VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 9.1s

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. *-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. *-commutative 99.7%

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

   9. associate-*r/ 99.8%

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

   10. *-rgt-identity 99.8%

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

   11. tan-neg 99.8%

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

   12. distribute-neg-frac2 99.8%

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

   13. distribute-neg-frac 99.8%

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

   14. 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: 98.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.225:\\ \;\;\;\;\frac{1 - x}{\tan B}\\ \mathbf{elif}\;x \leq 1.6:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\tan B} \cdot \left(-1 + \frac{1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= x -0.225)
   (/ (- 1.0 x) (tan B))
   (if (<= x 1.6)
     (- (/ 1.0 (sin B)) (/ x B))
     (* (/ x (tan B)) (+ -1.0 (/ 1.0 x))))))

C

double code(double B, double x) {
	double tmp;
	if (x <= -0.225) {
		tmp = (1.0 - x) / tan(B);
	} else if (x <= 1.6) {
		tmp = (1.0 / sin(B)) - (x / B);
	} else {
		tmp = (x / tan(B)) * (-1.0 + (1.0 / 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 <= (-0.225d0)) then
        tmp = (1.0d0 - x) / tan(b)
    else if (x <= 1.6d0) then
        tmp = (1.0d0 / sin(b)) - (x / b)
    else
        tmp = (x / tan(b)) * ((-1.0d0) + (1.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (x <= -0.225) {
		tmp = (1.0 - x) / Math.tan(B);
	} else if (x <= 1.6) {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	} else {
		tmp = (x / Math.tan(B)) * (-1.0 + (1.0 / x));
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if x <= -0.225:
		tmp = (1.0 - x) / math.tan(B)
	elif x <= 1.6:
		tmp = (1.0 / math.sin(B)) - (x / B)
	else:
		tmp = (x / math.tan(B)) * (-1.0 + (1.0 / x))
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (x <= -0.225)
		tmp = Float64(Float64(1.0 - x) / tan(B));
	elseif (x <= 1.6)
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	else
		tmp = Float64(Float64(x / tan(B)) * Float64(-1.0 + Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (x <= -0.225)
		tmp = (1.0 - x) / tan(B);
	elseif (x <= 1.6)
		tmp = (1.0 / sin(B)) - (x / B);
	else
		tmp = (x / tan(B)) * (-1.0 + (1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[x, -0.225], N[(N[(1.0 - x), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.6], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.225000000000000006

   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%

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

      2. distribute-lft-neg-in 99.6%

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

      3. distribute-neg-frac 99.6%

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

      4. metadata-eval 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. +-commutative 99.6%

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

      2. associate-*l/ 99.8%

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

      3. neg-mul-1 99.8%

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

      4. distribute-neg-frac 99.8%

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

      5. div-inv 99.6%

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

      6. sub-neg 99.6%

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

      7. div-inv 99.8%

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

      8. clear-num 99.7%

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

      9. frac-sub 85.6%

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

      10. *-un-lft-identity 85.6%

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

      11. metadata-eval 85.6%

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

      12. div-inv 85.6%

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

      13. /-rgt-identity 85.6%

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

   6. Applied egg-rr 85.6%

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

      1. associate-/r* 99.7%

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

      2. 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%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in B around 0 97.8%

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

       1. div-sub 97.8%

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

       2. div-inv 97.8%

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

       3. clear-num 97.8%

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

       4. clear-num 97.9%

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

   11. Applied egg-rr 97.9%

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

       1. sub-neg 97.9%

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

       2. neg-mul-1 97.9%

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

       3. distribute-rgt-in 97.9%

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

       4. +-commutative 97.9%

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

       5. *-commutative 97.9%

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

       6. associate-*r/ 97.9%

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

       7. /-rgt-identity 97.9%

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

       8. associate-/r/ 97.8%

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

       9. +-commutative 97.8%

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

       10. metadata-eval 97.8%

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

       11. sub-neg 97.8%

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

       12. div-sub 97.8%

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

       13. *-inverses 97.8%

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

       14. remove-double-div 97.9%

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

   13. Simplified 97.9%

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

   if -0.225000000000000006 < x < 1.6000000000000001

   1. Initial program 99.8%

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

      1. distribute-lft-neg-in 99.8%

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

      2. +-commutative 99.8%

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

      3. *-commutative 99.8%

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

      4. remove-double-neg 99.8%

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

      5. distribute-frac-neg2 99.8%

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

      6. tan-neg 99.8%

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

      7. cancel-sign-sub-inv 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-*r/ 99.8%

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

      10. *-rgt-identity 99.8%

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

      11. tan-neg 99.8%

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

      12. distribute-neg-frac2 99.8%

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

      13. distribute-neg-frac 99.8%

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

      14. 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. Step-by-step derivation

      1. expm1-log1p-u 86.4%

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

      2. expm1-undefine 39.4%

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

   6. Applied egg-rr 39.4%

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

      1. expm1-define 86.4%

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

   8. Simplified 86.4%

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

   9. Taylor expanded in B around 0 98.3%

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

   if 1.6000000000000001 < 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%

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

      2. distribute-lft-neg-in 99.6%

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

      3. distribute-neg-frac 99.6%

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

      4. metadata-eval 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. +-commutative 99.6%

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

      2. associate-*l/ 99.8%

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

      3. neg-mul-1 99.8%

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

      4. distribute-neg-frac 99.8%

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

      5. div-inv 99.6%

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

      6. sub-neg 99.6%

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

      7. div-inv 99.8%

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

      8. clear-num 99.6%

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

      9. frac-sub 91.9%

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

      10. *-un-lft-identity 91.9%

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

      11. metadata-eval 91.9%

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

      12. div-inv 91.9%

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

      13. /-rgt-identity 91.9%

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

   6. Applied egg-rr 91.9%

      \[\leadsto \color{blue}{\frac{\frac{\tan B}{x} - \sin B}{\sin B \cdot \frac{\tan B}{x}}} \]
   7. 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%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in B around 0 99.2%

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

       1. clear-num 99.2%

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

       2. associate-/r/ 99.2%

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

       3. clear-num 99.4%

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

       4. sub-neg 99.4%

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

       5. metadata-eval 99.4%

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

       6. +-commutative 99.4%

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

   11. Applied egg-rr 99.4%

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

Alternative 3: 98.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \lor \neg \left(x \leq 2.7\right):\\ \;\;\;\;\frac{1 - x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.1) (not (<= x 2.7)))
   (/ (- 1.0 x) (tan B))
   (- (/ 1.0 (sin B)) (/ x B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.1) || !(x <= 2.7)) {
		tmp = (1.0 - x) / tan(B);
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.1d0)) .or. (.not. (x <= 2.7d0))) then
        tmp = (1.0d0 - x) / tan(b)
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -1.1) || !(x <= 2.7)) {
		tmp = (1.0 - x) / Math.tan(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -1.1) or not (x <= 2.7):
		tmp = (1.0 - x) / math.tan(B)
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -1.1) || !(x <= 2.7))
		tmp = Float64(Float64(1.0 - x) / tan(B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -1.1) || ~((x <= 2.7)))
		tmp = (1.0 - x) / tan(B);
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -1.1], N[Not[LessEqual[x, 2.7]], $MachinePrecision]], N[(N[(1.0 - x), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1000000000000001 or 2.7000000000000002 < 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%

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

      2. distribute-lft-neg-in 99.6%

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

      3. distribute-neg-frac 99.6%

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

      4. metadata-eval 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. +-commutative 99.6%

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

      2. associate-*l/ 99.8%

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

      3. neg-mul-1 99.8%

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

      4. distribute-neg-frac 99.8%

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

      5. div-inv 99.6%

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

      6. sub-neg 99.6%

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

      7. div-inv 99.8%

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

      8. clear-num 99.6%

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

      9. frac-sub 89.0%

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

      10. *-un-lft-identity 89.0%

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

      11. metadata-eval 89.0%

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

      12. div-inv 89.0%

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

      13. /-rgt-identity 89.0%

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

   6. Applied egg-rr 89.0%

      \[\leadsto \color{blue}{\frac{\frac{\tan B}{x} - \sin B}{\sin B \cdot \frac{\tan B}{x}}} \]
   7. 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%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in B around 0 98.6%

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

       1. div-sub 84.5%

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

       2. div-inv 83.6%

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

       3. clear-num 83.6%

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

       4. clear-num 83.8%

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

   11. Applied egg-rr 83.8%

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

       1. sub-neg 83.8%

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

       2. neg-mul-1 83.8%

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

       3. distribute-rgt-in 98.7%

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

       4. +-commutative 98.7%

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

       5. *-commutative 98.7%

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

       6. associate-*r/ 98.7%

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

       7. /-rgt-identity 98.7%

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

       8. associate-/r/ 98.6%

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

       9. +-commutative 98.6%

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

       10. metadata-eval 98.6%

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

       11. sub-neg 98.6%

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

       12. div-sub 98.6%

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

       13. *-inverses 98.6%

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

       14. remove-double-div 98.7%

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

   13. Simplified 98.7%

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

   if -1.1000000000000001 < x < 2.7000000000000002

   1. Initial program 99.8%

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

      1. distribute-lft-neg-in 99.8%

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

      2. +-commutative 99.8%

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

      3. *-commutative 99.8%

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

      4. remove-double-neg 99.8%

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

      5. distribute-frac-neg2 99.8%

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

      6. tan-neg 99.8%

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

      7. cancel-sign-sub-inv 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-*r/ 99.8%

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

      10. *-rgt-identity 99.8%

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

      11. tan-neg 99.8%

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

      12. distribute-neg-frac2 99.8%

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

      13. distribute-neg-frac 99.8%

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

      14. 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. Step-by-step derivation

      1. expm1-log1p-u 86.4%

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

      2. expm1-undefine 39.4%

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

   6. Applied egg-rr 39.4%

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

      1. expm1-define 86.4%

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

   8. Simplified 86.4%

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

   9. Taylor expanded in B around 0 98.3%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \lor \neg \left(x \leq 2.7\right):\\ \;\;\;\;\frac{1 - x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.000135 \lor \neg \left(x \leq 0.36\right):\\ \;\;\;\;\frac{1 - x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -0.000135) (not (<= x 0.36)))
   (/ (- 1.0 x) (tan B))
   (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -0.000135) || !(x <= 0.36)) {
		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 <= (-0.000135d0)) .or. (.not. (x <= 0.36d0))) 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 <= -0.000135) || !(x <= 0.36)) {
		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 <= -0.000135) or not (x <= 0.36):
		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 <= -0.000135) || !(x <= 0.36))
		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 <= -0.000135) || ~((x <= 0.36)))
		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, -0.000135], N[Not[LessEqual[x, 0.36]], $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 < -1.35000000000000002e-4 or 0.35999999999999999 < 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%

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

      2. distribute-lft-neg-in 99.6%

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

      3. distribute-neg-frac 99.6%

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

      4. metadata-eval 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. +-commutative 99.6%

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

      2. associate-*l/ 99.8%

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

      3. neg-mul-1 99.8%

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

      4. distribute-neg-frac 99.8%

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

      5. div-inv 99.6%

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

      6. sub-neg 99.6%

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

      7. div-inv 99.8%

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

      8. clear-num 99.6%

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

      9. frac-sub 89.0%

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

      10. *-un-lft-identity 89.0%

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

      11. metadata-eval 89.0%

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

      12. div-inv 89.0%

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

      13. /-rgt-identity 89.0%

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

   6. Applied egg-rr 89.0%

      \[\leadsto \color{blue}{\frac{\frac{\tan B}{x} - \sin B}{\sin B \cdot \frac{\tan B}{x}}} \]
   7. 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%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in B around 0 98.6%

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

       1. div-sub 84.5%

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

       2. div-inv 83.6%

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

       3. clear-num 83.6%

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

       4. clear-num 83.8%

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

   11. Applied egg-rr 83.8%

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

       1. sub-neg 83.8%

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

       2. neg-mul-1 83.8%

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

       3. distribute-rgt-in 98.7%

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

       4. +-commutative 98.7%

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

       5. *-commutative 98.7%

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

       6. associate-*r/ 98.7%

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

       7. /-rgt-identity 98.7%

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

       8. associate-/r/ 98.6%

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

       9. +-commutative 98.6%

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

       10. metadata-eval 98.6%

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

       11. sub-neg 98.6%

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

       12. div-sub 98.6%

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

       13. *-inverses 98.6%

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

       14. remove-double-div 98.7%

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

   13. Simplified 98.7%

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

   if -1.35000000000000002e-4 < x < 0.35999999999999999

   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 98.1%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.000135 \lor \neg \left(x \leq 0.36\right):\\ \;\;\;\;\frac{1 - x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.25) (not (<= x 1.0))) (/ x (- (tan B))) (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.25) || !(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 <= (-1.25d0)) .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 <= -1.25) || !(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 <= -1.25) 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 <= -1.25) || !(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 <= -1.25) || ~((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, -1.25], 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 < -1.25 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%

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

      2. distribute-lft-neg-in 99.6%

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

      3. distribute-neg-frac 99.6%

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

      4. metadata-eval 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. +-commutative 99.6%

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

      2. associate-*l/ 99.8%

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

      3. neg-mul-1 99.8%

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

      4. distribute-neg-frac 99.8%

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

      5. div-inv 99.6%

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

      6. sub-neg 99.6%

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

      7. div-inv 99.8%

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

      8. clear-num 99.6%

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

      9. frac-sub 89.0%

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

      10. *-un-lft-identity 89.0%

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

      11. metadata-eval 89.0%

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

      12. div-inv 89.0%

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

      13. /-rgt-identity 89.0%

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

   6. Applied egg-rr 89.0%

      \[\leadsto \color{blue}{\frac{\frac{\tan B}{x} - \sin B}{\sin B \cdot \frac{\tan B}{x}}} \]
   7. 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%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in x around inf 97.1%

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

       1. associate-/r/ 97.1%

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

       2. *-commutative 97.1%

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

   11. Applied egg-rr 97.1%

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

       1. *-commutative 97.1%

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

       2. associate-*l/ 97.2%

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

       3. mul-1-neg 97.2%

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

   13. Simplified 97.2%

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

   if -1.25 < 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 98.1%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 7.6 \cdot 10^{-10}:\\ \;\;\;\;\frac{1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B 7.6e-10) (/ (- 1.0 x) B) (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if (B <= 7.6e-10) {
		tmp = (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 <= 7.6d-10) then
        tmp = (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 <= 7.6e-10) {
		tmp = (1.0 - x) / B;
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if B <= 7.6e-10:
		tmp = (1.0 - x) / B
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (B <= 7.6e-10)
		tmp = 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 <= 7.6e-10)
		tmp = (1.0 - x) / B;
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[B, 7.6e-10], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 7.5999999999999996e-10

   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 66.6%

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

   if 7.5999999999999996e-10 < B

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 64.7%

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

Alternative 7: 50.0% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 7.2 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 7.2e-6))) (/ x (- B)) (/ 1.0 B)))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 7.2e-6)) {
		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 <= 7.2d-6))) 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 <= 7.2e-6)) {
		tmp = x / -B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -1.0) or not (x <= 7.2e-6):
		tmp = x / -B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 7.2e-6))
		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 <= -1.0) || ~((x <= 7.2e-6)))
		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, 7.2e-6]], $MachinePrecision]], N[(x / (-B)), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 7.19999999999999967e-6 < 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 51.4%

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

   4. Taylor expanded in x around inf 50.1%

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

      1. neg-mul-1 50.1%

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

   6. Simplified 50.1%

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

   if -1 < x < 7.19999999999999967e-6

   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 52.1%

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

   4. Taylor expanded in x around 0 52.1%

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

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 7.2 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.1% 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. Add Preprocessing

Alternative 9: 26.4% 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 31.1%

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

Reproduce

herbie shell --seed 2024136 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))