VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 9.9s

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. cancel-sign-sub-inv 99.7%

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

   4. *-commutative 99.7%

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

   5. *-commutative 99.7%

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

   6. associate-*r/ 99.8%

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

   7. *-rgt-identity 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. Final simplification 99.8%

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

Alternative 2: 98.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1 + \frac{1}{x}}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= x -5.2e-6)
   (- (/ 1.0 B) (/ x (tan B)))
   (if (<= x 2.5e-8) (/ 1.0 (sin B)) (* x (/ (+ -1.0 (/ 1.0 x)) (tan B))))))

C

double code(double B, double x) {
	double tmp;
	if (x <= -5.2e-6) {
		tmp = (1.0 / B) - (x / tan(B));
	} else if (x <= 2.5e-8) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = x * ((-1.0 + (1.0 / x)) / tan(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 <= (-5.2d-6)) then
        tmp = (1.0d0 / b) - (x / tan(b))
    else if (x <= 2.5d-8) then
        tmp = 1.0d0 / sin(b)
    else
        tmp = x * (((-1.0d0) + (1.0d0 / x)) / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (x <= -5.2e-6) {
		tmp = (1.0 / B) - (x / Math.tan(B));
	} else if (x <= 2.5e-8) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = x * ((-1.0 + (1.0 / x)) / Math.tan(B));
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if x <= -5.2e-6:
		tmp = (1.0 / B) - (x / math.tan(B))
	elif x <= 2.5e-8:
		tmp = 1.0 / math.sin(B)
	else:
		tmp = x * ((-1.0 + (1.0 / x)) / math.tan(B))
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (x <= -5.2e-6)
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	elseif (x <= 2.5e-8)
		tmp = Float64(1.0 / sin(B));
	else
		tmp = Float64(x * Float64(Float64(-1.0 + Float64(1.0 / x)) / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (x <= -5.2e-6)
		tmp = (1.0 / B) - (x / tan(B));
	elseif (x <= 2.5e-8)
		tmp = 1.0 / sin(B);
	else
		tmp = x * ((-1.0 + (1.0 / x)) / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[x, -5.2e-6], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.5e-8], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(-1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.20000000000000019e-6

   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. cancel-sign-sub-inv 99.7%

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

      4. *-commutative 99.7%

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

      5. *-commutative 99.7%

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

      6. associate-*r/ 99.7%

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

      7. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in B around 0 99.3%

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

   if -5.20000000000000019e-6 < x < 2.4999999999999999e-8

   1. Initial program 99.8%

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

      1. distribute-lft-neg-in 99.8%

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

      2. +-commutative 99.8%

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

      3. distribute-lft-neg-in 99.8%

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

      4. distribute-rgt-neg-in 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 99.4%

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

   if 2.4999999999999999e-8 < x

   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. distribute-lft-neg-in 99.7%

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

      4. distribute-rgt-neg-in 99.7%

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

   3. Simplified 99.7%

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

      1. distribute-rgt-neg-out 99.7%

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

      2. div-inv 99.8%

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

      3. sub-neg 99.8%

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

      4. clear-num 99.8%

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

      5. frac-sub 90.1%

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

      6. *-un-lft-identity 90.1%

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

      7. *-commutative 90.1%

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

      8. *-un-lft-identity 90.1%

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

   5. Applied egg-rr 90.1%

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

      1. associate-/r* 99.7%

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

      2. associate-/r/ 99.7%

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

      3. div-sub 99.7%

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

      4. *-inverses 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in B around 0 98.8%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1 + \frac{1}{x}}{\tan B}\\ \end{array} \]

Alternative 3: 98.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + \frac{1}{x}}{\frac{\tan B}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= x -1.25e-5)
   (- (/ 1.0 B) (/ x (tan B)))
   (if (<= x 2.5e-8) (/ 1.0 (sin B)) (/ (+ -1.0 (/ 1.0 x)) (/ (tan B) x)))))

C

double code(double B, double x) {
	double tmp;
	if (x <= -1.25e-5) {
		tmp = (1.0 / B) - (x / tan(B));
	} else if (x <= 2.5e-8) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = (-1.0 + (1.0 / x)) / (tan(B) / x);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.25d-5)) then
        tmp = (1.0d0 / b) - (x / tan(b))
    else if (x <= 2.5d-8) then
        tmp = 1.0d0 / sin(b)
    else
        tmp = ((-1.0d0) + (1.0d0 / x)) / (tan(b) / x)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (x <= -1.25e-5) {
		tmp = (1.0 / B) - (x / Math.tan(B));
	} else if (x <= 2.5e-8) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = (-1.0 + (1.0 / x)) / (Math.tan(B) / x);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if x <= -1.25e-5:
		tmp = (1.0 / B) - (x / math.tan(B))
	elif x <= 2.5e-8:
		tmp = 1.0 / math.sin(B)
	else:
		tmp = (-1.0 + (1.0 / x)) / (math.tan(B) / x)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (x <= -1.25e-5)
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	elseif (x <= 2.5e-8)
		tmp = Float64(1.0 / sin(B));
	else
		tmp = Float64(Float64(-1.0 + Float64(1.0 / x)) / Float64(tan(B) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (x <= -1.25e-5)
		tmp = (1.0 / B) - (x / tan(B));
	elseif (x <= 2.5e-8)
		tmp = 1.0 / sin(B);
	else
		tmp = (-1.0 + (1.0 / x)) / (tan(B) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[x, -1.25e-5], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.5e-8], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25000000000000006e-5

   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. cancel-sign-sub-inv 99.7%

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

      4. *-commutative 99.7%

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

      5. *-commutative 99.7%

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

      6. associate-*r/ 99.7%

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

      7. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in B around 0 99.3%

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

   if -1.25000000000000006e-5 < x < 2.4999999999999999e-8

   1. Initial program 99.8%

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

      1. distribute-lft-neg-in 99.8%

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

      2. +-commutative 99.8%

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

      3. distribute-lft-neg-in 99.8%

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

      4. distribute-rgt-neg-in 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 99.4%

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

   if 2.4999999999999999e-8 < x

   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. distribute-lft-neg-in 99.7%

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

      4. distribute-rgt-neg-in 99.7%

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

   3. Simplified 99.7%

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

      1. distribute-rgt-neg-out 99.7%

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

      2. div-inv 99.8%

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

      3. sub-neg 99.8%

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

      4. clear-num 99.8%

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

      5. frac-sub 90.1%

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

      6. *-un-lft-identity 90.1%

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

      7. *-commutative 90.1%

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

      8. *-un-lft-identity 90.1%

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

   5. Applied egg-rr 90.1%

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

      1. associate-/r* 99.7%

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

      2. associate-/r/ 99.7%

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

      3. div-sub 99.7%

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

      4. *-inverses 99.7%

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

   7. Simplified 99.7%

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

      1. expm1-log1p-u 50.7%

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

      2. expm1-udef 50.7%

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

      3. *-commutative 50.7%

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

      4. sub-neg 50.7%

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

      5. associate-/l/ 50.7%

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

      6. metadata-eval 50.7%

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

   9. Applied egg-rr 50.7%

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

       1. expm1-def 50.7%

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

       2. expm1-log1p 99.7%

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

       3. associate-*r/ 99.8%

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

       4. *-commutative 99.8%

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

       5. associate-/l* 99.8%

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

       6. +-commutative 99.8%

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

   11. Simplified 99.8%

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

   12. Taylor expanded in B around 0 98.8%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 + \frac{1}{x}}{\frac{\tan B}{x}}\\ \end{array} \]

Alternative 4: 98.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-5} \lor \neg \left(x \leq 7 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.6e-5) (not (<= x 7e-8)))
   (- (/ 1.0 B) (/ x (tan B)))
   (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.6e-5) || !(x <= 7e-8)) {
		tmp = (1.0 / B) - (x / tan(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 <= (-1.6d-5)) .or. (.not. (x <= 7d-8))) then
        tmp = (1.0d0 / b) - (x / tan(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 <= -1.6e-5) || !(x <= 7e-8)) {
		tmp = (1.0 / B) - (x / Math.tan(B));
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -1.6e-5) or not (x <= 7e-8):
		tmp = (1.0 / B) - (x / math.tan(B))
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -1.6e-5) || !(x <= 7e-8))
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -1.6e-5) || ~((x <= 7e-8)))
		tmp = (1.0 / B) - (x / tan(B));
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -1.6e-5], N[Not[LessEqual[x, 7e-8]], $MachinePrecision]], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.59999999999999993e-5 or 7.00000000000000048e-8 < x

   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. cancel-sign-sub-inv 99.7%

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

      4. *-commutative 99.7%

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

      5. *-commutative 99.7%

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

      6. associate-*r/ 99.8%

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

      7. *-rgt-identity 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. Taylor expanded in B around 0 99.0%

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

   if -1.59999999999999993e-5 < x < 7.00000000000000048e-8

   1. Initial program 99.8%

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

      1. distribute-lft-neg-in 99.8%

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

      2. +-commutative 99.8%

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

      3. distribute-lft-neg-in 99.8%

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

      4. distribute-rgt-neg-in 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 99.4%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-5} \lor \neg \left(x \leq 7 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 5: 73.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 - x}{B}\\ \mathbf{if}\;x \leq -7.5 \cdot 10^{-6}:\\ \;\;\;\;B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + t_0\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (/ (- 1.0 x) B)))
   (if (<= x -7.5e-6)
     (+ (* B (+ 0.16666666666666666 (* x 0.3333333333333333))) t_0)
     (if (<= x 6.6e-8) (/ 1.0 (sin B)) t_0))))

C

double code(double B, double x) {
	double t_0 = (1.0 - x) / B;
	double tmp;
	if (x <= -7.5e-6) {
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + t_0;
	} else if (x <= 6.6e-8) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 - x) / b
    if (x <= (-7.5d-6)) then
        tmp = (b * (0.16666666666666666d0 + (x * 0.3333333333333333d0))) + t_0
    else if (x <= 6.6d-8) then
        tmp = 1.0d0 / sin(b)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double t_0 = (1.0 - x) / B;
	double tmp;
	if (x <= -7.5e-6) {
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + t_0;
	} else if (x <= 6.6e-8) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = (1.0 - x) / B
	tmp = 0
	if x <= -7.5e-6:
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + t_0
	elif x <= 6.6e-8:
		tmp = 1.0 / math.sin(B)
	else:
		tmp = t_0
	return tmp

Julia

function code(B, x)
	t_0 = Float64(Float64(1.0 - x) / B)
	tmp = 0.0
	if (x <= -7.5e-6)
		tmp = Float64(Float64(B * Float64(0.16666666666666666 + Float64(x * 0.3333333333333333))) + t_0);
	elseif (x <= 6.6e-8)
		tmp = Float64(1.0 / sin(B));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	t_0 = (1.0 - x) / B;
	tmp = 0.0;
	if (x <= -7.5e-6)
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + t_0;
	elseif (x <= 6.6e-8)
		tmp = 1.0 / sin(B);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := Block[{t$95$0 = N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]}, If[LessEqual[x, -7.5e-6], N[(N[(B * N[(0.16666666666666666 + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[x, 6.6e-8], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.50000000000000019e-6

   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. distribute-lft-neg-in 99.7%

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

      4. distribute-rgt-neg-in 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in B around 0 44.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B} + \left(B \cdot \left(0.16666666666666666 + 0.3333333333333333 \cdot x\right) + \frac{1}{B}\right)} \]
   5. Step-by-step derivation

      1. +-commutative 44.0%

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

      2. mul-1-neg 44.0%

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

      3. sub-neg 44.0%

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

      4. associate--l+ 44.0%

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

      5. *-commutative 44.0%

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

      6. div-sub 44.0%

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

   6. Simplified 44.0%

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

   if -7.50000000000000019e-6 < x < 6.59999999999999954e-8

   1. Initial program 99.8%

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

      1. distribute-lft-neg-in 99.8%

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

      2. +-commutative 99.8%

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

      3. distribute-lft-neg-in 99.8%

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

      4. distribute-rgt-neg-in 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 99.4%

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

   if 6.59999999999999954e-8 < x

   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. distribute-lft-neg-in 99.7%

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

      4. distribute-rgt-neg-in 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in B around 0 60.3%

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

      1. neg-mul-1 60.3%

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

      2. sub-neg 60.3%

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

   6. Simplified 60.3%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-6}:\\ \;\;\;\;B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]

Alternative 6: 51.0% accurate, 16.2× speedup

Math

\[\begin{array}{l} \\ B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (+ (* B (+ 0.16666666666666666 (* x 0.3333333333333333))) (/ (- 1.0 x) B)))

C

double code(double B, double x) {
	return (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B);
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = (b * (0.16666666666666666d0 + (x * 0.3333333333333333d0))) + ((1.0d0 - x) / b)
end function

Java

public static double code(double B, double x) {
	return (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B);
}

Python

def code(B, x):
	return (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B)

Julia

function code(B, x)
	return Float64(Float64(B * Float64(0.16666666666666666 + Float64(x * 0.3333333333333333))) + Float64(Float64(1.0 - x) / B))
end

MATLAB

function tmp = code(B, x)
	tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B);
end

Wolfram

code[B_, x_] := N[(N[(B * N[(0.16666666666666666 + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / B), $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. distribute-lft-neg-in 99.7%

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

   4. distribute-rgt-neg-in 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in B around 0 46.7%

   \[\leadsto \color{blue}{-1 \cdot \frac{x}{B} + \left(B \cdot \left(0.16666666666666666 + 0.3333333333333333 \cdot x\right) + \frac{1}{B}\right)} \]
5. Step-by-step derivation

   1. +-commutative 46.7%

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

   2. mul-1-neg 46.7%

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

   3. sub-neg 46.7%

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

   4. associate--l+ 46.7%

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

   5. *-commutative 46.7%

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

   6. div-sub 46.7%

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

6. Simplified 46.7%

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

7. Final simplification 46.7%

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

Alternative 7: 51.2% accurate, 16.2× speedup

Math

\[\begin{array}{l} \\ \frac{1}{B} + \left(x \cdot \left(B \cdot 0.3333333333333333\right) - \frac{x}{B}\right) \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (+ (/ 1.0 B) (- (* x (* B 0.3333333333333333)) (/ x B))))

C

double code(double B, double x) {
	return (1.0 / B) + ((x * (B * 0.3333333333333333)) - (x / B));
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = (1.0d0 / b) + ((x * (b * 0.3333333333333333d0)) - (x / b))
end function

Java

public static double code(double B, double x) {
	return (1.0 / B) + ((x * (B * 0.3333333333333333)) - (x / B));
}

Python

def code(B, x):
	return (1.0 / B) + ((x * (B * 0.3333333333333333)) - (x / B))

Julia

function code(B, x)
	return Float64(Float64(1.0 / B) + Float64(Float64(x * Float64(B * 0.3333333333333333)) - Float64(x / B)))
end

MATLAB

function tmp = code(B, x)
	tmp = (1.0 / B) + ((x * (B * 0.3333333333333333)) - (x / B));
end

Wolfram

code[B_, x_] := N[(N[(1.0 / B), $MachinePrecision] + N[(N[(x * N[(B * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - N[(x / 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. distribute-lft-neg-in 99.7%

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

   4. distribute-rgt-neg-in 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in B around 0 71.2%

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

5. Taylor expanded in B around 0 47.0%

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

   1. +-commutative 47.0%

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

   2. mul-1-neg 47.0%

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

   3. unsub-neg 47.0%

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

   4. *-commutative 47.0%

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

   5. *-commutative 47.0%

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

   6. associate-*l* 47.0%

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

7. Simplified 47.0%

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

8. Final simplification 47.0%

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

Alternative 8: 49.6% accurate, 25.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{-10} \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.02e-10) (not (<= x 1.0))) (/ (- x) B) (/ 1.0 B)))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.02e-10) || !(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 <= (-1.02d-10)) .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 <= -1.02e-10) || !(x <= 1.0)) {
		tmp = -x / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -1.02e-10) 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 <= -1.02e-10) || !(x <= 1.0))
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(1.0 / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -1.02e-10) || ~((x <= 1.0)))
		tmp = -x / B;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -1.02e-10], 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 < -1.01999999999999997e-10 or 1 < x

   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. cancel-sign-sub-inv 99.7%

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

      4. *-commutative 99.7%

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

      5. *-commutative 99.7%

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

      6. associate-*r/ 99.8%

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

      7. *-rgt-identity 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. Step-by-step derivation

      1. add-sqr-sqrt 46.0%

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

      2. pow2 46.0%

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

      3. inv-pow 46.0%

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

      4. sqrt-pow1 46.0%

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

      5. metadata-eval 46.0%

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

   5. Applied egg-rr 46.0%

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

   6. Taylor expanded in B around 0 49.8%

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

   7. Taylor expanded in x around inf 49.5%

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

   8. Taylor expanded in B around 0 49.4%

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

      1. associate-*r/ 49.4%

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

      2. neg-mul-1 49.4%

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

   10. Simplified 49.4%

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

   if -1.01999999999999997e-10 < x < 1

   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. distribute-lft-neg-in 99.7%

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

      4. distribute-rgt-neg-in 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in B around 0 43.4%

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

   5. Taylor expanded in x around 0 42.5%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{-10} \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]

Alternative 9: 50.8% 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. 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. distribute-lft-neg-in 99.7%

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

   4. distribute-rgt-neg-in 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in B around 0 46.6%

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

   1. neg-mul-1 46.6%

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

   2. sub-neg 46.6%

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

6. Simplified 46.6%

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

7. Final simplification 46.6%

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

Alternative 10: 3.2% 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. 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. cancel-sign-sub-inv 99.7%

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

   4. *-commutative 99.7%

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

   5. *-commutative 99.7%

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

   6. associate-*r/ 99.8%

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

   7. *-rgt-identity 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. Step-by-step derivation

   1. add-sqr-sqrt 49.3%

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

   2. pow2 49.3%

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

   3. inv-pow 49.3%

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

   4. sqrt-pow1 49.3%

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

   5. metadata-eval 49.3%

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

5. Applied egg-rr 49.3%

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

6. Taylor expanded in B around 0 46.7%

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

7. Taylor expanded in B around inf 2.7%

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

8. Taylor expanded in x around 0 3.0%

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

   1. *-commutative 3.0%

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

10. Simplified 3.0%

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

11. Final simplification 3.0%

    \[\leadsto B \cdot 0.16666666666666666 \]

Alternative 11: 26.9% 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. 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. distribute-lft-neg-in 99.7%

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

   4. distribute-rgt-neg-in 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in B around 0 71.2%

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

5. Taylor expanded in x around 0 22.3%

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

6. Final simplification 22.3%

   \[\leadsto \frac{1}{B} \]

Reproduce

herbie shell --seed 2023309 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))