VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 9.1s

Alternatives: 8

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 99.6%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
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. cancel-sign-sub-inv 99.6%

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

   4. *-commutative 99.6%

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

   5. *-commutative 99.6%

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

   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. Add Preprocessing

5. Final simplification 99.7%

   \[\leadsto \frac{1}{\sin B} - \frac{x}{\tan B} \]
6. Add Preprocessing

Alternative 2: 98.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.8 \cdot 10^{-12} \lor \neg \left(x \leq 0.05\right):\\ \;\;\;\;\frac{1 - x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -9.8e-12) (not (<= x 0.05)))
   (/ (- 1.0 x) (tan B))
   (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -9.8e-12) || !(x <= 0.05)) {
		tmp = (1.0 - 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 <= (-9.8d-12)) .or. (.not. (x <= 0.05d0))) then
        tmp = (1.0d0 - 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 <= -9.8e-12) || !(x <= 0.05)) {
		tmp = (1.0 - x) / Math.tan(B);
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -9.8e-12) or not (x <= 0.05):
		tmp = (1.0 - x) / math.tan(B)
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -9.8e-12) || !(x <= 0.05))
		tmp = Float64(Float64(1.0 - 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 <= -9.8e-12) || ~((x <= 0.05)))
		tmp = (1.0 - x) / tan(B);
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -9.8e-12], N[Not[LessEqual[x, 0.05]], $MachinePrecision]], N[(N[(1.0 - x), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.79999999999999944e-12 or 0.050000000000000003 < x

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      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 99.6%

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

      5. frac-sub 85.0%

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

      6. *-un-lft-identity 85.0%

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

      7. *-commutative 85.0%

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

      8. *-un-lft-identity 85.0%

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

   4. Applied egg-rr 85.0%

      \[\leadsto \color{blue}{\frac{\frac{\tan B}{x} - \sin B}{\sin B \cdot \frac{\tan B}{x}}} \]
   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 99.6%

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

      3. *-inverses 99.6%

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

   6. Simplified 99.6%

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

      1. div-inv 99.6%

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

      2. clear-num 99.8%

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

      3. div-inv 99.5%

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

      4. associate-*r* 99.6%

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

      5. sub-neg 99.6%

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

      6. associate-/l/ 99.3%

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

      7. associate-/r* 99.6%

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

      8. metadata-eval 99.6%

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

   8. Applied egg-rr 99.6%

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

   9. Taylor expanded in B around 0 97.7%

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

       1. expm1-log1p-u 47.3%

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

       2. expm1-udef 47.3%

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

       3. un-div-inv 47.3%

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

       4. *-commutative 47.3%

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

       5. associate-/l* 47.3%

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

   11. Applied egg-rr 47.3%

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

       1. expm1-def 47.3%

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

       2. expm1-log1p 97.6%

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

       3. associate-/l* 97.9%

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

       4. distribute-rgt-in 97.9%

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

       5. lft-mult-inverse 97.9%

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

       6. neg-mul-1 97.9%

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

       7. sub-neg 97.9%

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

   13. Simplified 97.9%

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

   if -9.79999999999999944e-12 < x < 0.050000000000000003

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 99.1%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.8 \cdot 10^{-12} \lor \neg \left(x \leq 0.05\right):\\ \;\;\;\;\frac{1 - x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.8 \cdot 10^{-12}:\\ \;\;\;\;\frac{x \cdot \left(\frac{1}{x} + -1\right)}{\tan B}\\ \mathbf{elif}\;x \leq 0.05:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= x -9.8e-12)
   (/ (* x (+ (/ 1.0 x) -1.0)) (tan B))
   (if (<= x 0.05) (/ 1.0 (sin B)) (/ (- 1.0 x) (tan B)))))

C

double code(double B, double x) {
	double tmp;
	if (x <= -9.8e-12) {
		tmp = (x * ((1.0 / x) + -1.0)) / tan(B);
	} else if (x <= 0.05) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = (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 <= (-9.8d-12)) then
        tmp = (x * ((1.0d0 / x) + (-1.0d0))) / tan(b)
    else if (x <= 0.05d0) then
        tmp = 1.0d0 / sin(b)
    else
        tmp = (1.0d0 - x) / tan(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (x <= -9.8e-12) {
		tmp = (x * ((1.0 / x) + -1.0)) / Math.tan(B);
	} else if (x <= 0.05) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = (1.0 - x) / Math.tan(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if x <= -9.8e-12:
		tmp = (x * ((1.0 / x) + -1.0)) / math.tan(B)
	elif x <= 0.05:
		tmp = 1.0 / math.sin(B)
	else:
		tmp = (1.0 - x) / math.tan(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (x <= -9.8e-12)
		tmp = Float64(Float64(x * Float64(Float64(1.0 / x) + -1.0)) / tan(B));
	elseif (x <= 0.05)
		tmp = Float64(1.0 / sin(B));
	else
		tmp = Float64(Float64(1.0 - x) / tan(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (x <= -9.8e-12)
		tmp = (x * ((1.0 / x) + -1.0)) / tan(B);
	elseif (x <= 0.05)
		tmp = 1.0 / sin(B);
	else
		tmp = (1.0 - x) / tan(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[x, -9.8e-12], N[(N[(x * N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.05], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.79999999999999944e-12

   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 99.8%

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

      3. sub-neg 99.8%

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

      4. clear-num 99.7%

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

      5. frac-sub 83.7%

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

      6. *-un-lft-identity 83.7%

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

      7. *-commutative 83.7%

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

      8. *-un-lft-identity 83.7%

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

   4. Applied egg-rr 83.7%

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

      1. div-inv 99.7%

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

      2. clear-num 99.8%

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

      3. div-inv 99.5%

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

      4. associate-*r* 99.6%

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

      5. sub-neg 99.6%

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

      6. associate-/l/ 99.0%

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

      7. associate-/r* 99.6%

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

      8. metadata-eval 99.6%

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

   8. Applied egg-rr 99.6%

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

   9. Taylor expanded in B around 0 97.1%

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

       1. un-div-inv 97.3%

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

       2. *-commutative 97.3%

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

   11. Applied egg-rr 97.3%

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

   if -9.79999999999999944e-12 < x < 0.050000000000000003

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 99.1%

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

   if 0.050000000000000003 < x

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      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 86.4%

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

      6. *-un-lft-identity 86.4%

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

      7. *-commutative 86.4%

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

      8. *-un-lft-identity 86.4%

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

   4. Applied egg-rr 86.4%

      \[\leadsto \color{blue}{\frac{\frac{\tan B}{x} - \sin B}{\sin B \cdot \frac{\tan B}{x}}} \]
   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 99.6%

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

      3. *-inverses 99.6%

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

   6. Simplified 99.6%

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

      1. div-inv 99.6%

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

      2. clear-num 99.7%

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

      3. div-inv 99.5%

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

      4. associate-*r* 99.6%

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

      5. sub-neg 99.6%

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

      6. associate-/l/ 99.6%

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

      7. associate-/r* 99.6%

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

      8. metadata-eval 99.6%

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

   8. Applied egg-rr 99.6%

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

   9. Taylor expanded in B around 0 98.3%

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

       1. expm1-log1p-u 46.5%

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

       2. expm1-udef 46.5%

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

       3. un-div-inv 46.5%

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

       4. *-commutative 46.5%

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

       5. associate-/l* 46.5%

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

   11. Applied egg-rr 46.5%

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

       1. expm1-def 46.5%

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

       2. expm1-log1p 98.4%

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

       3. associate-/l* 98.4%

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

       4. distribute-rgt-in 98.4%

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

       5. lft-mult-inverse 98.4%

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

       6. neg-mul-1 98.4%

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

       7. sub-neg 98.4%

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

   13. Simplified 98.4%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.8 \cdot 10^{-12}:\\ \;\;\;\;\frac{x \cdot \left(\frac{1}{x} + -1\right)}{\tan B}\\ \mathbf{elif}\;x \leq 0.05:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 400:\\ \;\;\;\;B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B 400.0)
   (+ (* B (+ 0.16666666666666666 (* x 0.3333333333333333))) (/ (- 1.0 x) B))
   (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if (B <= 400.0) {
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - 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 <= 400.0d0) then
        tmp = (b * (0.16666666666666666d0 + (x * 0.3333333333333333d0))) + ((1.0d0 - 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 <= 400.0) {
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B);
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if B <= 400.0:
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B)
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (B <= 400.0)
		tmp = Float64(Float64(B * Float64(0.16666666666666666 + Float64(x * 0.3333333333333333))) + Float64(Float64(1.0 - 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 <= 400.0)
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B);
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[B, 400.0], N[(N[(B * N[(0.16666666666666666 + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 400

   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 66.2%

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

      1. associate--l+ 66.2%

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

      2. *-commutative 66.2%

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

      3. div-sub 66.2%

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

   5. Simplified 66.2%

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

   if 400 < B

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0 42.1%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 400:\\ \;\;\;\;B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 50.7% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.4 \cdot 10^{-7} \lor \neg \left(x \leq 15600\right):\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -7.4e-7) (not (<= x 15600.0))) (- (/ x B)) (/ 1.0 B)))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -7.4e-7) || !(x <= 15600.0)) {
		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 <= (-7.4d-7)) .or. (.not. (x <= 15600.0d0))) 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 <= -7.4e-7) || !(x <= 15600.0)) {
		tmp = -(x / B);
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -7.4e-7) or not (x <= 15600.0):
		tmp = -(x / B)
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -7.4e-7) || !(x <= 15600.0))
		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 <= -7.4e-7) || ~((x <= 15600.0)))
		tmp = -(x / B);
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -7.4e-7], N[Not[LessEqual[x, 15600.0]], $MachinePrecision]], (-N[(x / B), $MachinePrecision]), N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.40000000000000009e-7 or 15600 < 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 50.4%

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

   4. Taylor expanded in x around inf 49.4%

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

      1. neg-mul-1 49.4%

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

      2. distribute-neg-frac 49.4%

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

   6. Simplified 49.4%

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

   if -7.40000000000000009e-7 < x < 15600

   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 56.0%

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

      1. div-inv 56.0%

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

      2. add-sqr-sqrt 23.8%

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

      3. associate-/l* 23.8%

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

   5. Applied egg-rr 23.8%

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

   6. Taylor expanded in x around 0 55.2%

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

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.4 \cdot 10^{-7} \lor \neg \left(x \leq 15600\right):\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.3% accurate, 19.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 76.2%

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

4. Taylor expanded in B around 0 53.2%

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

   1. associate--l+ 53.2%

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

   2. div-sub 53.2%

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

6. Simplified 53.2%

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

7. Final simplification 53.2%

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

Alternative 7: 51.9% 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 99.6%

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

3. Taylor expanded in B around 0 53.1%

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

4. Final simplification 53.1%

   \[\leadsto \frac{1 - x}{B} \]
5. Add Preprocessing

Alternative 8: 26.7% 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 99.6%

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

3. Taylor expanded in B around 0 76.2%

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

   1. div-inv 76.2%

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

   2. add-sqr-sqrt 35.9%

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

   3. associate-/l* 36.0%

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

5. Applied egg-rr 36.0%

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

6. Taylor expanded in x around 0 30.1%

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

7. Final simplification 30.1%

   \[\leadsto \frac{1}{B} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024019 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))