VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 9.3s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sin B} - \frac{x}{\tan B} \end{array} \]

FPCore

(FPCore (B x) :precision binary64 (- (/ 1.0 (sin B)) (/ x (tan B))))

C

double code(double B, double x) {
	return (1.0 / sin(B)) - (x / tan(B));
}

Fortran

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

Java

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

Python

def code(B, x):
	return (1.0 / math.sin(B)) - (x / math.tan(B))

Julia

function code(B, x)
	return Float64(Float64(1.0 / sin(B)) - Float64(x / tan(B)))
end

MATLAB

function tmp = code(B, x)
	tmp = (1.0 / sin(B)) - (x / tan(B));
end

Wolfram

code[B_, x_] := N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

   1. distribute-lft-neg-in 99.7%

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

   2. +-commutative 99.7%

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

   3. *-commutative 99.7%

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

   4. remove-double-neg 99.7%

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

   5. distribute-frac-neg2 99.7%

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

   6. tan-neg 99.7%

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

   7. cancel-sign-sub-inv 99.7%

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

   8. *-commutative 99.7%

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

   9. associate-*r/ 99.8%

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

   10. *-rgt-identity 99.8%

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

   11. tan-neg 99.8%

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

   12. distribute-neg-frac2 99.8%

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

   13. distribute-neg-frac 99.8%

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

   14. remove-double-neg 99.8%

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

3. Simplified 99.8%

   \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{\tan B}} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{+20} \lor \neg \left(x \leq 2600000\right):\\ \;\;\;\;\cos B \cdot \frac{x}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -3.4e+20) (not (<= x 2600000.0)))
   (* (cos B) (/ x (- (sin B))))
   (/ (- 1.0 x) (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -3.4e+20) || !(x <= 2600000.0)) {
		tmp = cos(B) * (x / -sin(B));
	} else {
		tmp = (1.0 - x) / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-3.4d+20)) .or. (.not. (x <= 2600000.0d0))) then
        tmp = cos(b) * (x / -sin(b))
    else
        tmp = (1.0d0 - x) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -3.4e+20) || !(x <= 2600000.0)) {
		tmp = Math.cos(B) * (x / -Math.sin(B));
	} else {
		tmp = (1.0 - x) / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -3.4e+20) or not (x <= 2600000.0):
		tmp = math.cos(B) * (x / -math.sin(B))
	else:
		tmp = (1.0 - x) / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -3.4e+20) || !(x <= 2600000.0))
		tmp = Float64(cos(B) * Float64(x / Float64(-sin(B))));
	else
		tmp = Float64(Float64(1.0 - x) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -3.4e+20) || ~((x <= 2600000.0)))
		tmp = cos(B) * (x / -sin(B));
	else
		tmp = (1.0 - x) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -3.4e+20], N[Not[LessEqual[x, 2600000.0]], $MachinePrecision]], N[(N[Cos[B], $MachinePrecision] * N[(x / (-N[Sin[B], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.4e20 or 2.6e6 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in B around inf 99.6%

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

      1. sub-div 99.7%

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

      2. div-inv 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in x around inf 99.2%

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

      1. mul-1-neg 99.2%

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

      2. associate-*l/ 99.3%

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

      3. *-commutative 99.3%

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

      4. distribute-rgt-neg-in 99.3%

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

      5. distribute-neg-frac2 99.3%

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

   8. Simplified 99.3%

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

   if -3.4e20 < x < 2.6e6

   1. Initial program 99.8%

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

   3. Taylor expanded in B around inf 99.8%

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

   4. Taylor expanded in x around 0 99.8%

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

      1. +-commutative 99.8%

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

      2. mul-1-neg 99.8%

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

      3. sub-neg 99.8%

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

      4. div-sub 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in B around 0 97.9%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{+20} \lor \neg \left(x \leq 2600000\right):\\ \;\;\;\;\cos B \cdot \frac{x}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{+20} \lor \neg \left(x \leq 850000\right):\\ \;\;\;\;x \cdot \frac{\cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -3.4e+20) (not (<= x 850000.0)))
   (* x (/ (cos B) (- (sin B))))
   (/ (- 1.0 x) (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -3.4e+20) || !(x <= 850000.0)) {
		tmp = x * (cos(B) / -sin(B));
	} else {
		tmp = (1.0 - x) / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-3.4d+20)) .or. (.not. (x <= 850000.0d0))) then
        tmp = x * (cos(b) / -sin(b))
    else
        tmp = (1.0d0 - x) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -3.4e+20) || !(x <= 850000.0)) {
		tmp = x * (Math.cos(B) / -Math.sin(B));
	} else {
		tmp = (1.0 - x) / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -3.4e+20) or not (x <= 850000.0):
		tmp = x * (math.cos(B) / -math.sin(B))
	else:
		tmp = (1.0 - x) / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -3.4e+20) || !(x <= 850000.0))
		tmp = Float64(x * Float64(cos(B) / Float64(-sin(B))));
	else
		tmp = Float64(Float64(1.0 - x) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -3.4e+20) || ~((x <= 850000.0)))
		tmp = x * (cos(B) / -sin(B));
	else
		tmp = (1.0 - x) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -3.4e+20], N[Not[LessEqual[x, 850000.0]], $MachinePrecision]], N[(x * N[(N[Cos[B], $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.4e20 or 8.5e5 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf 99.2%

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

      1. mul-1-neg 99.2%

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

      2. associate-/l* 99.1%

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

      3. distribute-rgt-neg-in 99.1%

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

      4. distribute-neg-frac 99.1%

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

   5. Simplified 99.1%

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

   if -3.4e20 < x < 8.5e5

   1. Initial program 99.8%

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

   3. Taylor expanded in B around inf 99.8%

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

   4. Taylor expanded in x around 0 99.8%

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

      1. +-commutative 99.8%

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

      2. mul-1-neg 99.8%

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

      3. sub-neg 99.8%

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

      4. div-sub 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in B around 0 97.9%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{+20} \lor \neg \left(x \leq 850000\right):\\ \;\;\;\;x \cdot \frac{\cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -21.5 \lor \neg \left(x \leq 12500000\right):\\ \;\;\;\;\frac{x}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -21.5) (not (<= x 12500000.0)))
   (/ x (- (sin B)))
   (/ (+ 1.0 x) (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -21.5) || !(x <= 12500000.0)) {
		tmp = x / -sin(B);
	} else {
		tmp = (1.0 + x) / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-21.5d0)) .or. (.not. (x <= 12500000.0d0))) then
        tmp = x / -sin(b)
    else
        tmp = (1.0d0 + x) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -21.5) || !(x <= 12500000.0)) {
		tmp = x / -Math.sin(B);
	} else {
		tmp = (1.0 + x) / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -21.5) or not (x <= 12500000.0):
		tmp = x / -math.sin(B)
	else:
		tmp = (1.0 + x) / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -21.5) || !(x <= 12500000.0))
		tmp = Float64(x / Float64(-sin(B)));
	else
		tmp = Float64(Float64(1.0 + x) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -21.5) || ~((x <= 12500000.0)))
		tmp = x / -sin(B);
	else
		tmp = (1.0 + x) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -21.5], N[Not[LessEqual[x, 12500000.0]], $MachinePrecision]], N[(x / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], N[(N[(1.0 + x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -21.5 or 1.25e7 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in B around inf 99.7%

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

   4. Taylor expanded in x around 0 99.7%

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

      1. +-commutative 99.7%

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

      2. mul-1-neg 99.7%

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

      3. sub-neg 99.7%

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

      4. div-sub 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in x around inf 99.2%

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

      1. associate-*r/ 99.2%

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

      2. associate-*r* 99.2%

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

      3. neg-mul-1 99.2%

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

   9. Simplified 99.2%

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

   10. Taylor expanded in B around 0 52.1%

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

       1. mul-1-neg 52.1%

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

   12. Simplified 52.1%

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

   if -21.5 < x < 1.25e7

   1. Initial program 99.8%

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

   3. Taylor expanded in B around inf 99.8%

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

   4. Taylor expanded in x around 0 99.8%

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

      1. +-commutative 99.8%

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

      2. mul-1-neg 99.8%

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

      3. sub-neg 99.8%

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

      4. div-sub 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in B around 0 97.1%

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

      1. *-un-lft-identity 97.1%

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

      2. *-un-lft-identity 97.1%

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

      3. *-un-lft-identity 97.1%

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

      4. sub-neg 97.1%

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

      5. add-sqr-sqrt 48.6%

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

      6. sqrt-unprod 96.9%

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

      7. sqr-neg 96.9%

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

      8. sqrt-unprod 48.3%

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

      9. add-sqr-sqrt 96.3%

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

   9. Applied egg-rr 96.3%

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

       1. *-lft-identity 96.3%

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

   11. Simplified 96.3%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -21.5 \lor \neg \left(x \leq 12500000\right):\\ \;\;\;\;\frac{x}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 12500000\right):\\ \;\;\;\;\frac{x}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 12500000.0)))
   (/ x (- (sin B)))
   (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 12500000.0)) {
		tmp = x / -sin(B);
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 12500000.0d0))) then
        tmp = x / -sin(b)
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 12500000.0)) {
		tmp = x / -Math.sin(B);
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -1.0) or not (x <= 12500000.0):
		tmp = x / -math.sin(B)
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 12500000.0))
		tmp = Float64(x / Float64(-sin(B)));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 12500000.0)))
		tmp = x / -sin(B);
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 12500000.0]], $MachinePrecision]], N[(x / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.25e7 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in B around inf 99.7%

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

   4. Taylor expanded in x around 0 99.7%

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

      1. +-commutative 99.7%

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

      2. mul-1-neg 99.7%

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

      3. sub-neg 99.7%

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

      4. div-sub 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in x around inf 99.2%

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

      1. associate-*r/ 99.2%

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

      2. associate-*r* 99.2%

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

      3. neg-mul-1 99.2%

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

   9. Simplified 99.2%

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

   10. Taylor expanded in B around 0 52.1%

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

       1. mul-1-neg 52.1%

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

   12. Simplified 52.1%

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

   if -1 < x < 1.25e7

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 96.2%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 12500000\right):\\ \;\;\;\;\frac{x}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 2600:\\ \;\;\;\;B \cdot 0.16666666666666666 + \left(\frac{1}{B} + x \cdot \left(B \cdot 0.3333333333333333 - \frac{1}{B}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B 2600.0)
   (+
    (* B 0.16666666666666666)
    (+ (/ 1.0 B) (* x (- (* B 0.3333333333333333) (/ 1.0 B)))))
   (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if (B <= 2600.0) {
		tmp = (B * 0.16666666666666666) + ((1.0 / B) + (x * ((B * 0.3333333333333333) - (1.0 / B))));
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (b <= 2600.0d0) then
        tmp = (b * 0.16666666666666666d0) + ((1.0d0 / b) + (x * ((b * 0.3333333333333333d0) - (1.0d0 / b))))
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (B <= 2600.0) {
		tmp = (B * 0.16666666666666666) + ((1.0 / B) + (x * ((B * 0.3333333333333333) - (1.0 / B))));
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if B <= 2600.0:
		tmp = (B * 0.16666666666666666) + ((1.0 / B) + (x * ((B * 0.3333333333333333) - (1.0 / B))))
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (B <= 2600.0)
		tmp = Float64(Float64(B * 0.16666666666666666) + Float64(Float64(1.0 / B) + Float64(x * Float64(Float64(B * 0.3333333333333333) - Float64(1.0 / B)))));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (B <= 2600.0)
		tmp = (B * 0.16666666666666666) + ((1.0 / B) + (x * ((B * 0.3333333333333333) - (1.0 / B))));
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[B, 2600.0], N[(N[(B * 0.16666666666666666), $MachinePrecision] + N[(N[(1.0 / B), $MachinePrecision] + N[(x * N[(N[(B * 0.3333333333333333), $MachinePrecision] - N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 2600

   1. Initial program 99.8%

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

   3. Taylor expanded in B around 0 63.8%

      \[\leadsto \color{blue}{\frac{\left(1 + {B}^{2} \cdot \left(0.16666666666666666 + 0.3333333333333333 \cdot x\right)\right) - x}{B}} \]

   4. Taylor expanded in x around 0 63.9%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot B + \left(x \cdot \left(0.3333333333333333 \cdot B - \frac{1}{B}\right) + \frac{1}{B}\right)} \]

   if 2600 < B

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0 48.8%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2600:\\ \;\;\;\;B \cdot 0.16666666666666666 + \left(\frac{1}{B} + x \cdot \left(B \cdot 0.3333333333333333 - \frac{1}{B}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{1 - x}{\sin B} \end{array} \]

FPCore

(FPCore (B x) :precision binary64 (/ (- 1.0 x) (sin B)))

C

double code(double B, double x) {
	return (1.0 - x) / sin(B);
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = (1.0d0 - x) / sin(b)
end function

Java

public static double code(double B, double x) {
	return (1.0 - x) / Math.sin(B);
}

Python

def code(B, x):
	return (1.0 - x) / math.sin(B)

Julia

function code(B, x)
	return Float64(Float64(1.0 - x) / sin(B))
end

MATLAB

function tmp = code(B, x)
	tmp = (1.0 - x) / sin(B);
end

Wolfram

code[B_, x_] := N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in B around inf 99.7%

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

4. Taylor expanded in x around 0 99.7%

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

   1. +-commutative 99.7%

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

   2. mul-1-neg 99.7%

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

   3. sub-neg 99.7%

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

   4. div-sub 99.7%

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

6. Simplified 99.7%

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

7. Taylor expanded in B around 0 76.6%

   \[\leadsto \frac{1 - \color{blue}{x}}{\sin B} \]
8. Add Preprocessing

Alternative 8: 49.3% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.06 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -0.06) (not (<= x 1.0))) (/ x (- B)) (/ 1.0 B)))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -0.06) || !(x <= 1.0)) {
		tmp = x / -B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-0.06d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = x / -b
    else
        tmp = 1.0d0 / b
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -0.06) || !(x <= 1.0)) {
		tmp = x / -B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -0.06) or not (x <= 1.0):
		tmp = x / -B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -0.06) || !(x <= 1.0))
		tmp = Float64(x / Float64(-B));
	else
		tmp = Float64(1.0 / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -0.06) || ~((x <= 1.0)))
		tmp = x / -B;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -0.06], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(x / (-B)), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.059999999999999998 or 1 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in B around 0 46.8%

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

   4. Taylor expanded in x around inf 46.3%

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

      1. neg-mul-1 46.3%

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

      2. distribute-neg-frac2 46.3%

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

   6. Simplified 46.3%

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

   if -0.059999999999999998 < x < 1

   1. Initial program 99.8%

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

   3. Taylor expanded in B around 0 49.8%

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

   4. Taylor expanded in x around 0 48.8%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.06 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 50.4% accurate, 42.0× speedup

Math

\[\begin{array}{l} \\ \frac{1 - x}{B} \end{array} \]

FPCore

(FPCore (B x) :precision binary64 (/ (- 1.0 x) B))

C

double code(double B, double x) {
	return (1.0 - x) / B;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = (1.0d0 - x) / b
end function

Java

public static double code(double B, double x) {
	return (1.0 - x) / B;
}

Python

def code(B, x):
	return (1.0 - x) / B

Julia

function code(B, x)
	return Float64(Float64(1.0 - x) / B)
end

MATLAB

function tmp = code(B, x)
	tmp = (1.0 - x) / B;
end

Wolfram

code[B_, x_] := N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in B around 0 48.4%

   \[\leadsto \color{blue}{\frac{1 - x}{B}} \]
4. Add Preprocessing

Alternative 10: 26.4% accurate, 70.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{B} \end{array} \]

FPCore

(FPCore (B x) :precision binary64 (/ 1.0 B))

C

double code(double B, double x) {
	return 1.0 / B;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = 1.0d0 / b
end function

Java

public static double code(double B, double x) {
	return 1.0 / B;
}

Python

def code(B, x):
	return 1.0 / B

Julia

function code(B, x)
	return Float64(1.0 / B)
end

MATLAB

function tmp = code(B, x)
	tmp = 1.0 / B;
end

Wolfram

code[B_, x_] := N[(1.0 / B), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in B around 0 48.4%

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

4. Taylor expanded in x around 0 27.3%

   \[\leadsto \color{blue}{\frac{1}{B}} \]
5. Add Preprocessing

Alternative 11: 3.1% accurate, 70.0× speedup

Math

\[\begin{array}{l} \\ B \cdot 0.16666666666666666 \end{array} \]

FPCore

(FPCore (B x) :precision binary64 (* B 0.16666666666666666))

C

double code(double B, double x) {
	return B * 0.16666666666666666;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = b * 0.16666666666666666d0
end function

Java

public static double code(double B, double x) {
	return B * 0.16666666666666666;
}

Python

def code(B, x):
	return B * 0.16666666666666666

Julia

function code(B, x)
	return Float64(B * 0.16666666666666666)
end

MATLAB

function tmp = code(B, x)
	tmp = B * 0.16666666666666666;
end

Wolfram

code[B_, x_] := N[(B * 0.16666666666666666), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in B around 0 48.2%

   \[\leadsto \color{blue}{\frac{\left(1 + {B}^{2} \cdot \left(0.16666666666666666 + 0.3333333333333333 \cdot x\right)\right) - x}{B}} \]

4. Taylor expanded in x around 0 48.1%

   \[\leadsto \frac{\left(1 + \color{blue}{0.16666666666666666 \cdot {B}^{2}}\right) - x}{B} \]
5. Step-by-step derivation

   1. *-commutative 48.1%

      \[\leadsto \frac{\left(1 + \color{blue}{{B}^{2} \cdot 0.16666666666666666}\right) - x}{B} \]

6. Simplified 48.1%

   \[\leadsto \frac{\left(1 + \color{blue}{{B}^{2} \cdot 0.16666666666666666}\right) - x}{B} \]

7. Taylor expanded in B around inf 3.0%

   \[\leadsto \color{blue}{0.16666666666666666 \cdot B} \]
8. Step-by-step derivation

   1. *-commutative 3.0%

      \[\leadsto \color{blue}{B \cdot 0.16666666666666666} \]

9. Simplified 3.0%

   \[\leadsto \color{blue}{B \cdot 0.16666666666666666} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024088 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))