VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 8.9s

Alternatives: 10

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: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.6 \cdot 10^{+18} \lor \neg \left(x \leq 28000000\right):\\ \;\;\;\;\cos B \cdot \left(-\frac{x}{\sin B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -9.6e+18) (not (<= x 28000000.0)))
   (* (cos B) (- (/ x (sin B))))
   (- (/ 1.0 (sin B)) (/ x B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -9.6e+18) || !(x <= 28000000.0)) {
		tmp = cos(B) * -(x / sin(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 <= (-9.6d+18)) .or. (.not. (x <= 28000000.0d0))) then
        tmp = cos(b) * -(x / sin(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 <= -9.6e+18) || !(x <= 28000000.0)) {
		tmp = Math.cos(B) * -(x / Math.sin(B));
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -9.6e+18) or not (x <= 28000000.0):
		tmp = math.cos(B) * -(x / math.sin(B))
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -9.6e+18) || !(x <= 28000000.0))
		tmp = Float64(cos(B) * Float64(-Float64(x / sin(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 <= -9.6e+18) || ~((x <= 28000000.0)))
		tmp = cos(B) * -(x / sin(B));
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -9.6e+18], N[Not[LessEqual[x, 28000000.0]], $MachinePrecision]], N[(N[Cos[B], $MachinePrecision] * (-N[(x / N[Sin[B], $MachinePrecision]), $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 < -9.6e18 or 2.8e7 < x

   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 99.7%

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

      1. mul-1-neg 99.7%

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

      2. associate-*l/ 99.7%

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

      3. *-commutative 99.7%

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

      4. distribute-rgt-neg-in 99.7%

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

   5. Simplified 99.7%

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

   if -9.6e18 < x < 2.8e7

   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 98.6%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.6 \cdot 10^{+18} \lor \neg \left(x \leq 28000000\right):\\ \;\;\;\;\cos B \cdot \left(-\frac{x}{\sin B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.6 \cdot 10^{+18}:\\ \;\;\;\;\cos B \cdot \left(-\frac{x}{\sin B}\right)\\ \mathbf{elif}\;x \leq 140000000:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\frac{\sin B}{\cos B}}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= x -9.6e+18)
   (* (cos B) (- (/ x (sin B))))
   (if (<= x 140000000.0)
     (- (/ 1.0 (sin B)) (/ x B))
     (/ (- x) (/ (sin B) (cos B))))))

C

double code(double B, double x) {
	double tmp;
	if (x <= -9.6e+18) {
		tmp = cos(B) * -(x / sin(B));
	} else if (x <= 140000000.0) {
		tmp = (1.0 / sin(B)) - (x / B);
	} else {
		tmp = -x / (sin(B) / cos(B));
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-9.6d+18)) then
        tmp = cos(b) * -(x / sin(b))
    else if (x <= 140000000.0d0) then
        tmp = (1.0d0 / sin(b)) - (x / b)
    else
        tmp = -x / (sin(b) / cos(b))
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (x <= -9.6e+18) {
		tmp = Math.cos(B) * -(x / Math.sin(B));
	} else if (x <= 140000000.0) {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	} else {
		tmp = -x / (Math.sin(B) / Math.cos(B));
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if x <= -9.6e+18:
		tmp = math.cos(B) * -(x / math.sin(B))
	elif x <= 140000000.0:
		tmp = (1.0 / math.sin(B)) - (x / B)
	else:
		tmp = -x / (math.sin(B) / math.cos(B))
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (x <= -9.6e+18)
		tmp = Float64(cos(B) * Float64(-Float64(x / sin(B))));
	elseif (x <= 140000000.0)
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	else
		tmp = Float64(Float64(-x) / Float64(sin(B) / cos(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (x <= -9.6e+18)
		tmp = cos(B) * -(x / sin(B));
	elseif (x <= 140000000.0)
		tmp = (1.0 / sin(B)) - (x / B);
	else
		tmp = -x / (sin(B) / cos(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[x, -9.6e+18], N[(N[Cos[B], $MachinePrecision] * (-N[(x / N[Sin[B], $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[x, 140000000.0], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[((-x) / N[(N[Sin[B], $MachinePrecision] / N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.6e18

   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 99.6%

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

      1. mul-1-neg 99.6%

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

      2. associate-*l/ 99.7%

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

      3. *-commutative 99.7%

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

      4. distribute-rgt-neg-in 99.7%

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

   5. Simplified 99.7%

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

   if -9.6e18 < x < 1.4e8

   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 98.6%

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

   if 1.4e8 < 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/ 99.9%

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

      7. *-rgt-identity 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. Step-by-step derivation

      1. div-inv 99.6%

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

      2. *-commutative 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. associate-*l/ 99.9%

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

      2. *-un-lft-identity 99.9%

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

      3. frac-sub 90.3%

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

      4. *-un-lft-identity 90.3%

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

   8. Applied egg-rr 90.3%

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

   9. Taylor expanded in x around inf 99.7%

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

       1. mul-1-neg 99.7%

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

       2. associate-/l* 99.8%

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

       3. distribute-neg-frac 99.8%

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

   11. Simplified 99.8%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.6 \cdot 10^{+18}:\\ \;\;\;\;\cos B \cdot \left(-\frac{x}{\sin B}\right)\\ \mathbf{elif}\;x \leq 140000000:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\frac{\sin B}{\cos B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{+21} \lor \neg \left(x \leq 4.9 \cdot 10^{+58}\right):\\ \;\;\;\;\left(B \cdot 0.16666666666666666 + \frac{1}{B}\right) - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -3.6e+21) (not (<= x 4.9e+58)))
   (- (+ (* B 0.16666666666666666) (/ 1.0 B)) (/ x (tan B)))
   (- (/ 1.0 (sin B)) (/ x B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -3.6e+21) || !(x <= 4.9e+58)) {
		tmp = ((B * 0.16666666666666666) + (1.0 / B)) - (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 <= (-3.6d+21)) .or. (.not. (x <= 4.9d+58))) then
        tmp = ((b * 0.16666666666666666d0) + (1.0d0 / b)) - (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 <= -3.6e+21) || !(x <= 4.9e+58)) {
		tmp = ((B * 0.16666666666666666) + (1.0 / B)) - (x / Math.tan(B));
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -3.6e+21) or not (x <= 4.9e+58):
		tmp = ((B * 0.16666666666666666) + (1.0 / B)) - (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 <= -3.6e+21) || !(x <= 4.9e+58))
		tmp = Float64(Float64(Float64(B * 0.16666666666666666) + Float64(1.0 / B)) - Float64(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 <= -3.6e+21) || ~((x <= 4.9e+58)))
		tmp = ((B * 0.16666666666666666) + (1.0 / B)) - (x / tan(B));
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -3.6e+21], N[Not[LessEqual[x, 4.9e+58]], $MachinePrecision]], N[(N[(N[(B * 0.16666666666666666), $MachinePrecision] + N[(1.0 / B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $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 < -3.6e21 or 4.90000000000000018e58 < x

   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 86.8%

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

   if -3.6e21 < x < 4.90000000000000018e58

   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 94.8%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{+21} \lor \neg \left(x \leq 4.9 \cdot 10^{+58}\right):\\ \;\;\;\;\left(B \cdot 0.16666666666666666 + \frac{1}{B}\right) - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 10000:\\ \;\;\;\;B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B 10000.0)
   (+ (* B (+ 0.16666666666666666 (* x 0.3333333333333333))) (/ (- 1.0 x) B))
   (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if (B <= 10000.0) {
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B);
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (b <= 10000.0d0) then
        tmp = (b * (0.16666666666666666d0 + (x * 0.3333333333333333d0))) + ((1.0d0 - x) / b)
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (B <= 10000.0) {
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B);
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if B <= 10000.0:
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B)
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (B <= 10000.0)
		tmp = Float64(Float64(B * Float64(0.16666666666666666 + Float64(x * 0.3333333333333333))) + Float64(Float64(1.0 - x) / B));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (B <= 10000.0)
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B);
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[B, 10000.0], N[(N[(B * N[(0.16666666666666666 + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1e4

   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 67.7%

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

      1. associate--l+ 67.7%

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

      2. *-commutative 67.7%

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

      3. div-sub 67.7%

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

   5. Simplified 67.7%

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

   if 1e4 < B

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 54.6%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 10000:\\ \;\;\;\;B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in B around 0 78.8%

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

4. Final simplification 78.8%

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

Alternative 7: 50.8% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-8} \lor \neg \left(x \leq 29.5\right):\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.25e-8) (not (<= x 29.5))) (- (/ x B)) (/ 1.0 B)))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.25e-8) || !(x <= 29.5)) {
		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 <= (-1.25d-8)) .or. (.not. (x <= 29.5d0))) 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 <= -1.25e-8) || !(x <= 29.5)) {
		tmp = -(x / B);
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -1.25e-8) or not (x <= 29.5):
		tmp = -(x / B)
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -1.25e-8) || !(x <= 29.5))
		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 <= -1.25e-8) || ~((x <= 29.5)))
		tmp = -(x / B);
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -1.25e-8], N[Not[LessEqual[x, 29.5]], $MachinePrecision]], (-N[(x / B), $MachinePrecision]), N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.2499999999999999e-8 or 29.5 < x

   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 55.8%

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

   4. Taylor expanded in x around inf 54.6%

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

      1. associate-*r/ 54.6%

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

      2. neg-mul-1 54.6%

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

   6. Simplified 54.6%

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

   if -1.2499999999999999e-8 < x < 29.5

   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 48.9%

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

   4. Taylor expanded in x around 0 47.8%

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

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-8} \lor \neg \left(x \leq 29.5\right):\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.9% accurate, 23.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in B around 0 78.8%

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

4. Taylor expanded in B around 0 52.3%

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

   1. associate--l+ 52.3%

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

   2. *-commutative 52.3%

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

   3. div-sub 52.3%

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

6. Simplified 52.3%

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

7. Final simplification 52.3%

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

Alternative 9: 51.8% 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 52.2%

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

4. Final simplification 52.2%

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

Alternative 10: 27.4% 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 52.2%

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

4. Taylor expanded in x around 0 26.4%

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

5. Final simplification 26.4%

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

Reproduce

herbie shell --seed 2024024 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))