VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 9.2s

Alternatives: 12

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. +-commutative 99.7%

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

   2. unsub-neg 99.7%

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

   3. associate-*r/ 99.8%

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

   4. *-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: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;x \leq 105000:\\ \;\;\;\;\frac{x - \frac{B}{\sin B}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\cos B \cdot \frac{x}{-\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= x -3.2)
   (- (/ 1.0 B) (/ x (tan B)))
   (if (<= x 105000.0)
     (/ (- x (/ B (sin B))) (- B))
     (* (cos B) (/ x (- (sin B)))))))

C

double code(double B, double x) {
	double tmp;
	if (x <= -3.2) {
		tmp = (1.0 / B) - (x / tan(B));
	} else if (x <= 105000.0) {
		tmp = (x - (B / sin(B))) / -B;
	} else {
		tmp = cos(B) * (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 <= (-3.2d0)) then
        tmp = (1.0d0 / b) - (x / tan(b))
    else if (x <= 105000.0d0) then
        tmp = (x - (b / sin(b))) / -b
    else
        tmp = cos(b) * (x / -sin(b))
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (x <= -3.2) {
		tmp = (1.0 / B) - (x / Math.tan(B));
	} else if (x <= 105000.0) {
		tmp = (x - (B / Math.sin(B))) / -B;
	} else {
		tmp = Math.cos(B) * (x / -Math.sin(B));
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if x <= -3.2:
		tmp = (1.0 / B) - (x / math.tan(B))
	elif x <= 105000.0:
		tmp = (x - (B / math.sin(B))) / -B
	else:
		tmp = math.cos(B) * (x / -math.sin(B))
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (x <= -3.2)
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	elseif (x <= 105000.0)
		tmp = Float64(Float64(x - Float64(B / sin(B))) / Float64(-B));
	else
		tmp = Float64(cos(B) * Float64(x / Float64(-sin(B))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (x <= -3.2)
		tmp = (1.0 / B) - (x / tan(B));
	elseif (x <= 105000.0)
		tmp = (x - (B / sin(B))) / -B;
	else
		tmp = cos(B) * (x / -sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[x, -3.2], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 105000.0], N[(N[(x - N[(B / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-B)), $MachinePrecision], N[(N[Cos[B], $MachinePrecision] * N[(x / (-N[Sin[B], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.2000000000000002

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. unsub-neg 99.7%

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

      3. associate-*r/ 99.8%

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

      4. *-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-cbrt-cube 92.6%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{1}{\sin B} \cdot \frac{1}{\sin B}\right) \cdot \frac{1}{\sin B}}} - \frac{x}{\tan B} \]

      2. pow3 92.6%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{1}{\sin B}\right)}^{3}}} - \frac{x}{\tan B} \]

      3. inv-pow 92.6%

         \[\leadsto \sqrt[3]{{\color{blue}{\left({\sin B}^{-1}\right)}}^{3}} - \frac{x}{\tan B} \]

      4. pow-pow 92.6%

         \[\leadsto \sqrt[3]{\color{blue}{{\sin B}^{\left(-1 \cdot 3\right)}}} - \frac{x}{\tan B} \]

      5. metadata-eval 92.6%

         \[\leadsto \sqrt[3]{{\sin B}^{\color{blue}{-3}}} - \frac{x}{\tan B} \]

   5. Applied egg-rr 92.6%

      \[\leadsto \color{blue}{\sqrt[3]{{\sin B}^{-3}}} - \frac{x}{\tan B} \]

   6. Taylor expanded in B around 0 98.1%

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

   if -3.2000000000000002 < x < 105000

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. unsub-neg 99.8%

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

      3. associate-*r/ 99.8%

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

      4. *-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%

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

      1. frac-sub 73.2%

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

      2. *-un-lft-identity 73.2%

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

   6. Applied egg-rr 73.2%

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

      1. frac-2neg 73.2%

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

      2. div-inv 72.8%

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

      3. *-commutative 72.8%

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

      4. distribute-rgt-neg-in 72.8%

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

   8. Applied egg-rr 72.8%

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

      1. associate-*r/ 73.2%

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

      2. *-rgt-identity 73.2%

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

      3. distribute-neg-frac 73.2%

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

      4. associate-/r* 99.0%

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

      5. distribute-neg-frac 99.0%

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

      6. div-sub 99.0%

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

      7. associate-/l* 99.0%

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

      8. *-inverses 99.0%

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

   10. Simplified 99.0%

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

   if 105000 < x

   1. Initial program 99.5%

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

      1. distribute-lft-neg-in 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in B around 0 69.3%

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

   5. Taylor expanded in x around inf 99.6%

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

      1. mul-1-neg 99.6%

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

      2. associate-*r/ 99.7%

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

      3. distribute-rgt-neg-in 99.7%

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

      4. neg-mul-1 99.7%

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

      5. associate-*r/ 99.7%

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

      6. *-commutative 99.7%

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

      7. remove-double-neg 99.7%

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

      8. neg-mul-1 99.7%

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

      9. times-frac 99.3%

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

      10. neg-mul-1 99.3%

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

      11. associate-/r* 99.3%

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

      12. metadata-eval 99.3%

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

      13. times-frac 99.7%

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

      14. *-rgt-identity 99.7%

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

      15. neg-mul-1 99.7%

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

   7. Simplified 99.7%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;x \leq 105000:\\ \;\;\;\;\frac{x - \frac{B}{\sin B}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\cos B \cdot \frac{x}{-\sin B}\\ \end{array} \]

Alternative 3: 98.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \lor \neg \left(x \leq 0.00065\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \frac{B}{\sin B}}{-B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.7) (not (<= x 0.00065)))
   (- (/ 1.0 B) (/ x (tan B)))
   (/ (- x (/ B (sin B))) (- B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.7) || !(x <= 0.00065)) {
		tmp = (1.0 / B) - (x / tan(B));
	} else {
		tmp = (x - (B / sin(B))) / -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 <= (-1.7d0)) .or. (.not. (x <= 0.00065d0))) then
        tmp = (1.0d0 / b) - (x / tan(b))
    else
        tmp = (x - (b / sin(b))) / -b
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -1.7) || !(x <= 0.00065)) {
		tmp = (1.0 / B) - (x / Math.tan(B));
	} else {
		tmp = (x - (B / Math.sin(B))) / -B;
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -1.7) or not (x <= 0.00065):
		tmp = (1.0 / B) - (x / math.tan(B))
	else:
		tmp = (x - (B / math.sin(B))) / -B
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -1.7) || !(x <= 0.00065))
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(x - Float64(B / sin(B))) / Float64(-B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -1.7) || ~((x <= 0.00065)))
		tmp = (1.0 / B) - (x / tan(B));
	else
		tmp = (x - (B / sin(B))) / -B;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -1.7], N[Not[LessEqual[x, 0.00065]], $MachinePrecision]], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x - N[(B / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-B)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.69999999999999996 or 6.4999999999999997e-4 < x

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unsub-neg 99.6%

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

      3. associate-*r/ 99.8%

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

      4. *-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-cbrt-cube 79.5%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{1}{\sin B} \cdot \frac{1}{\sin B}\right) \cdot \frac{1}{\sin B}}} - \frac{x}{\tan B} \]

      2. pow3 79.5%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{1}{\sin B}\right)}^{3}}} - \frac{x}{\tan B} \]

      3. inv-pow 79.5%

         \[\leadsto \sqrt[3]{{\color{blue}{\left({\sin B}^{-1}\right)}}^{3}} - \frac{x}{\tan B} \]

      4. pow-pow 79.5%

         \[\leadsto \sqrt[3]{\color{blue}{{\sin B}^{\left(-1 \cdot 3\right)}}} - \frac{x}{\tan B} \]

      5. metadata-eval 79.5%

         \[\leadsto \sqrt[3]{{\sin B}^{\color{blue}{-3}}} - \frac{x}{\tan B} \]

   5. Applied egg-rr 79.5%

      \[\leadsto \color{blue}{\sqrt[3]{{\sin B}^{-3}}} - \frac{x}{\tan B} \]

   6. Taylor expanded in B around 0 98.8%

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

   if -1.69999999999999996 < x < 6.4999999999999997e-4

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. unsub-neg 99.8%

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

      3. associate-*r/ 99.8%

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

      4. *-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%

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

      1. frac-sub 73.7%

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

      2. *-un-lft-identity 73.7%

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

   6. Applied egg-rr 73.7%

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

      1. frac-2neg 73.7%

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

      2. div-inv 73.3%

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

      3. *-commutative 73.3%

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

      4. distribute-rgt-neg-in 73.3%

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

   8. Applied egg-rr 73.3%

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

      1. associate-*r/ 73.7%

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

      2. *-rgt-identity 73.7%

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

      3. distribute-neg-frac 73.7%

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

      4. associate-/r* 99.0%

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

      5. distribute-neg-frac 99.0%

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

      6. div-sub 99.0%

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

      7. associate-/l* 99.0%

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

      8. *-inverses 99.0%

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

   10. Simplified 99.0%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \lor \neg \left(x \leq 0.00065\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \frac{B}{\sin B}}{-B}\\ \end{array} \]

Alternative 4: 98.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-5} \lor \neg \left(x \leq 1.15 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.1e-5) (not (<= x 1.15e-5)))
   (- (/ 1.0 B) (/ x (tan B)))
   (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.1e-5) || !(x <= 1.15e-5)) {
		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 <= (-1.1d-5)) .or. (.not. (x <= 1.15d-5))) 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 <= -1.1e-5) || !(x <= 1.15e-5)) {
		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 <= -1.1e-5) or not (x <= 1.15e-5):
		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 <= -1.1e-5) || !(x <= 1.15e-5))
		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 <= -1.1e-5) || ~((x <= 1.15e-5)))
		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, -1.1e-5], N[Not[LessEqual[x, 1.15e-5]], $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 < -1.1e-5 or 1.15e-5 < x

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unsub-neg 99.6%

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

      3. associate-*r/ 99.8%

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

      4. *-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-cbrt-cube 79.5%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{1}{\sin B} \cdot \frac{1}{\sin B}\right) \cdot \frac{1}{\sin B}}} - \frac{x}{\tan B} \]

      2. pow3 79.5%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{1}{\sin B}\right)}^{3}}} - \frac{x}{\tan B} \]

      3. inv-pow 79.5%

         \[\leadsto \sqrt[3]{{\color{blue}{\left({\sin B}^{-1}\right)}}^{3}} - \frac{x}{\tan B} \]

      4. pow-pow 79.5%

         \[\leadsto \sqrt[3]{\color{blue}{{\sin B}^{\left(-1 \cdot 3\right)}}} - \frac{x}{\tan B} \]

      5. metadata-eval 79.5%

         \[\leadsto \sqrt[3]{{\sin B}^{\color{blue}{-3}}} - \frac{x}{\tan B} \]

   5. Applied egg-rr 79.5%

      \[\leadsto \color{blue}{\sqrt[3]{{\sin B}^{-3}}} - \frac{x}{\tan B} \]

   6. Taylor expanded in B around 0 98.8%

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

   if -1.1e-5 < x < 1.15e-5

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. unsub-neg 99.8%

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

      3. associate-*r/ 99.8%

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

      4. *-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%

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

   5. Taylor expanded in B around inf 98.9%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-5} \lor \neg \left(x \leq 1.15 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 5: 98.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.2 \lor \neg \left(x \leq 0.00065\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -2.2) (not (<= x 0.00065)))
   (- (/ 1.0 B) (/ x (tan B)))
   (- (/ 1.0 (sin B)) (/ x B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -2.2) || !(x <= 0.00065)) {
		tmp = (1.0 / B) - (x / tan(B));
	} else {
		tmp = (1.0 / sin(B)) - (x / 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 <= (-2.2d0)) .or. (.not. (x <= 0.00065d0))) then
        tmp = (1.0d0 / b) - (x / tan(b))
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -2.2) || !(x <= 0.00065)) {
		tmp = (1.0 / B) - (x / Math.tan(B));
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -2.2) or not (x <= 0.00065):
		tmp = (1.0 / B) - (x / math.tan(B))
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -2.2) || !(x <= 0.00065))
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -2.2) || ~((x <= 0.00065)))
		tmp = (1.0 / B) - (x / tan(B));
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -2.2], N[Not[LessEqual[x, 0.00065]], $MachinePrecision]], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.2000000000000002 or 6.4999999999999997e-4 < x

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unsub-neg 99.6%

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

      3. associate-*r/ 99.8%

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

      4. *-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-cbrt-cube 79.5%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{1}{\sin B} \cdot \frac{1}{\sin B}\right) \cdot \frac{1}{\sin B}}} - \frac{x}{\tan B} \]

      2. pow3 79.5%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{1}{\sin B}\right)}^{3}}} - \frac{x}{\tan B} \]

      3. inv-pow 79.5%

         \[\leadsto \sqrt[3]{{\color{blue}{\left({\sin B}^{-1}\right)}}^{3}} - \frac{x}{\tan B} \]

      4. pow-pow 79.5%

         \[\leadsto \sqrt[3]{\color{blue}{{\sin B}^{\left(-1 \cdot 3\right)}}} - \frac{x}{\tan B} \]

      5. metadata-eval 79.5%

         \[\leadsto \sqrt[3]{{\sin B}^{\color{blue}{-3}}} - \frac{x}{\tan B} \]

   5. Applied egg-rr 79.5%

      \[\leadsto \color{blue}{\sqrt[3]{{\sin B}^{-3}}} - \frac{x}{\tan B} \]

   6. Taylor expanded in B around 0 98.8%

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

   if -2.2000000000000002 < x < 6.4999999999999997e-4

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. unsub-neg 99.8%

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

      3. associate-*r/ 99.8%

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

      4. *-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%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \lor \neg \left(x \leq 0.00065\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 6: 73.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -400000000:\\ \;\;\;\;\frac{-x}{B} - B \cdot \left(x \cdot -0.3333333333333333\right)\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{B} + \left(B \cdot 0.16666666666666666 - \frac{-1}{B}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= x -400000000.0)
   (- (/ (- x) B) (* B (* x -0.3333333333333333)))
   (if (<= x 1.3e-7)
     (/ 1.0 (sin B))
     (+ (* x (/ -1.0 B)) (- (* B 0.16666666666666666) (/ -1.0 B))))))

C

double code(double B, double x) {
	double tmp;
	if (x <= -400000000.0) {
		tmp = (-x / B) - (B * (x * -0.3333333333333333));
	} else if (x <= 1.3e-7) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = (x * (-1.0 / B)) + ((B * 0.16666666666666666) - (-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 <= (-400000000.0d0)) then
        tmp = (-x / b) - (b * (x * (-0.3333333333333333d0)))
    else if (x <= 1.3d-7) then
        tmp = 1.0d0 / sin(b)
    else
        tmp = (x * ((-1.0d0) / b)) + ((b * 0.16666666666666666d0) - ((-1.0d0) / b))
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (x <= -400000000.0) {
		tmp = (-x / B) - (B * (x * -0.3333333333333333));
	} else if (x <= 1.3e-7) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = (x * (-1.0 / B)) + ((B * 0.16666666666666666) - (-1.0 / B));
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if x <= -400000000.0:
		tmp = (-x / B) - (B * (x * -0.3333333333333333))
	elif x <= 1.3e-7:
		tmp = 1.0 / math.sin(B)
	else:
		tmp = (x * (-1.0 / B)) + ((B * 0.16666666666666666) - (-1.0 / B))
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (x <= -400000000.0)
		tmp = Float64(Float64(Float64(-x) / B) - Float64(B * Float64(x * -0.3333333333333333)));
	elseif (x <= 1.3e-7)
		tmp = Float64(1.0 / sin(B));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / B)) + Float64(Float64(B * 0.16666666666666666) - Float64(-1.0 / B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (x <= -400000000.0)
		tmp = (-x / B) - (B * (x * -0.3333333333333333));
	elseif (x <= 1.3e-7)
		tmp = 1.0 / sin(B);
	else
		tmp = (x * (-1.0 / B)) + ((B * 0.16666666666666666) - (-1.0 / B));
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[x, -400000000.0], N[(N[((-x) / B), $MachinePrecision] - N[(B * N[(x * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.3e-7], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / B), $MachinePrecision]), $MachinePrecision] + N[(N[(B * 0.16666666666666666), $MachinePrecision] - N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4e8

   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%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in B around 0 71.9%

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

   5. Taylor expanded in x around inf 99.6%

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

      1. mul-1-neg 99.6%

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

      2. associate-*r/ 99.6%

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

      3. distribute-rgt-neg-in 99.6%

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

      4. neg-mul-1 99.6%

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

      5. associate-*r/ 99.6%

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

      6. *-commutative 99.6%

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

      7. remove-double-neg 99.6%

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

      8. neg-mul-1 99.6%

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

      9. times-frac 99.4%

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

      10. neg-mul-1 99.4%

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

      11. associate-/r* 99.4%

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

      12. metadata-eval 99.4%

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

      13. times-frac 99.6%

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

      14. *-rgt-identity 99.6%

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

      15. neg-mul-1 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in B around 0 48.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B} + -1 \cdot \left(\left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right) \cdot B\right)} \]
   9. Step-by-step derivation

      1. distribute-lft-out 48.2%

         \[\leadsto \color{blue}{-1 \cdot \left(\frac{x}{B} + \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right) \cdot B\right)} \]

      2. *-commutative 48.2%

         \[\leadsto -1 \cdot \left(\frac{x}{B} + \color{blue}{B \cdot \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right)}\right) \]

      3. distribute-rgt-out-- 48.2%

         \[\leadsto -1 \cdot \left(\frac{x}{B} + B \cdot \color{blue}{\left(x \cdot \left(-0.5 - -0.16666666666666666\right)\right)}\right) \]

      4. metadata-eval 48.2%

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

   10. Simplified 48.2%

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

   if -4e8 < x < 1.29999999999999999e-7

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. unsub-neg 99.8%

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

      3. associate-*r/ 99.8%

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

      4. *-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 96.8%

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

   5. Taylor expanded in B around inf 96.7%

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

   if 1.29999999999999999e-7 < x

   1. Initial program 99.5%

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

      1. distribute-lft-neg-in 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in B around 0 69.9%

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

   5. Taylor expanded in B around 0 56.4%

      \[\leadsto \left(-x\right) \cdot \color{blue}{\frac{1}{B}} + \left(0.16666666666666666 \cdot B + \frac{1}{B}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -400000000:\\ \;\;\;\;\frac{-x}{B} - B \cdot \left(x \cdot -0.3333333333333333\right)\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{B} + \left(B \cdot 0.16666666666666666 - \frac{-1}{B}\right)\\ \end{array} \]

Alternative 7: 50.8% accurate, 14.0× speedup

Math

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

FPCore

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

C

double code(double B, double x) {
	return ((1.0 / B) + (B * (0.16666666666666666 + (x * 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) + (b * (0.16666666666666666d0 + (x * 0.3333333333333333d0)))) - (x / b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

3. Simplified 99.7%

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

4. Taylor expanded in B around 0 57.4%

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

5. Taylor expanded in B around 0 47.5%

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

6. Final simplification 47.5%

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

Alternative 8: 50.7% accurate, 16.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[B_, x_] := N[(N[(x * N[(-1.0 / B), $MachinePrecision]), $MachinePrecision] + N[(N[(B * 0.16666666666666666), $MachinePrecision] - N[(-1.0 / 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%

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

3. Simplified 99.7%

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

4. Taylor expanded in B around 0 57.4%

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

5. Taylor expanded in B around 0 47.3%

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

6. Final simplification 47.3%

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

Alternative 9: 49.3% accurate, 25.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10000000 \lor \neg \left(x \leq 1.55 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -10000000.0) (not (<= x 1.55e+14))) (/ (- x) B) (/ 1.0 B)))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -10000000.0) || !(x <= 1.55e+14)) {
		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 <= (-10000000.0d0)) .or. (.not. (x <= 1.55d+14))) 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 <= -10000000.0) || !(x <= 1.55e+14)) {
		tmp = -x / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -10000000.0) or not (x <= 1.55e+14):
		tmp = -x / B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -10000000.0) || !(x <= 1.55e+14))
		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 <= -10000000.0) || ~((x <= 1.55e+14)))
		tmp = -x / B;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -10000000.0], N[Not[LessEqual[x, 1.55e+14]], $MachinePrecision]], N[((-x) / B), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1e7 or 1.55e14 < x

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unsub-neg 99.6%

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

      3. associate-*r/ 99.8%

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

      4. *-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 51.1%

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

   5. Taylor expanded in x around inf 50.8%

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

      1. associate-*r/ 50.8%

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

      2. mul-1-neg 50.8%

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

   7. Simplified 50.8%

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

   if -1e7 < x < 1.55e14

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. unsub-neg 99.8%

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

      3. associate-*r/ 99.8%

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

      4. *-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-cbrt-cube 70.3%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{1}{\sin B} \cdot \frac{1}{\sin B}\right) \cdot \frac{1}{\sin B}}} - \frac{x}{\tan B} \]

      2. pow3 70.3%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{1}{\sin B}\right)}^{3}}} - \frac{x}{\tan B} \]

      3. inv-pow 70.3%

         \[\leadsto \sqrt[3]{{\color{blue}{\left({\sin B}^{-1}\right)}}^{3}} - \frac{x}{\tan B} \]

      4. pow-pow 70.2%

         \[\leadsto \sqrt[3]{\color{blue}{{\sin B}^{\left(-1 \cdot 3\right)}}} - \frac{x}{\tan B} \]

      5. metadata-eval 70.2%

         \[\leadsto \sqrt[3]{{\sin B}^{\color{blue}{-3}}} - \frac{x}{\tan B} \]

   5. Applied egg-rr 70.2%

      \[\leadsto \color{blue}{\sqrt[3]{{\sin B}^{-3}}} - \frac{x}{\tan B} \]

   6. Taylor expanded in B around 0 45.0%

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

   7. Taylor expanded in x around 0 42.9%

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

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10000000 \lor \neg \left(x \leq 1.55 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]

Alternative 10: 50.6% 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. +-commutative 99.7%

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

   2. unsub-neg 99.7%

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

   3. associate-*r/ 99.8%

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

   4. *-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 73.4%

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

5. Taylor expanded in B around 0 47.2%

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

6. Final simplification 47.2%

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

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. Step-by-step derivation

   1. distribute-lft-neg-in 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in B around 0 57.4%

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

5. Taylor expanded in B around inf 3.0%

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

   1. *-commutative 3.0%

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

7. Simplified 3.0%

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

8. Final simplification 3.0%

   \[\leadsto B \cdot 0.16666666666666666 \]

Alternative 12: 26.3% 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. +-commutative 99.7%

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

   2. unsub-neg 99.7%

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

   3. associate-*r/ 99.8%

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

   4. *-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-cbrt-cube 75.0%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{1}{\sin B} \cdot \frac{1}{\sin B}\right) \cdot \frac{1}{\sin B}}} - \frac{x}{\tan B} \]

   2. pow3 75.0%

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{1}{\sin B}\right)}^{3}}} - \frac{x}{\tan B} \]

   3. inv-pow 75.0%

      \[\leadsto \sqrt[3]{{\color{blue}{\left({\sin B}^{-1}\right)}}^{3}} - \frac{x}{\tan B} \]

   4. pow-pow 75.0%

      \[\leadsto \sqrt[3]{\color{blue}{{\sin B}^{\left(-1 \cdot 3\right)}}} - \frac{x}{\tan B} \]

   5. metadata-eval 75.0%

      \[\leadsto \sqrt[3]{{\sin B}^{\color{blue}{-3}}} - \frac{x}{\tan B} \]

5. Applied egg-rr 75.0%

   \[\leadsto \color{blue}{\sqrt[3]{{\sin B}^{-3}}} - \frac{x}{\tan B} \]

6. Taylor expanded in B around 0 72.9%

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

7. Taylor expanded in x around 0 22.3%

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

8. Final simplification 22.3%

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

Reproduce

herbie shell --seed 2023222 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))