F < 27

   1. Initial program 79.6%

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

   3. Simplified 79.7%

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

   5. Taylor expanded in x around 0 99.7%

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

      1. associate-*l/ 99.7%

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

      2. *-lft-identity 99.7%

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

      3. +-commutative 99.7%

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

      4. unpow2 99.7%

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

      5. fma-undefine 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in F around 0 99.8%

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

      1. *-commutative 99.8%

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

   10. Simplified 99.8%

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

   if 27 < F

   1. Initial program 58.2%

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

      1. associate-*l/ 68.0%

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

      2. +-commutative 68.0%

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

      3. *-commutative 68.0%

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

      4. fma-undefine 68.0%

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

      5. fma-define 68.0%

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

      6. metadata-eval 68.0%

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

      7. metadata-eval 68.0%

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

      8. associate-*r/ 68.0%

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

      9. clear-num 68.0%

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

      10. un-div-inv 68.1%

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

      11. fma-define 68.1%

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

      12. fma-undefine 68.1%

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

      13. *-commutative 68.1%

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

      14. fma-define 68.1%

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

      15. fma-define 68.1%

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

   4. Applied egg-rr 68.1%

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

   5. Taylor expanded in F around inf 98.2%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1300000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 27:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\frac{\sin B}{\frac{1 + -0.5 \cdot \frac{2 + x \cdot 2}{{F}^{2}}}{F}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1300000:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 5.8 \cdot 10^{-14}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1300000.0)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 5.8e-14)
       (- (* F (/ (sqrt 0.5) (sin B))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1300000.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 5.8e-14) {
		tmp = (F * (sqrt(0.5) / sin(B))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1300000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 5.8d-14) then
        tmp = (f * (sqrt(0.5d0) / sin(b))) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1300000.0) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 5.8e-14) {
		tmp = (F * (Math.sqrt(0.5) / Math.sin(B))) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1300000.0:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 5.8e-14:
		tmp = (F * (math.sqrt(0.5) / math.sin(B))) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1300000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 5.8e-14)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / sin(B))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1300000.0)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 5.8e-14)
		tmp = (F * (sqrt(0.5) / sin(B))) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1300000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 5.8e-14], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.3e6

   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 67.1%

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

   5. Taylor expanded in x around 0 70.7%

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

      1. associate-*l/ 70.7%

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

      2. *-lft-identity 70.7%

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

      3. +-commutative 70.7%

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

      4. unpow2 70.7%

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

      5. fma-undefine 70.7%

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

   7. Simplified 70.7%

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

   8. Taylor expanded in F around -inf 99.8%

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

   if -1.3e6 < F < 5.8000000000000005e-14

   1. Initial program 79.4%

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

   3. Simplified 79.5%

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

   5. Taylor expanded in x around 0 99.7%

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

      1. associate-*l/ 99.7%

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

      2. *-lft-identity 99.7%

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

      3. +-commutative 99.7%

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

      4. unpow2 99.7%

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

      5. fma-undefine 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in F around 0 99.7%

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

   if 5.8000000000000005e-14 < F

   1. Initial program 58.8%

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

   3. Simplified 68.4%

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

   5. Taylor expanded in x around 0 76.2%

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

      1. associate-*l/ 76.2%

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

      2. *-lft-identity 76.2%

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

      3. +-commutative 76.2%

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

      4. unpow2 76.2%

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

      5. fma-undefine 76.2%

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

   7. Simplified 76.2%

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

   8. Taylor expanded in F around inf 97.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1300000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 5.8 \cdot 10^{-14}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1300000:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 5.8 \cdot 10^{-14}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1300000.0)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 5.8e-14)
       (- (/ (* F (sqrt 0.5)) (sin B)) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1300000.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 5.8e-14) {
		tmp = ((F * sqrt(0.5)) / sin(B)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1300000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 5.8d-14) then
        tmp = ((f * sqrt(0.5d0)) / sin(b)) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1300000.0) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 5.8e-14) {
		tmp = ((F * Math.sqrt(0.5)) / Math.sin(B)) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1300000.0:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 5.8e-14:
		tmp = ((F * math.sqrt(0.5)) / math.sin(B)) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1300000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 5.8e-14)
		tmp = Float64(Float64(Float64(F * sqrt(0.5)) / sin(B)) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1300000.0)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 5.8e-14)
		tmp = ((F * sqrt(0.5)) / sin(B)) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1300000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 5.8e-14], N[(N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[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 < -1.3e6

   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 67.1%

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

   5. Taylor expanded in x around 0 70.7%

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

      1. associate-*l/ 70.7%

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

      2. *-lft-identity 70.7%

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

      3. +-commutative 70.7%

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

      4. unpow2 70.7%

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

      5. fma-undefine 70.7%

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

   7. Simplified 70.7%

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

   8. Taylor expanded in F around -inf 99.8%

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

   if -1.3e6 < F < 5.8000000000000005e-14

   1. Initial program 79.4%

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

   3. Simplified 79.5%

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

   5. Taylor expanded in x around 0 99.7%

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

      1. associate-*l/ 99.7%

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

      2. *-lft-identity 99.7%

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

      3. +-commutative 99.7%

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

      4. unpow2 99.7%

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

      5. fma-undefine 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in F around 0 99.8%

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

      1. *-commutative 99.8%

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

   10. Simplified 99.8%

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

   if 5.8000000000000005e-14 < F

   1. Initial program 58.8%

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

   3. Simplified 68.4%

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

   5. Taylor expanded in x around 0 76.2%

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

      1. associate-*l/ 76.2%

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

      2. *-lft-identity 76.2%

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

      3. +-commutative 76.2%

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

      4. unpow2 76.2%

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

      5. fma-undefine 76.2%

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

   7. Simplified 76.2%

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

   8. Taylor expanded in F around inf 97.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1300000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 5.8 \cdot 10^{-14}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -6.4 \cdot 10^{-15}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 4.9 \cdot 10^{-124}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{B} - t\_0\\ \mathbf{elif}\;F \leq 9 \cdot 10^{-70}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 5.8 \cdot 10^{-14}:\\ \;\;\;\;F \cdot \left(\sqrt{0.5} \cdot \frac{1}{B}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -6.4e-15)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 4.9e-124)
       (- (/ (* F (sqrt 0.5)) B) t_0)
       (if (<= F 9e-70)
         (- (* (/ F (sin B)) (pow (+ (* x 2.0) (+ 2.0 (* F F))) -0.5)) (/ x B))
         (if (<= F 5.8e-14)
           (- (* F (* (sqrt 0.5) (/ 1.0 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 <= -6.4e-15) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 4.9e-124) {
		tmp = ((F * sqrt(0.5)) / B) - t_0;
	} else if (F <= 9e-70) {
		tmp = ((F / sin(B)) * pow(((x * 2.0) + (2.0 + (F * F))), -0.5)) - (x / B);
	} else if (F <= 5.8e-14) {
		tmp = (F * (sqrt(0.5) * (1.0 / 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 <= (-6.4d-15)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 4.9d-124) then
        tmp = ((f * sqrt(0.5d0)) / b) - t_0
    else if (f <= 9d-70) then
        tmp = ((f / sin(b)) * (((x * 2.0d0) + (2.0d0 + (f * f))) ** (-0.5d0))) - (x / b)
    else if (f <= 5.8d-14) then
        tmp = (f * (sqrt(0.5d0) * (1.0d0 / 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 <= -6.4e-15) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 4.9e-124) {
		tmp = ((F * Math.sqrt(0.5)) / B) - t_0;
	} else if (F <= 9e-70) {
		tmp = ((F / Math.sin(B)) * Math.pow(((x * 2.0) + (2.0 + (F * F))), -0.5)) - (x / B);
	} else if (F <= 5.8e-14) {
		tmp = (F * (Math.sqrt(0.5) * (1.0 / 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 <= -6.4e-15:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 4.9e-124:
		tmp = ((F * math.sqrt(0.5)) / B) - t_0
	elif F <= 9e-70:
		tmp = ((F / math.sin(B)) * math.pow(((x * 2.0) + (2.0 + (F * F))), -0.5)) - (x / B)
	elif F <= 5.8e-14:
		tmp = (F * (math.sqrt(0.5) * (1.0 / 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 <= -6.4e-15)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 4.9e-124)
		tmp = Float64(Float64(Float64(F * sqrt(0.5)) / B) - t_0);
	elseif (F <= 9e-70)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(x * 2.0) + Float64(2.0 + Float64(F * F))) ^ -0.5)) - Float64(x / B));
	elseif (F <= 5.8e-14)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) * Float64(1.0 / 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 <= -6.4e-15)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 4.9e-124)
		tmp = ((F * sqrt(0.5)) / B) - t_0;
	elseif (F <= 9e-70)
		tmp = ((F / sin(B)) * (((x * 2.0) + (2.0 + (F * F))) ^ -0.5)) - (x / B);
	elseif (F <= 5.8e-14)
		tmp = (F * (sqrt(0.5) * (1.0 / 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, -6.4e-15], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 4.9e-124], N[(N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 9e-70], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(x * 2.0), $MachinePrecision] + N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.8e-14], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -6.3999999999999999e-15

   1. Initial program 55.5%

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

   3. Simplified 65.9%

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

   5. Taylor expanded in x around 0 72.7%

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

      1. associate-*l/ 72.7%

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

      2. *-lft-identity 72.7%

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

      3. +-commutative 72.7%

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

      4. unpow2 72.7%

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

      5. fma-undefine 72.7%

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

   7. Simplified 72.7%

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

   8. Taylor expanded in F around -inf 98.3%

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

   if -6.3999999999999999e-15 < F < 4.89999999999999965e-124

   1. Initial program 80.4%

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

   3. Simplified 80.4%

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

   5. Taylor expanded in x around 0 99.7%

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

      1. associate-*l/ 99.7%

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

      2. *-lft-identity 99.7%

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

      3. +-commutative 99.7%

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

      4. unpow2 99.7%

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

      5. fma-undefine 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in B around 0 89.6%

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

      1. +-commutative 89.6%

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

      2. unpow2 89.6%

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

      3. fma-undefine 89.6%

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

   10. Simplified 89.6%

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

   11. Taylor expanded in F around 0 89.7%

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

   if 4.89999999999999965e-124 < F < 9.00000000000000044e-70

   1. Initial program 99.1%

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

   3. Taylor expanded in B around 0 88.6%

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

   if 9.00000000000000044e-70 < F < 5.8000000000000005e-14

   1. Initial program 56.7%

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

   3. Simplified 56.7%

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

   5. Taylor expanded in x around 0 99.8%

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

      1. associate-*l/ 99.6%

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

      2. *-lft-identity 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. fma-undefine 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in B around 0 86.7%

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

      1. +-commutative 86.7%

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

      2. unpow2 86.7%

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

      3. fma-undefine 86.7%

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

   10. Simplified 86.7%

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

   11. Taylor expanded in F around 0 86.7%

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

   if 5.8000000000000005e-14 < F

   1. Initial program 58.8%

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

   3. Simplified 68.4%

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

   5. Taylor expanded in x around 0 76.2%

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

      1. associate-*l/ 76.2%

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

      2. *-lft-identity 76.2%

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

      3. +-commutative 76.2%

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

      4. unpow2 76.2%

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

      5. fma-undefine 76.2%

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

   7. Simplified 76.2%

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

   8. Taylor expanded in F around inf 97.9%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6.4 \cdot 10^{-15}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 4.9 \cdot 10^{-124}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 9 \cdot 10^{-70}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(x \cdot 2 + \left(2 + F \cdot F\right)\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 5.8 \cdot 10^{-14}:\\ \;\;\;\;F \cdot \left(\sqrt{0.5} \cdot \frac{1}{B}\right) - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -6.4 \cdot 10^{-15}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 5.8 \cdot 10^{-14}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -6.4e-15)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 5.8e-14)
       (- (* F (/ (sqrt 0.5) B)) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -6.4e-15) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 5.8e-14) {
		tmp = (F * (sqrt(0.5) / B)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-6.4d-15)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 5.8d-14) then
        tmp = (f * (sqrt(0.5d0) / b)) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -6.4e-15) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 5.8e-14) {
		tmp = (F * (Math.sqrt(0.5) / B)) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -6.4e-15:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 5.8e-14:
		tmp = (F * (math.sqrt(0.5) / B)) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -6.4e-15)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 5.8e-14)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / B)) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -6.4e-15)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 5.8e-14)
		tmp = (F * (sqrt(0.5) / B)) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -6.4e-15], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 5.8e-14], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -6.3999999999999999e-15

   1. Initial program 55.5%

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

   3. Simplified 65.9%

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

   5. Taylor expanded in x around 0 72.7%

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

      1. associate-*l/ 72.7%

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

      2. *-lft-identity 72.7%

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

      3. +-commutative 72.7%

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

      4. unpow2 72.7%

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

      5. fma-undefine 72.7%

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

   7. Simplified 72.7%

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

   8. Taylor expanded in F around -inf 98.3%

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

   if -6.3999999999999999e-15 < F < 5.8000000000000005e-14

   1. Initial program 80.4%

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

   3. Simplified 80.5%

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

   5. Taylor expanded in x around 0 99.6%

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

      1. associate-*l/ 99.7%

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

      2. *-lft-identity 99.7%

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

      3. +-commutative 99.7%

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

      4. unpow2 99.7%

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

      5. fma-undefine 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in B around 0 85.6%

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

      1. +-commutative 85.6%

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

      2. unpow2 85.6%

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

      3. fma-undefine 85.6%

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

   10. Simplified 85.6%

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

   11. Taylor expanded in F around 0 85.6%

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

       1. associate-/l* 85.6%

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

   13. Simplified 85.6%

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

   if 5.8000000000000005e-14 < F

   1. Initial program 58.8%

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

   3. Simplified 68.4%

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

   5. Taylor expanded in x around 0 76.2%

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

      1. associate-*l/ 76.2%

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

      2. *-lft-identity 76.2%

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

      3. +-commutative 76.2%

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

      4. unpow2 76.2%

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

      5. fma-undefine 76.2%

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

   7. Simplified 76.2%

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

   8. Taylor expanded in F around inf 97.9%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6.4 \cdot 10^{-15}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 5.8 \cdot 10^{-14}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -6.4 \cdot 10^{-15}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 5.8 \cdot 10^{-14}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -6.4e-15)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 5.8e-14)
       (- (/ (* F (sqrt 0.5)) B) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -6.4e-15) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 5.8e-14) {
		tmp = ((F * sqrt(0.5)) / B) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-6.4d-15)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 5.8d-14) then
        tmp = ((f * sqrt(0.5d0)) / b) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -6.4e-15) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 5.8e-14) {
		tmp = ((F * Math.sqrt(0.5)) / B) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -6.4e-15:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 5.8e-14:
		tmp = ((F * math.sqrt(0.5)) / B) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -6.4e-15)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 5.8e-14)
		tmp = Float64(Float64(Float64(F * sqrt(0.5)) / B) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -6.4e-15)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 5.8e-14)
		tmp = ((F * sqrt(0.5)) / B) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -6.4e-15], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 5.8e-14], N[(N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / B), $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 < -6.3999999999999999e-15

   1. Initial program 55.5%

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

   3. Simplified 65.9%

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

   5. Taylor expanded in x around 0 72.7%

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

      1. associate-*l/ 72.7%

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

      2. *-lft-identity 72.7%

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

      3. +-commutative 72.7%

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

      4. unpow2 72.7%

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

      5. fma-undefine 72.7%

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

   7. Simplified 72.7%

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

   8. Taylor expanded in F around -inf 98.3%

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

   if -6.3999999999999999e-15 < F < 5.8000000000000005e-14

   1. Initial program 80.4%

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

   3. Simplified 80.5%

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

   5. Taylor expanded in x around 0 99.6%

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

      1. associate-*l/ 99.7%

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

      2. *-lft-identity 99.7%

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

      3. +-commutative 99.7%

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

      4. unpow2 99.7%

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

      5. fma-undefine 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in B around 0 85.6%

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

      1. +-commutative 85.6%

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

      2. unpow2 85.6%

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

      3. fma-undefine 85.6%

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

   10. Simplified 85.6%

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

   11. Taylor expanded in F around 0 85.6%

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

   if 5.8000000000000005e-14 < F

   1. Initial program 58.8%

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

   3. Simplified 68.4%

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

   5. Taylor expanded in x around 0 76.2%

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

      1. associate-*l/ 76.2%

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

      2. *-lft-identity 76.2%

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

      3. +-commutative 76.2%

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

      4. unpow2 76.2%

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

      5. fma-undefine 76.2%

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

   7. Simplified 76.2%

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

   8. Taylor expanded in F around inf 97.9%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6.4 \cdot 10^{-15}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 5.8 \cdot 10^{-14}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.9 \cdot 10^{-47}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1.6 \cdot 10^{-84}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.9e-47)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.6e-84) (/ x (- (tan B))) (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.9e-47) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.6e-84) {
		tmp = x / -tan(B);
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.9d-47)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.6d-84) then
        tmp = x / -tan(b)
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.9e-47) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.6e-84) {
		tmp = x / -Math.tan(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.9e-47:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.6e-84:
		tmp = x / -math.tan(B)
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.9e-47)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.6e-84)
		tmp = Float64(x / Float64(-tan(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.9e-47)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.6e-84)
		tmp = x / -tan(B);
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.9e-47], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.6e-84], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.90000000000000007e-47

   1. Initial program 60.9%

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

   3. Simplified 69.9%

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

   5. Taylor expanded in x around 0 75.9%

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

      1. associate-*l/ 76.0%

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

      2. *-lft-identity 76.0%

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

      3. +-commutative 76.0%

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

      4. unpow2 76.0%

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

      5. fma-undefine 76.0%

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

   7. Simplified 76.0%

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

   8. Taylor expanded in F around -inf 90.1%

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

   if -1.90000000000000007e-47 < F < 1.6e-84

   1. Initial program 80.0%

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

      1. div-inv 80.1%

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

   4. Applied egg-rr 80.1%

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

   5. Taylor expanded in F around -inf 48.3%

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

   6. Taylor expanded in x around inf 76.5%

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

      1. mul-1-neg 76.5%

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

      2. associate-/l* 76.5%

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

      3. distribute-lft-neg-in 76.5%

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

   8. Simplified 76.5%

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

      1. distribute-lft-neg-out 76.5%

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

      2. clear-num 76.4%

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

      3. tan-quot 76.5%

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

      4. div-inv 76.7%

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

      5. distribute-neg-frac2 76.7%

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

   10. Applied egg-rr 76.7%

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

   if 1.6e-84 < F

   1. Initial program 60.1%

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

   3. Simplified 68.6%

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

   5. Taylor expanded in x around 0 79.0%

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

      1. associate-*l/ 79.0%

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

      2. *-lft-identity 79.0%

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

      3. +-commutative 79.0%

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

      4. unpow2 79.0%

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

      5. fma-undefine 79.0%

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

   7. Simplified 79.0%

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

   8. Taylor expanded in F around inf 91.4%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.9 \cdot 10^{-47}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.6 \cdot 10^{-84}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.8 \cdot 10^{-47}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{-53}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 1.95 \cdot 10^{+132}:\\ \;\;\;\;\frac{1}{F \cdot \frac{\sin B}{F}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.8e-47)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 4.5e-53)
       (/ x (- (tan B)))
       (if (<= F 1.95e+132)
         (- (/ 1.0 (* F (/ (sin B) F))) (/ 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 <= -1.8e-47) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 4.5e-53) {
		tmp = x / -tan(B);
	} else if (F <= 1.95e+132) {
		tmp = (1.0 / (F * (sin(B) / F))) - (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 <= (-1.8d-47)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 4.5d-53) then
        tmp = x / -tan(b)
    else if (f <= 1.95d+132) then
        tmp = (1.0d0 / (f * (sin(b) / f))) - (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 <= -1.8e-47) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 4.5e-53) {
		tmp = x / -Math.tan(B);
	} else if (F <= 1.95e+132) {
		tmp = (1.0 / (F * (Math.sin(B) / F))) - (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 <= -1.8e-47:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 4.5e-53:
		tmp = x / -math.tan(B)
	elif F <= 1.95e+132:
		tmp = (1.0 / (F * (math.sin(B) / F))) - (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 <= -1.8e-47)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 4.5e-53)
		tmp = Float64(x / Float64(-tan(B)));
	elseif (F <= 1.95e+132)
		tmp = Float64(Float64(1.0 / Float64(F * Float64(sin(B) / F))) - 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 <= -1.8e-47)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 4.5e-53)
		tmp = x / -tan(B);
	elseif (F <= 1.95e+132)
		tmp = (1.0 / (F * (sin(B) / F))) - (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, -1.8e-47], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 4.5e-53], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 1.95e+132], N[(N[(1.0 / N[(F * N[(N[Sin[B], $MachinePrecision] / F), $MachinePrecision]), $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 < -1.79999999999999995e-47

   1. Initial program 60.9%

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

   3. Simplified 69.9%

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

   5. Taylor expanded in x around 0 75.9%

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

      1. associate-*l/ 76.0%

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

      2. *-lft-identity 76.0%

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

      3. +-commutative 76.0%

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

      4. unpow2 76.0%

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

      5. fma-undefine 76.0%

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

   7. Simplified 76.0%

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

   8. Taylor expanded in F around -inf 90.1%

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

   if -1.79999999999999995e-47 < F < 4.49999999999999985e-53

   1. Initial program 79.0%

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

      1. div-inv 79.2%

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

   4. Applied egg-rr 79.2%

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

   5. Taylor expanded in F around -inf 48.9%

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

   6. Taylor expanded in x around inf 75.9%

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

      1. mul-1-neg 75.9%

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

      2. associate-/l* 75.8%

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

      3. distribute-lft-neg-in 75.8%

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

   8. Simplified 75.8%

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

      1. distribute-lft-neg-out 75.8%

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

      2. clear-num 75.7%

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

      3. tan-quot 75.8%

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

      4. div-inv 76.0%

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

      5. distribute-neg-frac2 76.0%

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

   10. Applied egg-rr 76.0%

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

   if 4.49999999999999985e-53 < F < 1.95000000000000001e132

   1. Initial program 78.1%

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

   3. Taylor expanded in F around inf 80.8%

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

      1. clear-num 80.7%

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

      2. frac-times 80.8%

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

      3. metadata-eval 80.8%

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

   5. Applied egg-rr 80.8%

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

   6. Taylor expanded in B around 0 64.3%

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

   if 1.95000000000000001e132 < F

   1. Initial program 43.9%

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

   3. Simplified 56.4%

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

   5. Taylor expanded in x around 0 58.8%

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

      1. associate-*l/ 58.8%

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

      2. *-lft-identity 58.8%

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

      3. +-commutative 58.8%

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

      4. unpow2 58.8%

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

      5. fma-undefine 58.8%

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

   7. Simplified 58.8%

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

   8. Taylor expanded in B around 0 56.5%

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

      1. +-commutative 56.5%

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

      2. unpow2 56.5%

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

      3. fma-undefine 56.5%

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

   10. Simplified 56.5%

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

   11. Taylor expanded in F around inf 75.4%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.8 \cdot 10^{-47}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{-53}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 1.95 \cdot 10^{+132}:\\ \;\;\;\;\frac{1}{F \cdot \frac{\sin B}{F}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 72.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -7000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{-53}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 2.7 \cdot 10^{+112}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \frac{1}{F} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -7000000.0)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 4.5e-53)
     (/ x (- (tan B)))
     (if (<= F 2.7e+112)
       (- (* (/ F (sin B)) (/ 1.0 F)) (/ x B))
       (- (/ 1.0 B) (/ x (tan B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -7000000.0) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 4.5e-53) {
		tmp = x / -tan(B);
	} else if (F <= 2.7e+112) {
		tmp = ((F / sin(B)) * (1.0 / F)) - (x / B);
	} else {
		tmp = (1.0 / B) - (x / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-7000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 4.5d-53) then
        tmp = x / -tan(b)
    else if (f <= 2.7d+112) then
        tmp = ((f / sin(b)) * (1.0d0 / f)) - (x / b)
    else
        tmp = (1.0d0 / b) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -7000000.0) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 4.5e-53) {
		tmp = x / -Math.tan(B);
	} else if (F <= 2.7e+112) {
		tmp = ((F / Math.sin(B)) * (1.0 / F)) - (x / B);
	} else {
		tmp = (1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -7000000.0:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 4.5e-53:
		tmp = x / -math.tan(B)
	elif F <= 2.7e+112:
		tmp = ((F / math.sin(B)) * (1.0 / F)) - (x / B)
	else:
		tmp = (1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -7000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 4.5e-53)
		tmp = Float64(x / Float64(-tan(B)));
	elseif (F <= 2.7e+112)
		tmp = Float64(Float64(Float64(F / sin(B)) * Float64(1.0 / F)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -7000000.0)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 4.5e-53)
		tmp = x / -tan(B);
	elseif (F <= 2.7e+112)
		tmp = ((F / sin(B)) * (1.0 / F)) - (x / B);
	else
		tmp = (1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -7000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.5e-53], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 2.7e+112], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(1.0 / F), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -7e6

   1. Initial program 55.9%

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

   3. Taylor expanded in F around -inf 99.7%

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

   4. Taylor expanded in B around 0 84.1%

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

   if -7e6 < F < 4.49999999999999985e-53

   1. Initial program 79.4%

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

      1. div-inv 79.6%

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

   4. Applied egg-rr 79.6%

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

   5. Taylor expanded in F around -inf 48.6%

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

   6. Taylor expanded in x around inf 72.7%

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

      1. mul-1-neg 72.7%

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

      2. associate-/l* 72.6%

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

      3. distribute-lft-neg-in 72.6%

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

   8. Simplified 72.6%

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

      1. distribute-lft-neg-out 72.6%

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

      2. clear-num 72.5%

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

      3. tan-quot 72.6%

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

      4. div-inv 72.8%

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

      5. distribute-neg-frac2 72.8%

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

   10. Applied egg-rr 72.8%

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

   if 4.49999999999999985e-53 < F < 2.7000000000000001e112

   1. Initial program 79.0%

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

   3. Taylor expanded in F around inf 81.9%

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

   4. Taylor expanded in B around 0 63.9%

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

   if 2.7000000000000001e112 < F

   1. Initial program 45.5%

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

   3. Simplified 59.2%

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

   5. Taylor expanded in x around 0 61.5%

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

      1. associate-*l/ 61.5%

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

      2. *-lft-identity 61.5%

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

      3. +-commutative 61.5%

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

      4. unpow2 61.5%

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

      5. fma-undefine 61.5%

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

   7. Simplified 61.5%

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

   8. Taylor expanded in B around 0 57.3%

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

      1. +-commutative 57.3%

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

      2. unpow2 57.3%

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

      3. fma-undefine 57.3%

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

   10. Simplified 57.3%

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

   11. Taylor expanded in F around inf 75.0%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{-53}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 2.7 \cdot 10^{+112}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \frac{1}{F} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 72.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1300000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{-53}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 2.75 \cdot 10^{+132}:\\ \;\;\;\;\frac{1}{F \cdot \frac{\sin B}{F}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1300000.0)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 4.5e-53)
     (/ x (- (tan B)))
     (if (<= F 2.75e+132)
       (- (/ 1.0 (* F (/ (sin B) F))) (/ x B))
       (- (/ 1.0 B) (/ x (tan B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1300000.0) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 4.5e-53) {
		tmp = x / -tan(B);
	} else if (F <= 2.75e+132) {
		tmp = (1.0 / (F * (sin(B) / F))) - (x / B);
	} else {
		tmp = (1.0 / B) - (x / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1300000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 4.5d-53) then
        tmp = x / -tan(b)
    else if (f <= 2.75d+132) then
        tmp = (1.0d0 / (f * (sin(b) / f))) - (x / b)
    else
        tmp = (1.0d0 / b) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1300000.0) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 4.5e-53) {
		tmp = x / -Math.tan(B);
	} else if (F <= 2.75e+132) {
		tmp = (1.0 / (F * (Math.sin(B) / F))) - (x / B);
	} else {
		tmp = (1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1300000.0:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 4.5e-53:
		tmp = x / -math.tan(B)
	elif F <= 2.75e+132:
		tmp = (1.0 / (F * (math.sin(B) / F))) - (x / B)
	else:
		tmp = (1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1300000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 4.5e-53)
		tmp = Float64(x / Float64(-tan(B)));
	elseif (F <= 2.75e+132)
		tmp = Float64(Float64(1.0 / Float64(F * Float64(sin(B) / F))) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1300000.0)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 4.5e-53)
		tmp = x / -tan(B);
	elseif (F <= 2.75e+132)
		tmp = (1.0 / (F * (sin(B) / F))) - (x / B);
	else
		tmp = (1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1300000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.5e-53], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 2.75e+132], N[(N[(1.0 / N[(F * N[(N[Sin[B], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.3e6

   1. Initial program 55.9%

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

   3. Taylor expanded in F around -inf 99.7%

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

   4. Taylor expanded in B around 0 84.1%

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

   if -1.3e6 < F < 4.49999999999999985e-53

   1. Initial program 79.4%

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

      1. div-inv 79.6%

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

   4. Applied egg-rr 79.6%

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

   5. Taylor expanded in F around -inf 48.6%

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

   6. Taylor expanded in x around inf 72.7%

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

      1. mul-1-neg 72.7%

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

      2. associate-/l* 72.6%

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

      3. distribute-lft-neg-in 72.6%

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

   8. Simplified 72.6%

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

      1. distribute-lft-neg-out 72.6%

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

      2. clear-num 72.5%

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

      3. tan-quot 72.6%

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

      4. div-inv 72.8%

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

      5. distribute-neg-frac2 72.8%

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

   10. Applied egg-rr 72.8%

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

   if 4.49999999999999985e-53 < F < 2.75e132

   1. Initial program 78.1%

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

   3. Taylor expanded in F around inf 80.8%

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

      1. clear-num 80.7%

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

      2. frac-times 80.8%

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

      3. metadata-eval 80.8%

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

   5. Applied egg-rr 80.8%

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

   6. Taylor expanded in B around 0 64.3%

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

   if 2.75e132 < F

   1. Initial program 43.9%

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

   3. Simplified 56.4%

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

   5. Taylor expanded in x around 0 58.8%

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

      1. associate-*l/ 58.8%

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

      2. *-lft-identity 58.8%

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

      3. +-commutative 58.8%

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

      4. unpow2 58.8%

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

      5. fma-undefine 58.8%

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

   7. Simplified 58.8%

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

   8. Taylor expanded in B around 0 56.5%

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

      1. +-commutative 56.5%

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

      2. unpow2 56.5%

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

      3. fma-undefine 56.5%

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

   10. Simplified 56.5%

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

   11. Taylor expanded in F around inf 75.4%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1300000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{-53}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 2.75 \cdot 10^{+132}:\\ \;\;\;\;\frac{1}{F \cdot \frac{\sin B}{F}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 68.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -5.4 \cdot 10^{-48}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.55 \cdot 10^{+107}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 2.4 \cdot 10^{+132}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -5.4e-48)
   (- (/ -1.0 B) (/ x (tan B)))
   (if (<= F 2.55e+107)
     (/ x (- (tan B)))
     (if (<= F 2.4e+132) (/ 1.0 (sin B)) (* x (/ -1.0 (tan B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -5.4e-48) {
		tmp = (-1.0 / B) - (x / tan(B));
	} else if (F <= 2.55e+107) {
		tmp = x / -tan(B);
	} else if (F <= 2.4e+132) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = x * (-1.0 / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-5.4d-48)) then
        tmp = ((-1.0d0) / b) - (x / tan(b))
    else if (f <= 2.55d+107) then
        tmp = x / -tan(b)
    else if (f <= 2.4d+132) then
        tmp = 1.0d0 / sin(b)
    else
        tmp = x * ((-1.0d0) / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -5.4e-48) {
		tmp = (-1.0 / B) - (x / Math.tan(B));
	} else if (F <= 2.55e+107) {
		tmp = x / -Math.tan(B);
	} else if (F <= 2.4e+132) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = x * (-1.0 / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -5.4e-48:
		tmp = (-1.0 / B) - (x / math.tan(B))
	elif F <= 2.55e+107:
		tmp = x / -math.tan(B)
	elif F <= 2.4e+132:
		tmp = 1.0 / math.sin(B)
	else:
		tmp = x * (-1.0 / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -5.4e-48)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / tan(B)));
	elseif (F <= 2.55e+107)
		tmp = Float64(x / Float64(-tan(B)));
	elseif (F <= 2.4e+132)
		tmp = Float64(1.0 / sin(B));
	else
		tmp = Float64(x * Float64(-1.0 / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -5.4e-48)
		tmp = (-1.0 / B) - (x / tan(B));
	elseif (F <= 2.55e+107)
		tmp = x / -tan(B);
	elseif (F <= 2.4e+132)
		tmp = 1.0 / sin(B);
	else
		tmp = x * (-1.0 / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -5.4e-48], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.55e+107], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 2.4e+132], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -5.40000000000000023e-48

   1. Initial program 60.9%

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

   3. Simplified 69.9%

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

   5. Taylor expanded in x around 0 75.9%

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

      1. associate-*l/ 76.0%

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

      2. *-lft-identity 76.0%

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

      3. +-commutative 76.0%

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

      4. unpow2 76.0%

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

      5. fma-undefine 76.0%

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

   7. Simplified 76.0%

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

   8. Taylor expanded in B around 0 60.5%

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

      1. +-commutative 60.5%

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

      2. unpow2 60.5%

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

      3. fma-undefine 60.5%

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

   10. Simplified 60.5%

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

   11. Taylor expanded in F around -inf 68.2%

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

   if -5.40000000000000023e-48 < F < 2.5500000000000001e107

   1. Initial program 78.6%

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

      1. div-inv 78.8%

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

   4. Applied egg-rr 78.8%

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

   5. Taylor expanded in F around -inf 47.5%

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

   6. Taylor expanded in x around inf 69.6%

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

      1. mul-1-neg 69.6%

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

      2. associate-/l* 69.5%

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

      3. distribute-lft-neg-in 69.5%

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

   8. Simplified 69.5%

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

      1. distribute-lft-neg-out 69.5%

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

      2. clear-num 69.4%

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

      3. tan-quot 69.6%

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

      4. div-inv 69.7%

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

      5. distribute-neg-frac2 69.7%

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

   10. Applied egg-rr 69.7%

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

   if 2.5500000000000001e107 < F < 2.4000000000000001e132

   1. Initial program 83.9%

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

   3. Taylor expanded in F around -inf 1.3%

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

      1. div-inv 1.3%

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

      2. metadata-eval 1.3%

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

      3. rgt-mult-inverse 1.3%

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

      4. associate-*l/ 1.2%

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

      5. add-sqr-sqrt 0.4%

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

      6. sqrt-unprod 51.0%

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

      7. *-commutative 51.0%

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

      8. associate-*r/ 51.2%

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

      9. *-commutative 51.2%

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

      10. rgt-mult-inverse 51.2%

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

      11. associate-*l/ 51.2%

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

      12. rgt-mult-inverse 51.2%

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

      13. frac-times 51.2%

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

      14. metadata-eval 51.2%

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

      15. metadata-eval 51.2%

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

      16. frac-times 51.2%

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

      17. sqrt-unprod 66.1%

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

      18. add-sqr-sqrt 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. *-lft-identity 99.7%

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

      2. +-commutative 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around 0 99.7%

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

   if 2.4000000000000001e132 < F

   1. Initial program 43.9%

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

      1. div-inv 43.9%

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

   4. Applied egg-rr 43.9%

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

   5. Taylor expanded in F around -inf 43.0%

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

   6. Taylor expanded in x around inf 54.2%

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

      1. mul-1-neg 54.2%

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

      2. associate-/l* 54.4%

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

      3. distribute-lft-neg-in 54.4%

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

   8. Simplified 54.4%

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

      1. *-un-lft-identity 54.4%

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

      2. clear-num 54.3%

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

      3. tan-quot 54.2%

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

   10. Applied egg-rr 54.2%

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

       1. *-lft-identity 54.2%

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

   12. Simplified 54.2%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5.4 \cdot 10^{-48}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.55 \cdot 10^{+107}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 2.4 \cdot 10^{+132}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 73.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -6 \cdot 10^{-48}:\\ \;\;\;\;\frac{-1}{B} - t\_0\\ \mathbf{elif}\;F \leq 1.02 \cdot 10^{-170}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -6e-48)
     (- (/ -1.0 B) t_0)
     (if (<= F 1.02e-170) (/ x (- (tan 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 <= -6e-48) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= 1.02e-170) {
		tmp = x / -tan(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 <= (-6d-48)) then
        tmp = ((-1.0d0) / b) - t_0
    else if (f <= 1.02d-170) then
        tmp = x / -tan(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 <= -6e-48) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= 1.02e-170) {
		tmp = x / -Math.tan(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 <= -6e-48:
		tmp = (-1.0 / B) - t_0
	elif F <= 1.02e-170:
		tmp = x / -math.tan(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 <= -6e-48)
		tmp = Float64(Float64(-1.0 / B) - t_0);
	elseif (F <= 1.02e-170)
		tmp = Float64(x / Float64(-tan(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 <= -6e-48)
		tmp = (-1.0 / B) - t_0;
	elseif (F <= 1.02e-170)
		tmp = x / -tan(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, -6e-48], N[(N[(-1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.02e-170], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.9999999999999998e-48

   1. Initial program 60.9%

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

   3. Simplified 69.9%

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

   5. Taylor expanded in x around 0 75.9%

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

      1. associate-*l/ 76.0%

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

      2. *-lft-identity 76.0%

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

      3. +-commutative 76.0%

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

      4. unpow2 76.0%

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

      5. fma-undefine 76.0%

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

   7. Simplified 76.0%

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

   8. Taylor expanded in B around 0 60.5%

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

      1. +-commutative 60.5%

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

      2. unpow2 60.5%

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

      3. fma-undefine 60.5%

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

   10. Simplified 60.5%

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

   11. Taylor expanded in F around -inf 68.2%

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

   if -5.9999999999999998e-48 < F < 1.02e-170

   1. Initial program 78.9%

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

      1. div-inv 79.1%

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

   4. Applied egg-rr 79.1%

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

   5. Taylor expanded in F around -inf 48.4%

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

   6. Taylor expanded in x around inf 80.0%

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

      1. mul-1-neg 80.0%

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

      2. associate-/l* 79.9%

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

      3. distribute-lft-neg-in 79.9%

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

   8. Simplified 79.9%

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

      1. distribute-lft-neg-out 79.9%

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

      2. clear-num 79.9%

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

      3. tan-quot 80.0%

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

      4. div-inv 80.2%

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

      5. distribute-neg-frac2 80.2%

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

   10. Applied egg-rr 80.2%

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

   if 1.02e-170 < F

   1. Initial program 64.1%

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

   3. Simplified 71.3%

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

   5. Taylor expanded in x around 0 82.2%

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

      1. associate-*l/ 82.1%

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

      2. *-lft-identity 82.1%

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

      3. +-commutative 82.1%

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

      4. unpow2 82.1%

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

      5. fma-undefine 82.1%

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

   7. Simplified 82.1%

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

   8. Taylor expanded in B around 0 60.3%

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

      1. +-commutative 60.3%

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

      2. unpow2 60.3%

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

      3. fma-undefine 60.3%

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

   10. Simplified 60.3%

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

   11. Taylor expanded in F around inf 62.9%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6 \cdot 10^{-48}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.02 \cdot 10^{-170}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 73.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -105000000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{-168}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -105000000000.0)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 1.8e-168) (/ x (- (tan B))) (- (/ 1.0 B) (/ x (tan B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -105000000000.0) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 1.8e-168) {
		tmp = x / -tan(B);
	} else {
		tmp = (1.0 / B) - (x / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-105000000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 1.8d-168) then
        tmp = x / -tan(b)
    else
        tmp = (1.0d0 / b) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -105000000000.0) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 1.8e-168) {
		tmp = x / -Math.tan(B);
	} else {
		tmp = (1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -105000000000.0:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 1.8e-168:
		tmp = x / -math.tan(B)
	else:
		tmp = (1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -105000000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 1.8e-168)
		tmp = Float64(x / Float64(-tan(B)));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -105000000000.0)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 1.8e-168)
		tmp = x / -tan(B);
	else
		tmp = (1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -105000000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.8e-168], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.05e11

   1. Initial program 55.9%

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

   3. Taylor expanded in F around -inf 99.7%

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

   4. Taylor expanded in B around 0 84.1%

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

   if -1.05e11 < F < 1.7999999999999999e-168

   1. Initial program 79.4%

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

      1. div-inv 79.6%

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

   4. Applied egg-rr 79.6%

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

   5. Taylor expanded in F around -inf 48.2%

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

   6. Taylor expanded in x around inf 75.7%

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

      1. mul-1-neg 75.7%

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

      2. associate-/l* 75.7%

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

      3. distribute-lft-neg-in 75.7%

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

   8. Simplified 75.7%

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

      1. distribute-lft-neg-out 75.7%

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

      2. clear-num 75.6%

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

      3. tan-quot 75.7%

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

      4. div-inv 75.9%

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

      5. distribute-neg-frac2 75.9%

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

   10. Applied egg-rr 75.9%

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

   if 1.7999999999999999e-168 < F

   1. Initial program 64.1%

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

   3. Simplified 71.3%

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

   5. Taylor expanded in x around 0 82.2%

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

      1. associate-*l/ 82.1%

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

      2. *-lft-identity 82.1%

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

      3. +-commutative 82.1%

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

      4. unpow2 82.1%

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

      5. fma-undefine 82.1%

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

   7. Simplified 82.1%

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

   8. Taylor expanded in B around 0 60.3%

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

      1. +-commutative 60.3%

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

      2. unpow2 60.3%

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

      3. fma-undefine 60.3%

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

   10. Simplified 60.3%

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

   11. Taylor expanded in F around inf 62.9%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -105000000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{-168}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 44.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -5.5 \cdot 10^{-53}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 24:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -5.5e-53)
   (/ (- -1.0 x) B)
   (if (<= F 24.0) (/ x (- B)) (/ 1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -5.5e-53) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 24.0) {
		tmp = x / -B;
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-5.5d-53)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 24.0d0) then
        tmp = x / -b
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -5.5e-53) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 24.0) {
		tmp = x / -B;
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -5.5e-53:
		tmp = (-1.0 - x) / B
	elif F <= 24.0:
		tmp = x / -B
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -5.5e-53)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 24.0)
		tmp = Float64(x / Float64(-B));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -5.5e-53)
		tmp = (-1.0 - x) / B;
	elseif (F <= 24.0)
		tmp = x / -B;
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -5.5e-53], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 24.0], N[(x / (-B)), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.50000000000000023e-53

   1. Initial program 62.0%

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

   3. Taylor expanded in F around -inf 87.4%

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

   4. Taylor expanded in B around 0 49.0%

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

      1. mul-1-neg 49.0%

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

      2. distribute-neg-frac2 49.0%

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

   6. Simplified 49.0%

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

   7. Taylor expanded in B around 0 49.0%

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

      1. associate-*r/ 49.0%

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

      2. neg-mul-1 49.0%

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

      3. distribute-neg-in 49.0%

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

      4. metadata-eval 49.0%

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

      5. unsub-neg 49.0%

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

   9. Simplified 49.0%

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

   if -5.50000000000000023e-53 < F < 24

   1. Initial program 78.9%

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

   3. Taylor expanded in F around -inf 48.8%

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

   4. Taylor expanded in B around 0 30.8%

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

      1. mul-1-neg 30.8%

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

      2. distribute-neg-frac2 30.8%

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

   6. Simplified 30.8%

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

   7. Taylor expanded in x around inf 38.4%

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

      1. mul-1-neg 38.4%

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

      2. distribute-neg-frac2 38.4%

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

   9. Simplified 38.4%

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

   if 24 < F

   1. Initial program 58.2%

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

   3. Taylor expanded in F around -inf 47.0%

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

      1. div-inv 47.0%

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

      2. metadata-eval 47.0%

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

      3. rgt-mult-inverse 47.0%

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

      4. associate-*l/ 40.0%

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

      5. add-sqr-sqrt 19.4%

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

      6. sqrt-unprod 63.4%

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

      7. *-commutative 63.4%

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

      8. associate-*r/ 63.4%

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

      9. *-commutative 63.4%

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

      10. rgt-mult-inverse 63.4%

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

      11. associate-*l/ 64.8%

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

      12. rgt-mult-inverse 64.8%

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

      13. frac-times 64.7%

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

      14. metadata-eval 64.7%

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

      15. metadata-eval 64.7%

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

      16. frac-times 64.8%

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

      17. sqrt-unprod 54.7%

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

      18. add-sqr-sqrt 97.8%

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

   5. Applied egg-rr 47.3%

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

      1. *-lft-identity 47.3%

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

      2. +-commutative 47.3%

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

   7. Simplified 47.3%

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

   8. Taylor expanded in x around 0 49.3%

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

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5.5 \cdot 10^{-53}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 24:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 47.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 2.8 \cdot 10^{-51}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 2.8e-51) (/ (- -1.0 x) B) (/ x (- (tan B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 2.8e-51) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = x / -tan(B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (b <= 2.8d-51) then
        tmp = ((-1.0d0) - x) / b
    else
        tmp = x / -tan(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (B <= 2.8e-51) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = x / -Math.tan(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if B <= 2.8e-51:
		tmp = (-1.0 - x) / B
	else:
		tmp = x / -math.tan(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 2.8e-51)
		tmp = Float64(Float64(-1.0 - x) / B);
	else
		tmp = Float64(x / Float64(-tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (B <= 2.8e-51)
		tmp = (-1.0 - x) / B;
	else
		tmp = x / -tan(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 2.8e-51], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 2.8e-51

   1. Initial program 64.7%

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

   3. Taylor expanded in F around -inf 62.7%

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

   4. Taylor expanded in B around 0 43.9%

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

      1. mul-1-neg 43.9%

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

      2. distribute-neg-frac2 43.9%

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

   6. Simplified 43.9%

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

   7. Taylor expanded in B around 0 43.9%

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

      1. associate-*r/ 43.9%

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

      2. neg-mul-1 43.9%

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

      3. distribute-neg-in 43.9%

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

      4. metadata-eval 43.9%

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

      5. unsub-neg 43.9%

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

   9. Simplified 43.9%

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

   if 2.8e-51 < B

   1. Initial program 76.5%

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

      1. div-inv 76.5%

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

   4. Applied egg-rr 76.5%

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

   5. Taylor expanded in F around -inf 50.2%

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

   6. Taylor expanded in x around inf 58.6%

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

      1. mul-1-neg 58.6%

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

      2. associate-/l* 58.6%

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

      3. distribute-lft-neg-in 58.6%

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

   8. Simplified 58.6%

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

      1. distribute-lft-neg-out 58.6%

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

      2. clear-num 58.6%

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

      3. tan-quot 58.7%

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

      4. div-inv 58.8%

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

      5. distribute-neg-frac2 58.8%

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

   10. Applied egg-rr 58.8%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.8 \cdot 10^{-51}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 43.7% accurate, 21.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.4 \cdot 10^{-54}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{-168}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.4e-54)
   (/ (- -1.0 x) B)
   (if (<= F 1.8e-168) (/ x (- B)) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.4e-54) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.8e-168) {
		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-54)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 1.8d-168) 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-54) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.8e-168) {
		tmp = x / -B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.4e-54:
		tmp = (-1.0 - x) / B
	elif F <= 1.8e-168:
		tmp = x / -B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.4e-54)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 1.8e-168)
		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-54)
		tmp = (-1.0 - x) / B;
	elseif (F <= 1.8e-168)
		tmp = x / -B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.4e-54], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.8e-168], N[(x / (-B)), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.4000000000000001e-54

   1. Initial program 62.0%

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

   3. Taylor expanded in F around -inf 87.4%

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

   4. Taylor expanded in B around 0 49.0%

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

      1. mul-1-neg 49.0%

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

      2. distribute-neg-frac2 49.0%

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

   6. Simplified 49.0%

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

   7. Taylor expanded in B around 0 49.0%

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

      1. associate-*r/ 49.0%

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

      2. neg-mul-1 49.0%

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

      3. distribute-neg-in 49.0%

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

      4. metadata-eval 49.0%

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

      5. unsub-neg 49.0%

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

   9. Simplified 49.0%

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

   if -1.4000000000000001e-54 < F < 1.7999999999999999e-168

   1. Initial program 78.5%

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

   3. Taylor expanded in F around -inf 49.3%

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

   4. Taylor expanded in B around 0 31.2%

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

      1. mul-1-neg 31.2%

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

      2. distribute-neg-frac2 31.2%

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

   6. Simplified 31.2%

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

   7. Taylor expanded in x around inf 40.9%

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

      1. mul-1-neg 40.9%

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

      2. distribute-neg-frac2 40.9%

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

   9. Simplified 40.9%

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

   if 1.7999999999999999e-168 < F

   1. Initial program 64.1%

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

   3. Taylor expanded in F around -inf 47.0%

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

      1. distribute-rgt-neg-in 47.0%

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

      2. fma-define 47.0%

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

      3. distribute-neg-frac 47.0%

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

      4. metadata-eval 47.0%

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

      5. add-sqr-sqrt 22.3%

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

      6. sqrt-unprod 55.5%

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

      7. frac-times 55.4%

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

      8. metadata-eval 55.4%

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

      9. metadata-eval 55.4%

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

      10. frac-times 55.5%

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

      11. rgt-mult-inverse 55.4%

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

      12. associate-*l/ 55.4%

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

      13. rgt-mult-inverse 55.4%

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

      14. associate-*l/ 55.4%

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

      15. sqrt-unprod 37.2%

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

      16. add-sqr-sqrt 75.1%

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

      17. associate-*l/ 84.8%

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

      18. rgt-mult-inverse 84.9%

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

   5. Applied egg-rr 84.9%

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

   6. Taylor expanded in B around 0 39.5%

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

      1. neg-mul-1 39.5%

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

      2. unsub-neg 39.5%

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

   8. Simplified 39.5%

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

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.4 \cdot 10^{-54}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{-168}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 38.0% accurate, 32.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.6 \cdot 10^{-52}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.6e-52) (/ (- -1.0 x) B) (/ x (- B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.6e-52) {
		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 <= (-1.6d-52)) 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 <= -1.6e-52) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = x / -B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.6e-52:
		tmp = (-1.0 - x) / B
	else:
		tmp = x / -B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.6e-52)
		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 <= -1.6e-52)
		tmp = (-1.0 - x) / B;
	else
		tmp = x / -B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.6e-52], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[(x / (-B)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -1.60000000000000005e-52

   1. Initial program 62.0%

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

   3. Taylor expanded in F around -inf 87.4%

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

   4. Taylor expanded in B around 0 49.0%

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

      1. mul-1-neg 49.0%

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

      2. distribute-neg-frac2 49.0%

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

   6. Simplified 49.0%

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

   7. Taylor expanded in B around 0 49.0%

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

      1. associate-*r/ 49.0%

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

      2. neg-mul-1 49.0%

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

      3. distribute-neg-in 49.0%

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

      4. metadata-eval 49.0%

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

      5. unsub-neg 49.0%

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

   9. Simplified 49.0%

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

   if -1.60000000000000005e-52 < F

   1. Initial program 71.0%

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

   3. Taylor expanded in F around -inf 48.1%

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

   4. Taylor expanded in B around 0 28.3%

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

      1. mul-1-neg 28.3%

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

      2. distribute-neg-frac2 28.3%

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

   6. Simplified 28.3%

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

   7. Taylor expanded in x around inf 33.1%

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

      1. mul-1-neg 33.1%

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

      2. distribute-neg-frac2 33.1%

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

   9. Simplified 33.1%

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

4. Final simplification 37.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.6 \cdot 10^{-52}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 31.9% accurate, 81.0× speedup

Math

\[\begin{array}{l} \\ \frac{x}{-B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (/ x (- B)))

C

double code(double F, double B, double x) {
	return x / -B;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = x / -b
end function

Java

public static double code(double F, double B, double x) {
	return x / -B;
}

Python

def code(F, B, x):
	return x / -B

Julia

function code(F, B, x)
	return Float64(x / Float64(-B))
end

MATLAB

function tmp = code(F, B, x)
	tmp = x / -B;
end

Wolfram

code[F_, B_, x_] := N[(x / (-B)), $MachinePrecision]

Derivation

1. Initial program 68.6%

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

3. Taylor expanded in F around -inf 58.5%

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

4. Taylor expanded in B around 0 33.8%

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

   1. mul-1-neg 33.8%

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

   2. distribute-neg-frac2 33.8%

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

6. Simplified 33.8%

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

7. Taylor expanded in x around inf 31.6%

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

   1. mul-1-neg 31.6%

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

   2. distribute-neg-frac2 31.6%

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

9. Simplified 31.6%

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

10. Final simplification 31.6%

    \[\leadsto \frac{x}{-B} \]
11. Add Preprocessing

Alternative 20: 9.3% 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 68.6%

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

3. Simplified 73.7%

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

5. Taylor expanded in x around 0 86.9%

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

   1. associate-*l/ 86.9%

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

   2. *-lft-identity 86.9%

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

   3. +-commutative 86.9%

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

   4. unpow2 86.9%

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

   5. fma-undefine 86.9%

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

7. Simplified 86.9%

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

8. Taylor expanded in B around 0 71.3%

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

   1. +-commutative 71.3%

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

   2. unpow2 71.3%

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

   3. fma-undefine 71.3%

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

10. Simplified 71.3%

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

11. Taylor expanded in F around inf 57.6%

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

12. Taylor expanded in x around 0 8.2%

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

13. Final simplification 8.2%

    \[\leadsto \frac{1}{B} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024067 
(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))))))