VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 8.5s

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: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2:\\ \;\;\;\;x \cdot \frac{\cos B}{-\sin B}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= x -1.2)
   (* x (/ (cos B) (- (sin B))))
   (if (<= x 6.8e-7) (/ 1.0 (sin B)) (- (/ 1.0 B) (/ x (tan B))))))

C

double code(double B, double x) {
	double tmp;
	if (x <= -1.2) {
		tmp = x * (cos(B) / -sin(B));
	} else if (x <= 6.8e-7) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = (1.0 / B) - (x / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.2d0)) then
        tmp = x * (cos(b) / -sin(b))
    else if (x <= 6.8d-7) then
        tmp = 1.0d0 / sin(b)
    else
        tmp = (1.0d0 / b) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (x <= -1.2) {
		tmp = x * (Math.cos(B) / -Math.sin(B));
	} else if (x <= 6.8e-7) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = (1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if x <= -1.2:
		tmp = x * (math.cos(B) / -math.sin(B))
	elif x <= 6.8e-7:
		tmp = 1.0 / math.sin(B)
	else:
		tmp = (1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (x <= -1.2)
		tmp = Float64(x * Float64(cos(B) / Float64(-sin(B))));
	elseif (x <= 6.8e-7)
		tmp = Float64(1.0 / sin(B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (x <= -1.2)
		tmp = x * (cos(B) / -sin(B));
	elseif (x <= 6.8e-7)
		tmp = 1.0 / sin(B);
	else
		tmp = (1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[x, -1.2], N[(x * N[(N[Cos[B], $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.8e-7], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.19999999999999996

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 99.1%

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

      1. mul-1-neg 99.1%

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

      2. associate-/l* 99.2%

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

      3. distribute-rgt-neg-in 99.2%

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

      4. distribute-neg-frac2 99.2%

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

   5. Simplified 99.2%

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

   if -1.19999999999999996 < x < 6.79999999999999948e-7

   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 98.0%

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

   if 6.79999999999999948e-7 < x

   1. Initial program 99.5%

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

      1. distribute-lft-neg-in 99.5%

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

      2. +-commutative 99.5%

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

      3. *-commutative 99.5%

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

      4. remove-double-neg 99.5%

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

      5. distribute-frac-neg2 99.5%

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

      6. tan-neg 99.5%

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

      7. cancel-sign-sub-inv 99.5%

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

      8. *-commutative 99.5%

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

      9. associate-*r/ 99.9%

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

      10. *-rgt-identity 99.9%

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

      11. tan-neg 99.9%

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

      12. distribute-neg-frac2 99.9%

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

      13. distribute-neg-frac 99.9%

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

      14. remove-double-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in B around 0 99.4%

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

Alternative 3: 98.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= x -1.75)
   (/ -1.0 (/ (tan B) x))
   (if (<= x 1.8e-6) (/ 1.0 (sin B)) (- (/ 1.0 B) (/ x (tan B))))))

C

double code(double B, double x) {
	double tmp;
	if (x <= -1.75) {
		tmp = -1.0 / (tan(B) / x);
	} else if (x <= 1.8e-6) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = (1.0 / B) - (x / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.75d0)) then
        tmp = (-1.0d0) / (tan(b) / x)
    else if (x <= 1.8d-6) then
        tmp = 1.0d0 / sin(b)
    else
        tmp = (1.0d0 / b) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (x <= -1.75) {
		tmp = -1.0 / (Math.tan(B) / x);
	} else if (x <= 1.8e-6) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = (1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if x <= -1.75:
		tmp = -1.0 / (math.tan(B) / x)
	elif x <= 1.8e-6:
		tmp = 1.0 / math.sin(B)
	else:
		tmp = (1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (x <= -1.75)
		tmp = Float64(-1.0 / Float64(tan(B) / x));
	elseif (x <= 1.8e-6)
		tmp = Float64(1.0 / sin(B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (x <= -1.75)
		tmp = -1.0 / (tan(B) / x);
	elseif (x <= 1.8e-6)
		tmp = 1.0 / sin(B);
	else
		tmp = (1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[x, -1.75], N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.8e-6], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.75

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 99.1%

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

      1. mul-1-neg 99.1%

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

      2. associate-/l* 99.2%

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

      3. distribute-rgt-neg-in 99.2%

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

      4. distribute-neg-frac2 99.2%

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

   5. Simplified 99.2%

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

      1. distribute-frac-neg2 99.2%

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

      2. clear-num 99.1%

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

      3. tan-quot 99.1%

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

      4. neg-sub0 99.1%

         \[\leadsto x \cdot \color{blue}{\left(0 - \frac{1}{\tan B}\right)} \]

   7. Applied egg-rr 99.1%

      \[\leadsto x \cdot \color{blue}{\left(0 - \frac{1}{\tan B}\right)} \]
   8. Step-by-step derivation

      1. neg-sub0 99.1%

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

      2. distribute-neg-frac 99.1%

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

      3. metadata-eval 99.1%

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

   9. Simplified 99.1%

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

       1. div-inv 99.1%

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

       2. mul-1-neg 99.1%

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

       3. distribute-rgt-neg-in 99.1%

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

       4. div-inv 99.1%

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

       5. remove-double-div 99.1%

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

       6. distribute-neg-frac 99.1%

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

       7. metadata-eval 99.1%

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

       8. clear-num 99.2%

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

   11. Applied egg-rr 99.2%

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

   if -1.75 < x < 1.79999999999999992e-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 x around 0 98.0%

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

   if 1.79999999999999992e-6 < x

   1. Initial program 99.5%

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

      1. distribute-lft-neg-in 99.5%

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

      2. +-commutative 99.5%

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

      3. *-commutative 99.5%

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

      4. remove-double-neg 99.5%

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

      5. distribute-frac-neg2 99.5%

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

      6. tan-neg 99.5%

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

      7. cancel-sign-sub-inv 99.5%

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

      8. *-commutative 99.5%

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

      9. associate-*r/ 99.9%

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

      10. *-rgt-identity 99.9%

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

      11. tan-neg 99.9%

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

      12. distribute-neg-frac2 99.9%

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

      13. distribute-neg-frac 99.9%

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

      14. remove-double-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in B around 0 99.4%

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

Alternative 4: 97.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \lor \neg \left(x \leq 1.05\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -2.0) (not (<= x 1.05))) (/ (- x) (tan B)) (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -2.0) || !(x <= 1.05)) {
		tmp = -x / tan(B);
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-2.0d0)) .or. (.not. (x <= 1.05d0))) then
        tmp = -x / tan(b)
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -2.0) || !(x <= 1.05)) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -2.0) or not (x <= 1.05):
		tmp = -x / math.tan(B)
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -2.0) || !(x <= 1.05))
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -2.0) || ~((x <= 1.05)))
		tmp = -x / tan(B);
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -2.0], N[Not[LessEqual[x, 1.05]], $MachinePrecision]], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2 or 1.05000000000000004 < 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 x around inf 97.6%

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

      1. mul-1-neg 97.6%

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

      2. associate-/l* 97.6%

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

      3. distribute-rgt-neg-in 97.6%

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

      4. distribute-neg-frac2 97.6%

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

   5. Simplified 97.6%

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

      1. distribute-frac-neg2 97.6%

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

      2. clear-num 97.5%

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

      3. tan-quot 97.5%

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

      4. neg-sub0 97.5%

         \[\leadsto x \cdot \color{blue}{\left(0 - \frac{1}{\tan B}\right)} \]

   7. Applied egg-rr 97.5%

      \[\leadsto x \cdot \color{blue}{\left(0 - \frac{1}{\tan B}\right)} \]
   8. Step-by-step derivation

      1. neg-sub0 97.5%

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

      2. distribute-neg-frac 97.5%

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

      3. metadata-eval 97.5%

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

   9. Simplified 97.5%

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

       1. associate-*r/ 97.7%

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

       2. *-commutative 97.7%

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

       3. neg-mul-1 97.7%

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

   11. Applied egg-rr 97.7%

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

   if -2 < x < 1.05000000000000004

   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 98.0%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \lor \neg \left(x \leq 1.05\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= x -1.7)
   (/ -1.0 (/ (tan B) x))
   (if (<= x 1.0) (/ 1.0 (sin B)) (/ (- x) (tan B)))))

C

double code(double B, double x) {
	double tmp;
	if (x <= -1.7) {
		tmp = -1.0 / (tan(B) / x);
	} else if (x <= 1.0) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = -x / tan(B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.7d0)) then
        tmp = (-1.0d0) / (tan(b) / x)
    else if (x <= 1.0d0) then
        tmp = 1.0d0 / sin(b)
    else
        tmp = -x / tan(b)
    end if
    code = tmp
end function

Java

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

Python

def code(B, x):
	tmp = 0
	if x <= -1.7:
		tmp = -1.0 / (math.tan(B) / x)
	elif x <= 1.0:
		tmp = 1.0 / math.sin(B)
	else:
		tmp = -x / math.tan(B)
	return tmp

Julia

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

MATLAB

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

Wolfram

code[B_, x_] := If[LessEqual[x, -1.7], N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.69999999999999996

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 99.1%

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

      1. mul-1-neg 99.1%

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

      2. associate-/l* 99.2%

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

      3. distribute-rgt-neg-in 99.2%

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

      4. distribute-neg-frac2 99.2%

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

   5. Simplified 99.2%

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

      1. distribute-frac-neg2 99.2%

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

      2. clear-num 99.1%

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

      3. tan-quot 99.1%

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

      4. neg-sub0 99.1%

         \[\leadsto x \cdot \color{blue}{\left(0 - \frac{1}{\tan B}\right)} \]

   7. Applied egg-rr 99.1%

      \[\leadsto x \cdot \color{blue}{\left(0 - \frac{1}{\tan B}\right)} \]
   8. Step-by-step derivation

      1. neg-sub0 99.1%

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

      2. distribute-neg-frac 99.1%

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

      3. metadata-eval 99.1%

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

   9. Simplified 99.1%

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

       1. div-inv 99.1%

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

       2. mul-1-neg 99.1%

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

       3. distribute-rgt-neg-in 99.1%

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

       4. div-inv 99.1%

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

       5. remove-double-div 99.1%

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

       6. distribute-neg-frac 99.1%

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

       7. metadata-eval 99.1%

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

       8. clear-num 99.2%

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

   11. Applied egg-rr 99.2%

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

   if -1.69999999999999996 < 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 98.0%

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

   if 1 < x

   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 inf 96.3%

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

      1. mul-1-neg 96.3%

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

      2. associate-/l* 96.2%

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

      3. distribute-rgt-neg-in 96.2%

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

      4. distribute-neg-frac2 96.2%

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

   5. Simplified 96.2%

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

      1. distribute-frac-neg2 96.2%

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

      2. clear-num 96.2%

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

      3. tan-quot 96.2%

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

      4. neg-sub0 96.2%

         \[\leadsto x \cdot \color{blue}{\left(0 - \frac{1}{\tan B}\right)} \]

   7. Applied egg-rr 96.2%

      \[\leadsto x \cdot \color{blue}{\left(0 - \frac{1}{\tan B}\right)} \]
   8. Step-by-step derivation

      1. neg-sub0 96.2%

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

      2. distribute-neg-frac 96.2%

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

      3. metadata-eval 96.2%

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

   9. Simplified 96.2%

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

       1. associate-*r/ 96.5%

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

       2. *-commutative 96.5%

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

       3. neg-mul-1 96.5%

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

   11. Applied egg-rr 96.5%

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

Alternative 6: 63.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.00031:\\ \;\;\;\;\frac{1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B 0.00031) (/ (- 1.0 x) B) (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if (B <= 0.00031) {
		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 <= 0.00031d0) 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 <= 0.00031) {
		tmp = (1.0 - x) / B;
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if B <= 0.00031:
		tmp = (1.0 - x) / B
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (B <= 0.00031)
		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 <= 0.00031)
		tmp = (1.0 - x) / B;
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[B, 0.00031], 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 < 3.1e-4

   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 66.4%

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

   if 3.1e-4 < 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 56.0%

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

Alternative 7: 49.7% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{+26} \lor \neg \left(x \leq 6.5 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -5.8e+26) (not (<= x 6.5e-12))) (/ x (- B)) (/ 1.0 B)))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -5.8e+26) || !(x <= 6.5e-12)) {
		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 <= (-5.8d+26)) .or. (.not. (x <= 6.5d-12))) 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 <= -5.8e+26) || !(x <= 6.5e-12)) {
		tmp = x / -B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -5.8e+26) or not (x <= 6.5e-12):
		tmp = x / -B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -5.8e+26) || !(x <= 6.5e-12))
		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 <= -5.8e+26) || ~((x <= 6.5e-12)))
		tmp = x / -B;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -5.8e+26], N[Not[LessEqual[x, 6.5e-12]], $MachinePrecision]], N[(x / (-B)), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.8e26 or 6.5000000000000002e-12 < 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 53.3%

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

   4. Taylor expanded in x around inf 51.8%

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

      1. neg-mul-1 51.8%

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

      2. distribute-neg-frac2 51.8%

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

   6. Simplified 51.8%

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

   if -5.8e26 < x < 6.5000000000000002e-12

   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 50.6%

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

   4. Taylor expanded in x around 0 50.3%

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

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{+26} \lor \neg \left(x \leq 6.5 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.5% 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 52.0%

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

Alternative 9: 26.7% accurate, 70.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in B around 0 52.0%

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

4. Taylor expanded in x around 0 25.5%

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

Reproduce

herbie shell --seed 2024096 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))