VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 9.6s

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. distribute-lft-neg-in 99.7%

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

   2. +-commutative 99.7%

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

   3. *-commutative 99.7%

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

   4. remove-double-neg 99.7%

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

   5. distribute-frac-neg2 99.7%

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

   6. tan-neg 99.7%

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

   7. cancel-sign-sub-inv 99.7%

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

   8. associate-*l/ 99.8%

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

   9. *-lft-identity 99.8%

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

   10. tan-neg 99.8%

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

   11. distribute-neg-frac2 99.8%

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

   12. distribute-neg-frac 99.8%

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

   13. remove-double-neg 99.8%

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

3. Simplified 99.8%

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

Alternative 2: 98.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-10} \lor \neg \left(x \leq 0.038\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -2.9e-10) (not (<= x 0.038)))
   (- (/ 1.0 B) (/ x (tan B)))
   (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -2.9e-10) || !(x <= 0.038)) {
		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 <= (-2.9d-10)) .or. (.not. (x <= 0.038d0))) 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 <= -2.9e-10) || !(x <= 0.038)) {
		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 <= -2.9e-10) or not (x <= 0.038):
		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 <= -2.9e-10) || !(x <= 0.038))
		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 <= -2.9e-10) || ~((x <= 0.038)))
		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, -2.9e-10], N[Not[LessEqual[x, 0.038]], $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 < -2.89999999999999981e-10 or 0.0379999999999999991 < x

   1. Initial program 99.6%

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

      1. distribute-lft-neg-in 99.6%

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

      2. +-commutative 99.6%

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

      3. *-commutative 99.6%

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

      4. remove-double-neg 99.6%

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

      5. distribute-frac-neg2 99.6%

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

      6. tan-neg 99.6%

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

      7. cancel-sign-sub-inv 99.6%

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

      8. associate-*l/ 99.7%

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

      9. *-lft-identity 99.7%

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

      10. tan-neg 99.7%

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

      11. distribute-neg-frac2 99.7%

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

      12. distribute-neg-frac 99.7%

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

      13. remove-double-neg 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. Add Preprocessing

   5. Taylor expanded in B around 0 99.0%

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

   if -2.89999999999999981e-10 < x < 0.0379999999999999991

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 97.5%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-10} \lor \neg \left(x \leq 0.038\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -2.6) (not (<= x 1.0))) (/ x (- (tan B))) (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -2.6) || !(x <= 1.0)) {
		tmp = 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 <= (-2.6d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = 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 <= -2.6) || !(x <= 1.0)) {
		tmp = x / -Math.tan(B);
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -2.6) or not (x <= 1.0):
		tmp = x / -math.tan(B)
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -2.6) || !(x <= 1.0))
		tmp = Float64(x / Float64(-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 <= -2.6) || ~((x <= 1.0)))
		tmp = x / -tan(B);
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -2.6], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.60000000000000009 or 1 < x

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. div-inv 99.7%

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

      3. sub-neg 99.7%

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

      4. clear-num 99.6%

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

      5. frac-sub 92.9%

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

      6. *-un-lft-identity 92.9%

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

      7. *-commutative 92.9%

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

      8. *-un-lft-identity 92.9%

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

   4. Applied egg-rr 92.9%

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

      1. associate-/r* 99.6%

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

      2. div-sub 99.7%

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

      3. *-inverses 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in x around inf 99.3%

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

      1. associate-/r/ 99.3%

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

      2. *-commutative 99.3%

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

   9. Applied egg-rr 99.3%

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

       1. associate-*r/ 99.4%

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

       2. associate-*l/ 99.4%

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

       3. *-commutative 99.4%

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

       4. neg-mul-1 99.4%

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

       5. distribute-neg-frac2 99.4%

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

   11. Simplified 99.4%

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

   if -2.60000000000000009 < x < 1

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 96.1%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.34:\\ \;\;\;\;\frac{\left(1 + \left(B \cdot B\right) \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right)\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B 0.34)
   (/
    (- (+ 1.0 (* (* B B) (+ 0.16666666666666666 (* x 0.3333333333333333)))) x)
    B)
   (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if (B <= 0.34) {
		tmp = ((1.0 + ((B * B) * (0.16666666666666666 + (x * 0.3333333333333333)))) - x) / B;
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (b <= 0.34d0) then
        tmp = ((1.0d0 + ((b * b) * (0.16666666666666666d0 + (x * 0.3333333333333333d0)))) - x) / b
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (B <= 0.34) {
		tmp = ((1.0 + ((B * B) * (0.16666666666666666 + (x * 0.3333333333333333)))) - x) / B;
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if B <= 0.34:
		tmp = ((1.0 + ((B * B) * (0.16666666666666666 + (x * 0.3333333333333333)))) - x) / B
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (B <= 0.34)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(B * B) * Float64(0.16666666666666666 + Float64(x * 0.3333333333333333)))) - x) / B);
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (B <= 0.34)
		tmp = ((1.0 + ((B * B) * (0.16666666666666666 + (x * 0.3333333333333333)))) - x) / B;
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[B, 0.34], N[(N[(N[(1.0 + N[(N[(B * B), $MachinePrecision] * N[(0.16666666666666666 + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 0.340000000000000024

   1. Initial program 99.8%

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

   3. Taylor expanded in B around 0 65.5%

      \[\leadsto \color{blue}{\frac{\left(1 + {B}^{2} \cdot \left(0.16666666666666666 + 0.3333333333333333 \cdot x\right)\right) - x}{B}} \]
   4. Step-by-step derivation

      1. unpow2 65.5%

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

   5. Applied egg-rr 65.5%

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

   if 0.340000000000000024 < B

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 58.0%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 0.34:\\ \;\;\;\;\frac{\left(1 + \left(B \cdot B\right) \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right)\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.3% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in B around 0 49.4%

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

4. Taylor expanded in x around 0 49.4%

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

5. Final simplification 49.4%

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

Alternative 6: 49.2% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+31}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= x -4e+31) (/ x (- B)) (if (<= x 1.0) (/ 1.0 B) (/ -1.0 (/ B x)))))

C

double code(double B, double x) {
	double tmp;
	if (x <= -4e+31) {
		tmp = x / -B;
	} else if (x <= 1.0) {
		tmp = 1.0 / B;
	} else {
		tmp = -1.0 / (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 <= (-4d+31)) then
        tmp = x / -b
    else if (x <= 1.0d0) then
        tmp = 1.0d0 / b
    else
        tmp = (-1.0d0) / (b / x)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (x <= -4e+31) {
		tmp = x / -B;
	} else if (x <= 1.0) {
		tmp = 1.0 / B;
	} else {
		tmp = -1.0 / (B / x);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if x <= -4e+31:
		tmp = x / -B
	elif x <= 1.0:
		tmp = 1.0 / B
	else:
		tmp = -1.0 / (B / x)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (x <= -4e+31)
		tmp = Float64(x / Float64(-B));
	elseif (x <= 1.0)
		tmp = Float64(1.0 / B);
	else
		tmp = Float64(-1.0 / Float64(B / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (x <= -4e+31)
		tmp = x / -B;
	elseif (x <= 1.0)
		tmp = 1.0 / B;
	else
		tmp = -1.0 / (B / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[x, -4e+31], N[(x / (-B)), $MachinePrecision], If[LessEqual[x, 1.0], N[(1.0 / B), $MachinePrecision], N[(-1.0 / N[(B / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.9999999999999999e31

   1. Initial program 99.6%

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

   3. Taylor expanded in B around 0 38.0%

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

   4. Taylor expanded in x around inf 38.0%

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

      1. associate-*r/ 38.0%

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

      2. neg-mul-1 38.0%

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

   6. Simplified 38.0%

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

   if -3.9999999999999999e31 < x < 1

   1. Initial program 99.8%

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

   3. Taylor expanded in B around 0 55.5%

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

   4. Taylor expanded in x around 0 53.7%

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

   if 1 < x

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. div-inv 99.7%

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

      3. sub-neg 99.7%

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

      4. clear-num 99.6%

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

      5. frac-sub 95.7%

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

      6. *-un-lft-identity 95.7%

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

      7. *-commutative 95.7%

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

      8. *-un-lft-identity 95.7%

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

   4. Applied egg-rr 95.7%

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

      1. associate-/r* 99.6%

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

      2. div-sub 99.7%

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

      3. *-inverses 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in x around inf 99.0%

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

   8. Taylor expanded in B around 0 43.5%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+31}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.3% accurate, 14.0× speedup

Math

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

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 + Float64(Float64(B * B) * Float64(0.16666666666666666 + Float64(x * 0.3333333333333333)))) - 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 + N[(N[(B * B), $MachinePrecision] * N[(0.16666666666666666 + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in B around 0 49.4%

   \[\leadsto \color{blue}{\frac{\left(1 + {B}^{2} \cdot \left(0.16666666666666666 + 0.3333333333333333 \cdot x\right)\right) - x}{B}} \]
4. Step-by-step derivation

   1. unpow2 49.4%

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

5. Applied egg-rr 49.4%

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

6. Final simplification 49.4%

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

Alternative 8: 49.2% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+31} \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -4e+31) (not (<= x 1.0))) (/ x (- B)) (/ 1.0 B)))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -4e+31) || !(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 <= (-4d+31)) .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 <= -4e+31) || !(x <= 1.0)) {
		tmp = x / -B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -4e+31) 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 <= -4e+31) || !(x <= 1.0))
		tmp = Float64(x / Float64(-B));
	else
		tmp = Float64(1.0 / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -4e+31) || ~((x <= 1.0)))
		tmp = x / -B;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -4e+31], 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 < -3.9999999999999999e31 or 1 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in B around 0 41.3%

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

   4. Taylor expanded in x around inf 40.9%

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

      1. associate-*r/ 40.9%

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

      2. neg-mul-1 40.9%

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

   6. Simplified 40.9%

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

   if -3.9999999999999999e31 < x < 1

   1. Initial program 99.8%

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

   3. Taylor expanded in B around 0 55.5%

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

   4. Taylor expanded in x around 0 53.7%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+31} \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 51.4% accurate, 23.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[B_, x_] := N[(N[(B * 0.16666666666666666), $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. Add Preprocessing

3. Taylor expanded in B around 0 49.4%

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

4. Taylor expanded in x around 0 49.4%

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

5. Taylor expanded in B around 0 49.0%

   \[\leadsto 0.16666666666666666 \cdot B + \color{blue}{\frac{1 + -1 \cdot x}{B}} \]
6. Step-by-step derivation

   1. neg-mul-1 49.0%

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

   2. sub-neg 49.0%

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

7. Simplified 49.0%

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

8. Final simplification 49.0%

   \[\leadsto B \cdot 0.16666666666666666 + \frac{1 - x}{B} \]
9. Add Preprocessing

Alternative 10: 51.2% accurate, 42.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in B around 0 48.7%

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

Alternative 11: 26.2% accurate, 70.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in B around 0 48.7%

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

4. Taylor expanded in x around 0 29.4%

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

Alternative 12: 3.1% accurate, 70.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in B around 0 49.4%

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

4. Taylor expanded in x around 0 49.4%

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

5. Taylor expanded in x around inf 22.3%

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

   1. *-commutative 22.3%

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

7. Simplified 22.3%

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

8. Taylor expanded in x around 0 3.3%

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

   1. *-commutative 3.3%

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

10. Simplified 3.3%

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

Reproduce

herbie shell --seed 2024132 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))