VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 14.7s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-neg-in 99.6%

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

   2. +-commutative 99.6%

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

   3. cancel-sign-sub-inv 99.6%

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

   4. *-commutative 99.6%

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

   5. *-commutative 99.6%

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

   6. associate-*r/ 99.7%

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

   7. *-rgt-identity 99.7%

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

3. Simplified 99.7%

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

4. Final simplification 99.7%

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

Alternative 2: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45:\\ \;\;\;\;\frac{\frac{1}{x} + -1}{\frac{\tan B}{x}}\\ \mathbf{elif}\;x \leq 1800000000:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(-\cos B\right)}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= x -1.45)
   (/ (+ (/ 1.0 x) -1.0) (/ (tan B) x))
   (if (<= x 1800000000.0)
     (- (/ 1.0 (sin B)) (/ x (sin B)))
     (/ (* x (- (cos B))) (sin B)))))

C

double code(double B, double x) {
	double tmp;
	if (x <= -1.45) {
		tmp = ((1.0 / x) + -1.0) / (tan(B) / x);
	} else if (x <= 1800000000.0) {
		tmp = (1.0 / sin(B)) - (x / sin(B));
	} else {
		tmp = (x * -cos(B)) / 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 <= (-1.45d0)) then
        tmp = ((1.0d0 / x) + (-1.0d0)) / (tan(b) / x)
    else if (x <= 1800000000.0d0) then
        tmp = (1.0d0 / sin(b)) - (x / sin(b))
    else
        tmp = (x * -cos(b)) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (x <= -1.45) {
		tmp = ((1.0 / x) + -1.0) / (Math.tan(B) / x);
	} else if (x <= 1800000000.0) {
		tmp = (1.0 / Math.sin(B)) - (x / Math.sin(B));
	} else {
		tmp = (x * -Math.cos(B)) / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if x <= -1.45:
		tmp = ((1.0 / x) + -1.0) / (math.tan(B) / x)
	elif x <= 1800000000.0:
		tmp = (1.0 / math.sin(B)) - (x / math.sin(B))
	else:
		tmp = (x * -math.cos(B)) / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (x <= -1.45)
		tmp = Float64(Float64(Float64(1.0 / x) + -1.0) / Float64(tan(B) / x));
	elseif (x <= 1800000000.0)
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / sin(B)));
	else
		tmp = Float64(Float64(x * Float64(-cos(B))) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (x <= -1.45)
		tmp = ((1.0 / x) + -1.0) / (tan(B) / x);
	elseif (x <= 1800000000.0)
		tmp = (1.0 / sin(B)) - (x / sin(B));
	else
		tmp = (x * -cos(B)) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[x, -1.45], N[(N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision] / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1800000000.0], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * (-N[Cos[B], $MachinePrecision])), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.44999999999999996

   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. distribute-lft-neg-in 99.5%

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

      4. distribute-rgt-neg-in 99.5%

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

   3. Simplified 99.5%

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

      1. distribute-rgt-neg-out 99.5%

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

      2. div-inv 99.8%

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

      3. sub-neg 99.8%

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

      4. clear-num 99.5%

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

      5. frac-sub 87.5%

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

      6. *-un-lft-identity 87.5%

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

      7. *-commutative 87.5%

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

      8. *-un-lft-identity 87.5%

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

   5. Applied egg-rr 87.5%

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

      1. associate-/r* 99.5%

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

      2. div-sub 99.5%

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

      3. *-inverses 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in B around 0 95.5%

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

   if -1.44999999999999996 < x < 1.8e9

   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/ 99.7%

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

      7. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 99.7%

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

   5. Taylor expanded in B around 0 98.6%

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

   if 1.8e9 < x

   1. Initial program 99.3%

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

      1. distribute-lft-neg-in 99.3%

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

      2. +-commutative 99.3%

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

      3. cancel-sign-sub-inv 99.3%

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

      4. *-commutative 99.3%

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

      5. *-commutative 99.3%

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

      6. associate-*r/ 99.6%

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

      7. *-rgt-identity 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 99.7%

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

   5. Taylor expanded in x around inf 99.7%

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

      1. mul-1-neg 99.7%

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

      2. associate-*r/ 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in x around 0 99.7%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45:\\ \;\;\;\;\frac{\frac{1}{x} + -1}{\frac{\tan B}{x}}\\ \mathbf{elif}\;x \leq 1800000000:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(-\cos B\right)}{\sin B}\\ \end{array} \]

Alternative 3: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4:\\ \;\;\;\;\frac{\frac{1}{x} + -1}{\frac{\tan B}{x}}\\ \mathbf{elif}\;x \leq 23000:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(-\cos B\right)}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= x -1.4)
   (/ (+ (/ 1.0 x) -1.0) (/ (tan B) x))
   (if (<= x 23000.0) (/ (- 1.0 x) (sin B)) (/ (* x (- (cos B))) (sin B)))))

C

double code(double B, double x) {
	double tmp;
	if (x <= -1.4) {
		tmp = ((1.0 / x) + -1.0) / (tan(B) / x);
	} else if (x <= 23000.0) {
		tmp = (1.0 - x) / sin(B);
	} else {
		tmp = (x * -cos(B)) / 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 <= (-1.4d0)) then
        tmp = ((1.0d0 / x) + (-1.0d0)) / (tan(b) / x)
    else if (x <= 23000.0d0) then
        tmp = (1.0d0 - x) / sin(b)
    else
        tmp = (x * -cos(b)) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (x <= -1.4) {
		tmp = ((1.0 / x) + -1.0) / (Math.tan(B) / x);
	} else if (x <= 23000.0) {
		tmp = (1.0 - x) / Math.sin(B);
	} else {
		tmp = (x * -Math.cos(B)) / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if x <= -1.4:
		tmp = ((1.0 / x) + -1.0) / (math.tan(B) / x)
	elif x <= 23000.0:
		tmp = (1.0 - x) / math.sin(B)
	else:
		tmp = (x * -math.cos(B)) / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (x <= -1.4)
		tmp = Float64(Float64(Float64(1.0 / x) + -1.0) / Float64(tan(B) / x));
	elseif (x <= 23000.0)
		tmp = Float64(Float64(1.0 - x) / sin(B));
	else
		tmp = Float64(Float64(x * Float64(-cos(B))) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (x <= -1.4)
		tmp = ((1.0 / x) + -1.0) / (tan(B) / x);
	elseif (x <= 23000.0)
		tmp = (1.0 - x) / sin(B);
	else
		tmp = (x * -cos(B)) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[x, -1.4], N[(N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision] / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 23000.0], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(x * (-N[Cos[B], $MachinePrecision])), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.3999999999999999

   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. distribute-lft-neg-in 99.5%

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

      4. distribute-rgt-neg-in 99.5%

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

   3. Simplified 99.5%

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

      1. distribute-rgt-neg-out 99.5%

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

      2. div-inv 99.8%

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

      3. sub-neg 99.8%

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

      4. clear-num 99.5%

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

      5. frac-sub 87.5%

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

      6. *-un-lft-identity 87.5%

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

      7. *-commutative 87.5%

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

      8. *-un-lft-identity 87.5%

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

   5. Applied egg-rr 87.5%

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

      1. associate-/r* 99.5%

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

      2. div-sub 99.5%

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

      3. *-inverses 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in B around 0 95.5%

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

   if -1.3999999999999999 < x < 23000

   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/ 99.7%

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

      7. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 99.7%

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

   5. Taylor expanded in x around 0 99.7%

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

      1. +-commutative 99.7%

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

      2. mul-1-neg 99.7%

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

      3. sub-neg 99.7%

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

      4. div-sub 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in B around 0 98.6%

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

   if 23000 < x

   1. Initial program 99.3%

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

      1. distribute-lft-neg-in 99.3%

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

      2. +-commutative 99.3%

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

      3. cancel-sign-sub-inv 99.3%

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

      4. *-commutative 99.3%

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

      5. *-commutative 99.3%

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

      6. associate-*r/ 99.6%

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

      7. *-rgt-identity 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 99.7%

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

   5. Taylor expanded in x around inf 99.7%

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

      1. mul-1-neg 99.7%

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

      2. associate-*r/ 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in x around 0 99.7%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4:\\ \;\;\;\;\frac{\frac{1}{x} + -1}{\frac{\tan B}{x}}\\ \mathbf{elif}\;x \leq 23000:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(-\cos B\right)}{\sin B}\\ \end{array} \]

Alternative 4: 98.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -500000000 \lor \neg \left(x \leq 65000000\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} \cdot \left(1 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -500000000.0) (not (<= x 65000000.0)))
   (/ (- x) (tan B))
   (* (/ 1.0 (sin B)) (- 1.0 x))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -500000000.0) || !(x <= 65000000.0)) {
		tmp = -x / tan(B);
	} else {
		tmp = (1.0 / sin(B)) * (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 <= (-500000000.0d0)) .or. (.not. (x <= 65000000.0d0))) then
        tmp = -x / tan(b)
    else
        tmp = (1.0d0 / sin(b)) * (1.0d0 - x)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -500000000.0) || !(x <= 65000000.0)) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (1.0 / Math.sin(B)) * (1.0 - x);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -500000000.0) or not (x <= 65000000.0):
		tmp = -x / math.tan(B)
	else:
		tmp = (1.0 / math.sin(B)) * (1.0 - x)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -500000000.0) || !(x <= 65000000.0))
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) * Float64(1.0 - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -500000000.0) || ~((x <= 65000000.0)))
		tmp = -x / tan(B);
	else
		tmp = (1.0 / sin(B)) * (1.0 - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -500000000.0], N[Not[LessEqual[x, 65000000.0]], $MachinePrecision]], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5e8 or 6.5e7 < x

   1. Initial program 99.4%

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

      1. distribute-lft-neg-in 99.4%

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

      2. +-commutative 99.4%

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

      3. cancel-sign-sub-inv 99.4%

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

      4. *-commutative 99.4%

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

      5. *-commutative 99.4%

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

      6. associate-*r/ 99.7%

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

      7. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 99.6%

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

   5. Taylor expanded in x around inf 99.6%

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

      1. mul-1-neg 99.6%

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

      2. associate-*r/ 99.4%

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

   7. Simplified 99.4%

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

      1. expm1-log1p-u 50.1%

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

      2. expm1-udef 50.1%

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

      3. clear-num 50.1%

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

      4. tan-quot 50.1%

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

   9. Applied egg-rr 50.1%

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

       1. expm1-def 50.1%

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

       2. expm1-log1p 99.4%

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

       3. associate-*r/ 99.7%

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

       4. *-rgt-identity 99.7%

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

   11. Simplified 99.7%

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

   if -5e8 < x < 6.5e7

   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/ 99.7%

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

      7. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 99.8%

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

   5. Taylor expanded in B around 0 96.0%

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

      1. *-un-lft-identity 96.0%

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

      2. div-inv 96.0%

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

      3. distribute-rgt-out-- 96.0%

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

   7. Applied egg-rr 96.0%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -500000000 \lor \neg \left(x \leq 65000000\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} \cdot \left(1 - x\right)\\ \end{array} \]

Alternative 5: 98.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45:\\ \;\;\;\;\frac{\frac{1}{x} + -1}{\frac{\tan B}{x}}\\ \mathbf{elif}\;x \leq 7400000:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= x -1.45)
   (/ (+ (/ 1.0 x) -1.0) (/ (tan B) x))
   (if (<= x 7400000.0) (/ (- 1.0 x) (sin B)) (/ (- x) (tan B)))))

C

double code(double B, double x) {
	double tmp;
	if (x <= -1.45) {
		tmp = ((1.0 / x) + -1.0) / (tan(B) / x);
	} else if (x <= 7400000.0) {
		tmp = (1.0 - x) / 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 <= (-1.45d0)) then
        tmp = ((1.0d0 / x) + (-1.0d0)) / (tan(b) / x)
    else if (x <= 7400000.0d0) then
        tmp = (1.0d0 - x) / 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 <= -1.45) {
		tmp = ((1.0 / x) + -1.0) / (Math.tan(B) / x);
	} else if (x <= 7400000.0) {
		tmp = (1.0 - x) / Math.sin(B);
	} else {
		tmp = -x / Math.tan(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if x <= -1.45:
		tmp = ((1.0 / x) + -1.0) / (math.tan(B) / x)
	elif x <= 7400000.0:
		tmp = (1.0 - x) / math.sin(B)
	else:
		tmp = -x / math.tan(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (x <= -1.45)
		tmp = Float64(Float64(Float64(1.0 / x) + -1.0) / Float64(tan(B) / x));
	elseif (x <= 7400000.0)
		tmp = Float64(Float64(1.0 - x) / 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 <= -1.45)
		tmp = ((1.0 / x) + -1.0) / (tan(B) / x);
	elseif (x <= 7400000.0)
		tmp = (1.0 - x) / sin(B);
	else
		tmp = -x / tan(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[x, -1.45], N[(N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision] / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7400000.0], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.44999999999999996

   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. distribute-lft-neg-in 99.5%

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

      4. distribute-rgt-neg-in 99.5%

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

   3. Simplified 99.5%

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

      1. distribute-rgt-neg-out 99.5%

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

      2. div-inv 99.8%

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

      3. sub-neg 99.8%

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

      4. clear-num 99.5%

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

      5. frac-sub 87.5%

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

      6. *-un-lft-identity 87.5%

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

      7. *-commutative 87.5%

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

      8. *-un-lft-identity 87.5%

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

   5. Applied egg-rr 87.5%

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

      1. associate-/r* 99.5%

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

      2. div-sub 99.5%

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

      3. *-inverses 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in B around 0 95.5%

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

   if -1.44999999999999996 < x < 7.4e6

   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/ 99.7%

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

      7. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 99.7%

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

   5. Taylor expanded in x around 0 99.7%

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

      1. +-commutative 99.7%

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

      2. mul-1-neg 99.7%

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

      3. sub-neg 99.7%

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

      4. div-sub 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in B around 0 98.6%

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

   if 7.4e6 < x

   1. Initial program 99.3%

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

      1. distribute-lft-neg-in 99.3%

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

      2. +-commutative 99.3%

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

      3. cancel-sign-sub-inv 99.3%

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

      4. *-commutative 99.3%

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

      5. *-commutative 99.3%

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

      6. associate-*r/ 99.6%

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

      7. *-rgt-identity 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 99.7%

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

   5. Taylor expanded in x around inf 99.7%

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

      1. mul-1-neg 99.7%

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

      2. associate-*r/ 99.3%

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

   7. Simplified 99.3%

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

      1. expm1-log1p-u 51.2%

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

      2. expm1-udef 51.2%

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

      3. clear-num 51.2%

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

      4. tan-quot 51.2%

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

   9. Applied egg-rr 51.2%

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

       1. expm1-def 51.2%

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

       2. expm1-log1p 99.3%

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

       3. associate-*r/ 99.6%

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

       4. *-rgt-identity 99.6%

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

   11. Simplified 99.6%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45:\\ \;\;\;\;\frac{\frac{1}{x} + -1}{\frac{\tan B}{x}}\\ \mathbf{elif}\;x \leq 7400000:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \end{array} \]

Alternative 6: 98.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -100000000 \lor \neg \left(x \leq 240000\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -100000000.0) (not (<= x 240000.0)))
   (/ (- x) (tan B))
   (/ (- 1.0 x) (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -100000000.0) || !(x <= 240000.0)) {
		tmp = -x / tan(B);
	} else {
		tmp = (1.0 - x) / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-100000000.0d0)) .or. (.not. (x <= 240000.0d0))) then
        tmp = -x / tan(b)
    else
        tmp = (1.0d0 - x) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -100000000.0) || !(x <= 240000.0)) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = (1.0 - x) / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -100000000.0) or not (x <= 240000.0):
		tmp = -x / math.tan(B)
	else:
		tmp = (1.0 - x) / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -100000000.0) || !(x <= 240000.0))
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(1.0 - x) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -100000000.0) || ~((x <= 240000.0)))
		tmp = -x / tan(B);
	else
		tmp = (1.0 - x) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -100000000.0], N[Not[LessEqual[x, 240000.0]], $MachinePrecision]], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1e8 or 2.4e5 < x

   1. Initial program 99.4%

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

      1. distribute-lft-neg-in 99.4%

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

      2. +-commutative 99.4%

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

      3. cancel-sign-sub-inv 99.4%

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

      4. *-commutative 99.4%

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

      5. *-commutative 99.4%

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

      6. associate-*r/ 99.7%

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

      7. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 99.6%

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

   5. Taylor expanded in x around inf 99.6%

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

      1. mul-1-neg 99.6%

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

      2. associate-*r/ 99.4%

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

   7. Simplified 99.4%

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

      1. expm1-log1p-u 50.1%

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

      2. expm1-udef 50.1%

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

      3. clear-num 50.1%

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

      4. tan-quot 50.1%

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

   9. Applied egg-rr 50.1%

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

       1. expm1-def 50.1%

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

       2. expm1-log1p 99.4%

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

       3. associate-*r/ 99.7%

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

       4. *-rgt-identity 99.7%

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

   11. Simplified 99.7%

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

   if -1e8 < x < 2.4e5

   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/ 99.7%

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

      7. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 99.8%

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

   5. Taylor expanded in x around 0 99.8%

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

      1. +-commutative 99.8%

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

      2. mul-1-neg 99.8%

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

      3. sub-neg 99.8%

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

      4. div-sub 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in B around 0 96.0%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -100000000 \lor \neg \left(x \leq 240000\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \]

Alternative 7: 97.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.5) (not (<= x 1.0))) (/ (- x) (tan B)) (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.5) || !(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 <= (-1.5d0)) .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 <= -1.5) || !(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 <= -1.5) 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 <= -1.5) || !(x <= 1.0))
		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 <= -1.5) || ~((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, -1.5], 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 < -1.5 or 1 < x

   1. Initial program 99.4%

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

      1. distribute-lft-neg-in 99.4%

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

      2. +-commutative 99.4%

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

      3. cancel-sign-sub-inv 99.4%

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

      4. *-commutative 99.4%

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

      5. *-commutative 99.4%

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

      6. associate-*r/ 99.7%

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

      7. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 99.7%

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

   5. Taylor expanded in x around inf 96.5%

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

      1. mul-1-neg 96.5%

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

      2. associate-*r/ 96.2%

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

   7. Simplified 96.2%

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

      1. expm1-log1p-u 48.2%

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

      2. expm1-udef 48.2%

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

      3. clear-num 48.2%

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

      4. tan-quot 48.2%

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

   9. Applied egg-rr 48.2%

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

       1. expm1-def 48.2%

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

       2. expm1-log1p 96.2%

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

       3. associate-*r/ 96.5%

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

       4. *-rgt-identity 96.5%

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

   11. Simplified 96.5%

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

   if -1.5 < x < 1

   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. distribute-lft-neg-in 99.7%

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

      4. distribute-rgt-neg-in 99.7%

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

   3. Simplified 99.7%

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

   4. 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 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 8: 74.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq -2.65 \cdot 10^{+20} \lor \neg \left(B \leq 0.054\right):\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= B -2.65e+20) (not (<= B 0.054)))
   (/ 1.0 (sin B))
   (+ (* B (+ 0.16666666666666666 (* x 0.3333333333333333))) (/ (- 1.0 x) B))))

C

double code(double B, double x) {
	double tmp;
	if ((B <= -2.65e+20) || !(B <= 0.054)) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - 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 ((b <= (-2.65d+20)) .or. (.not. (b <= 0.054d0))) then
        tmp = 1.0d0 / sin(b)
    else
        tmp = (b * (0.16666666666666666d0 + (x * 0.3333333333333333d0))) + ((1.0d0 - x) / b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((B <= -2.65e+20) || !(B <= 0.054)) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (B <= -2.65e+20) or not (B <= 0.054):
		tmp = 1.0 / math.sin(B)
	else:
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((B <= -2.65e+20) || !(B <= 0.054))
		tmp = Float64(1.0 / sin(B));
	else
		tmp = Float64(Float64(B * Float64(0.16666666666666666 + Float64(x * 0.3333333333333333))) + Float64(Float64(1.0 - x) / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((B <= -2.65e+20) || ~((B <= 0.054)))
		tmp = 1.0 / sin(B);
	else
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[B, -2.65e+20], N[Not[LessEqual[B, 0.054]], $MachinePrecision]], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(B * N[(0.16666666666666666 + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < -2.65e20 or 0.0539999999999999994 < B

   1. Initial program 99.3%

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

      1. distribute-lft-neg-in 99.3%

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

      2. +-commutative 99.3%

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

      3. distribute-lft-neg-in 99.3%

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

      4. distribute-rgt-neg-in 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x around 0 49.0%

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

   if -2.65e20 < B < 0.0539999999999999994

   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. distribute-lft-neg-in 99.8%

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

      4. distribute-rgt-neg-in 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in B around 0 96.4%

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

      1. +-commutative 96.4%

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

      2. mul-1-neg 96.4%

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

      3. sub-neg 96.4%

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

      4. associate--l+ 96.4%

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

      5. *-commutative 96.4%

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

      6. div-sub 96.4%

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

   6. Simplified 96.4%

      \[\leadsto \color{blue}{B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -2.65 \cdot 10^{+20} \lor \neg \left(B \leq 0.054\right):\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \end{array} \]

Alternative 9: 51.5% accurate, 16.2× speedup

Math

\[\begin{array}{l} \\ B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (+ (* B (+ 0.16666666666666666 (* x 0.3333333333333333))) (/ (- 1.0 x) B)))

C

double code(double B, double x) {
	return (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((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 + (x * 0.3333333333333333d0))) + ((1.0d0 - x) / b)
end function

Java

public static double code(double B, double x) {
	return (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B);
}

Python

def code(B, x):
	return (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B)

Julia

function code(B, x)
	return Float64(Float64(B * Float64(0.16666666666666666 + Float64(x * 0.3333333333333333))) + Float64(Float64(1.0 - x) / B))
end

MATLAB

function tmp = code(B, x)
	tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B);
end

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-neg-in 99.6%

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

   2. +-commutative 99.6%

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

   3. distribute-lft-neg-in 99.6%

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

   4. distribute-rgt-neg-in 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in B around 0 48.7%

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

   1. +-commutative 48.7%

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

   2. mul-1-neg 48.7%

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

   3. sub-neg 48.7%

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

   4. associate--l+ 48.7%

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

   5. *-commutative 48.7%

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

   6. div-sub 48.7%

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

6. Simplified 48.7%

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

7. Final simplification 48.7%

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

Alternative 10: 50.3% accurate, 25.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -5.5) (not (<= x 1.0))) (- (/ x B)) (/ 1.0 B)))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -5.5) || !(x <= 1.0)) {
		tmp = -(x / B);
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-5.5d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = -(x / b)
    else
        tmp = 1.0d0 / b
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -5.5) || !(x <= 1.0)) {
		tmp = -(x / B);
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -5.5) or not (x <= 1.0):
		tmp = -(x / B)
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -5.5) || !(x <= 1.0))
		tmp = Float64(-Float64(x / B));
	else
		tmp = Float64(1.0 / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -5.5) || ~((x <= 1.0)))
		tmp = -(x / B);
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -5.5], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], (-N[(x / B), $MachinePrecision]), N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.5 or 1 < x

   1. Initial program 99.4%

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

      1. distribute-lft-neg-in 99.4%

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

      2. +-commutative 99.4%

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

      3. distribute-lft-neg-in 99.4%

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

      4. distribute-rgt-neg-in 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in B around 0 47.6%

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

      1. neg-mul-1 47.6%

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

      2. sub-neg 47.6%

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

   6. Simplified 47.6%

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

   7. Taylor expanded in x around inf 46.6%

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

      1. neg-mul-1 46.6%

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

      2. distribute-neg-frac 46.6%

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

   9. Simplified 46.6%

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

   if -5.5 < x < 1

   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. distribute-lft-neg-in 99.7%

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

      4. distribute-rgt-neg-in 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in B around 0 49.2%

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

      1. neg-mul-1 49.2%

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

      2. sub-neg 49.2%

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

   6. Simplified 49.2%

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

   7. Taylor expanded in x around 0 48.9%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]

Alternative 11: 51.4% accurate, 42.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
2. 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. distribute-lft-neg-in 99.6%

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

   4. distribute-rgt-neg-in 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in B around 0 48.4%

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

   1. neg-mul-1 48.4%

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

   2. sub-neg 48.4%

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

6. Simplified 48.4%

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

7. Final simplification 48.4%

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

Alternative 12: 3.1% accurate, 70.0× speedup

Math

\[\begin{array}{l} \\ B \cdot 0.16666666666666666 \end{array} \]

FPCore

(FPCore (B x) :precision binary64 (* B 0.16666666666666666))

C

double code(double B, double x) {
	return B * 0.16666666666666666;
}

Fortran

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

Java

public static double code(double B, double x) {
	return B * 0.16666666666666666;
}

Python

def code(B, x):
	return B * 0.16666666666666666

Julia

function code(B, x)
	return Float64(B * 0.16666666666666666)
end

MATLAB

function tmp = code(B, x)
	tmp = B * 0.16666666666666666;
end

Wolfram

code[B_, x_] := N[(B * 0.16666666666666666), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-neg-in 99.6%

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

   2. +-commutative 99.6%

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

   3. cancel-sign-sub-inv 99.6%

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

   4. *-commutative 99.6%

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

   5. *-commutative 99.6%

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

   6. associate-*r/ 99.7%

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

   7. *-rgt-identity 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in B around 0 62.5%

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

5. Taylor expanded in B around inf 3.2%

   \[\leadsto \color{blue}{0.16666666666666666 \cdot B} \]
6. Step-by-step derivation

   1. *-commutative 3.2%

      \[\leadsto \color{blue}{B \cdot 0.16666666666666666} \]

7. Simplified 3.2%

   \[\leadsto \color{blue}{B \cdot 0.16666666666666666} \]

8. Final simplification 3.2%

   \[\leadsto B \cdot 0.16666666666666666 \]

Alternative 13: 26.3% accurate, 70.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
2. 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. distribute-lft-neg-in 99.6%

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

   4. distribute-rgt-neg-in 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in B around 0 48.4%

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

   1. neg-mul-1 48.4%

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

   2. sub-neg 48.4%

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

6. Simplified 48.4%

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

7. Taylor expanded in x around 0 25.6%

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

8. Final simplification 25.6%

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

Reproduce

herbie shell --seed 2023279 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))