VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 8.4s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. distribute-lft-neg-in 99.7%

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

   2. +-commutative 99.7%

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

   3. cancel-sign-sub-inv 99.7%

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

   4. *-commutative 99.7%

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

   5. *-commutative 99.7%

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

   6. associate-*r/ 99.8%

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

   7. *-rgt-identity 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 98.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.11:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;x \leq 1.05:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} + -1}{\frac{\tan B}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= x -0.11)
   (- (/ 1.0 B) (/ x (tan B)))
   (if (<= x 1.05)
     (/ (- 1.0 x) (sin B))
     (/ (+ (/ 1.0 x) -1.0) (/ (tan B) x)))))

C

double code(double B, double x) {
	double tmp;
	if (x <= -0.11) {
		tmp = (1.0 / B) - (x / tan(B));
	} else if (x <= 1.05) {
		tmp = (1.0 - x) / sin(B);
	} else {
		tmp = ((1.0 / x) + -1.0) / (tan(B) / x);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-0.11d0)) then
        tmp = (1.0d0 / b) - (x / tan(b))
    else if (x <= 1.05d0) then
        tmp = (1.0d0 - x) / sin(b)
    else
        tmp = ((1.0d0 / x) + (-1.0d0)) / (tan(b) / x)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (x <= -0.11) {
		tmp = (1.0 / B) - (x / Math.tan(B));
	} else if (x <= 1.05) {
		tmp = (1.0 - x) / Math.sin(B);
	} else {
		tmp = ((1.0 / x) + -1.0) / (Math.tan(B) / x);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if x <= -0.11:
		tmp = (1.0 / B) - (x / math.tan(B))
	elif x <= 1.05:
		tmp = (1.0 - x) / math.sin(B)
	else:
		tmp = ((1.0 / x) + -1.0) / (math.tan(B) / x)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (x <= -0.11)
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	elseif (x <= 1.05)
		tmp = Float64(Float64(1.0 - x) / sin(B));
	else
		tmp = Float64(Float64(Float64(1.0 / x) + -1.0) / Float64(tan(B) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (x <= -0.11)
		tmp = (1.0 / B) - (x / tan(B));
	elseif (x <= 1.05)
		tmp = (1.0 - x) / sin(B);
	else
		tmp = ((1.0 / x) + -1.0) / (tan(B) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[x, -0.11], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.05], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision] / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.110000000000000001

   1. Initial program 99.7%

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

      1. distribute-lft-neg-in 99.7%

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

      2. +-commutative 99.7%

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

      3. cancel-sign-sub-inv 99.7%

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

      4. *-commutative 99.7%

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

      5. *-commutative 99.7%

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

      6. associate-*r/ 99.8%

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

      7. *-rgt-identity 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in B around 0 99.8%

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

   if -0.110000000000000001 < x < 1.05000000000000004

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      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 99.8%

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

      5. frac-sub 89.6%

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

      6. *-un-lft-identity 89.6%

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

      7. *-commutative 89.6%

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

      8. *-un-lft-identity 89.6%

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

   4. Applied egg-rr 89.6%

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

      1. associate-/r* 99.5%

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

      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. Taylor expanded in B around inf 99.6%

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

      1. associate-*r* 99.7%

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

      2. sub-neg 99.7%

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

      3. associate-/r* 99.6%

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

      4. metadata-eval 99.6%

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

   9. Simplified 99.6%

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

       1. +-commutative 99.6%

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

       2. distribute-rgt-in 99.6%

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

       3. associate-*r* 99.6%

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

       4. neg-mul-1 99.6%

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

       5. associate-/r* 99.7%

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

       6. lft-mult-inverse 99.8%

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

   11. Applied egg-rr 99.8%

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

   12. Taylor expanded in B around 0 98.5%

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

       1. mul-1-neg 98.5%

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

       2. sub-neg 98.5%

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

   14. Simplified 98.5%

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

   if 1.05000000000000004 < 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 99.8%

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

      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 87.8%

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

      6. *-un-lft-identity 87.8%

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

      7. *-commutative 87.8%

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

      8. *-un-lft-identity 87.8%

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

   4. Applied egg-rr 87.8%

      \[\leadsto \color{blue}{\frac{\frac{\tan B}{x} - \sin B}{\sin B \cdot \frac{\tan B}{x}}} \]
   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. Taylor expanded in B around 0 97.4%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.11:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;x \leq 1.05:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} + -1}{\frac{\tan B}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.11 \lor \neg \left(x \leq 15.5\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -0.11) (not (<= x 15.5)))
   (- (/ 1.0 B) (/ x (tan B)))
   (/ (- 1.0 x) (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -0.11) || !(x <= 15.5)) {
		tmp = (1.0 / B) - (x / tan(B));
	} else {
		tmp = (1.0 - x) / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-0.11d0)) .or. (.not. (x <= 15.5d0))) then
        tmp = (1.0d0 / b) - (x / tan(b))
    else
        tmp = (1.0d0 - x) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -0.11) || !(x <= 15.5)) {
		tmp = (1.0 / B) - (x / Math.tan(B));
	} else {
		tmp = (1.0 - x) / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -0.11) or not (x <= 15.5):
		tmp = (1.0 / B) - (x / math.tan(B))
	else:
		tmp = (1.0 - x) / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -0.11) || !(x <= 15.5))
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(1.0 - x) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -0.11) || ~((x <= 15.5)))
		tmp = (1.0 / B) - (x / tan(B));
	else
		tmp = (1.0 - x) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -0.11], N[Not[LessEqual[x, 15.5]], $MachinePrecision]], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.110000000000000001 or 15.5 < x

   1. Initial program 99.6%

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

      1. 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/ 99.8%

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

      7. *-rgt-identity 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in B around 0 99.0%

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

   if -0.110000000000000001 < x < 15.5

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      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 99.8%

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

      5. frac-sub 89.6%

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

      6. *-un-lft-identity 89.6%

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

      7. *-commutative 89.6%

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

      8. *-un-lft-identity 89.6%

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

   4. Applied egg-rr 89.6%

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

      1. associate-/r* 99.5%

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

      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. Taylor expanded in B around inf 99.6%

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

      1. associate-*r* 99.7%

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

      2. sub-neg 99.7%

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

      3. associate-/r* 99.6%

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

      4. metadata-eval 99.6%

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

   9. Simplified 99.6%

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

       1. +-commutative 99.6%

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

       2. distribute-rgt-in 99.6%

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

       3. associate-*r* 99.6%

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

       4. neg-mul-1 99.6%

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

       5. associate-/r* 99.7%

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

       6. lft-mult-inverse 99.8%

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

   11. Applied egg-rr 99.8%

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

   12. Taylor expanded in B around 0 98.0%

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

       1. mul-1-neg 98.0%

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

       2. sub-neg 98.0%

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

   14. Simplified 98.0%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.11 \lor \neg \left(x \leq 15.5\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.15:\\ \;\;\;\;\left(\frac{1}{B} + B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right)\right) - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B 0.15)
   (-
    (+ (/ 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 <= 0.15) {
		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 <= 0.15d0) 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 <= 0.15) {
		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 <= 0.15:
		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 <= 0.15)
		tmp = Float64(Float64(Float64(1.0 / B) + Float64(B * Float64(0.16666666666666666 + Float64(x * 0.3333333333333333)))) - Float64(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 <= 0.15)
		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, 0.15], 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], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 0.149999999999999994

   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 67.3%

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

   if 0.149999999999999994 < 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 47.0%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 0.15:\\ \;\;\;\;\left(\frac{1}{B} + B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right)\right) - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. +-commutative 99.7%

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

   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 89.2%

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

   6. *-un-lft-identity 89.2%

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

   7. *-commutative 89.2%

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

   8. *-un-lft-identity 89.2%

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

4. Applied egg-rr 89.2%

   \[\leadsto \color{blue}{\frac{\frac{\tan B}{x} - \sin B}{\sin B \cdot \frac{\tan B}{x}}} \]
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. Taylor expanded in B around inf 99.6%

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

   1. associate-*r* 99.6%

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

   2. sub-neg 99.6%

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

   3. associate-/r* 99.6%

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

   4. metadata-eval 99.6%

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

9. Simplified 99.6%

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

    1. +-commutative 99.6%

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

    2. distribute-rgt-in 99.6%

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

    3. associate-*r* 99.6%

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

    4. neg-mul-1 99.6%

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

    5. associate-/r* 99.6%

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

    6. lft-mult-inverse 99.7%

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

11. Applied egg-rr 99.7%

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

12. Taylor expanded in B around 0 75.8%

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

    1. mul-1-neg 75.8%

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

    2. sub-neg 75.8%

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

14. Simplified 75.8%

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

15. Final simplification 75.8%

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

Alternative 6: 49.4% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 6.5 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 6.5e-9))) (/ (- x) B) (/ 1.0 B)))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 6.5e-9)) {
		tmp = -x / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 6.5d-9))) then
        tmp = -x / b
    else
        tmp = 1.0d0 / b
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 6.5e-9)) {
		tmp = -x / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -1.0) or not (x <= 6.5e-9):
		tmp = -x / B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 6.5e-9))
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(1.0 / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 6.5e-9)))
		tmp = -x / B;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 6.5e-9]], $MachinePrecision]], N[((-x) / B), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 6.5000000000000003e-9 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in B around 0 46.8%

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

   4. Taylor expanded in x around inf 44.6%

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

      1. neg-mul-1 44.6%

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

      2. distribute-neg-frac 44.6%

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

   6. Simplified 44.6%

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

   if -1 < x < 6.5000000000000003e-9

   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 56.7%

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

   4. Taylor expanded in x around 0 55.5%

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

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 6.5 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.0% accurate, 19.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. distribute-lft-neg-in 99.7%

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

   2. +-commutative 99.7%

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

   3. cancel-sign-sub-inv 99.7%

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

   4. *-commutative 99.7%

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

   5. *-commutative 99.7%

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

   6. associate-*r/ 99.8%

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

   7. *-rgt-identity 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in B around 0 77.2%

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

6. Taylor expanded in B around 0 52.6%

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

   1. associate--l+ 52.6%

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

   2. *-commutative 52.6%

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

   3. div-sub 52.6%

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

8. Simplified 52.6%

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

9. Final simplification 52.6%

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

Alternative 8: 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. Add Preprocessing

3. Taylor expanded in B around 0 51.8%

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

4. Final simplification 51.8%

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

Alternative 9: 26.6% 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 51.8%

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

4. Taylor expanded in x around 0 29.5%

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

5. Final simplification 29.5%

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

Reproduce

herbie shell --seed 2024041 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))