VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.7%

Time: 31.9s

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: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in x around inf 87.6%

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

4. Taylor expanded in x around 0 99.7%

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

   1. mul-1-neg 99.7%

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

   2. associate-/l* 99.7%

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

   3. distribute-rgt-neg-in 99.7%

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

   4. mul-1-neg 99.7%

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

   5. rgt-mult-inverse 99.6%

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

   6. associate-*r/ 87.5%

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

   7. associate-/r* 87.6%

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

   8. distribute-lft-in 87.6%

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

   9. associate-/r* 87.5%

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

   10. +-commutative 87.5%

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

   11. mul-1-neg 87.5%

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

   12. sub-neg 87.5%

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

   13. div-sub 87.5%

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

   14. associate-*r/ 99.7%

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

6. Simplified 99.7%

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

7. Taylor expanded in x around 0 99.7%

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

   1. neg-mul-1 99.7%

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

   2. sub-neg 99.7%

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

9. Simplified 99.7%

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

Alternative 2: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -750:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+14}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= x -750.0)
   (/ (* x (cos B)) (- (sin B)))
   (if (<= x 5e+14) (- (/ 1.0 (sin B)) (/ x B)) (/ x (- (tan B))))))

C

double code(double B, double x) {
	double tmp;
	if (x <= -750.0) {
		tmp = (x * cos(B)) / -sin(B);
	} else if (x <= 5e+14) {
		tmp = (1.0 / sin(B)) - (x / 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 <= (-750.0d0)) then
        tmp = (x * cos(b)) / -sin(b)
    else if (x <= 5d+14) then
        tmp = (1.0d0 / sin(b)) - (x / 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 <= -750.0) {
		tmp = (x * Math.cos(B)) / -Math.sin(B);
	} else if (x <= 5e+14) {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	} else {
		tmp = x / -Math.tan(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if x <= -750.0:
		tmp = (x * math.cos(B)) / -math.sin(B)
	elif x <= 5e+14:
		tmp = (1.0 / math.sin(B)) - (x / B)
	else:
		tmp = x / -math.tan(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (x <= -750.0)
		tmp = Float64(Float64(x * cos(B)) / Float64(-sin(B)));
	elseif (x <= 5e+14)
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	else
		tmp = Float64(x / Float64(-tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (x <= -750.0)
		tmp = (x * cos(B)) / -sin(B);
	elseif (x <= 5e+14)
		tmp = (1.0 / sin(B)) - (x / B);
	else
		tmp = x / -tan(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[x, -750.0], N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[x, 5e+14], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -750

   1. Initial program 99.4%

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

   3. Taylor expanded in x around inf 97.2%

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

      1. mul-1-neg 97.2%

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

      2. associate-/l* 97.2%

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

      3. distribute-lft-neg-in 97.2%

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

   5. Simplified 97.2%

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

      1. associate-*r/ 97.2%

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

      2. frac-2neg 97.2%

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

      3. add-sqr-sqrt 96.9%

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

      4. sqrt-unprod 65.8%

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

      5. sqr-neg 65.8%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 0.5%

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

      8. distribute-lft-neg-out 0.5%

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

      9. add-sqr-sqrt 0.5%

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

      10. sqrt-unprod 0.4%

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

      11. sqr-neg 0.4%

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

      12. sqrt-unprod 0.0%

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

      13. add-sqr-sqrt 97.2%

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

   7. Applied egg-rr 97.2%

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

   if -750 < x < 5e14

   1. Initial program 99.8%

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

      1. distribute-lft-neg-in 99.8%

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

      2. +-commutative 99.8%

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

      3. *-commutative 99.8%

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

      4. remove-double-neg 99.8%

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

      5. distribute-frac-neg2 99.8%

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

      6. tan-neg 99.8%

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

      7. cancel-sign-sub-inv 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-*r/ 99.8%

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

      10. *-rgt-identity 99.8%

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

      11. tan-neg 99.8%

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

      12. distribute-neg-frac2 99.8%

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

      13. distribute-neg-frac 99.8%

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

      14. remove-double-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in B around 0 98.3%

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

   if 5e14 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf 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* 99.6%

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

      3. distribute-lft-neg-in 99.6%

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

   5. Simplified 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. clear-num 99.5%

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

      3. tan-quot 99.6%

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

   7. Applied egg-rr 99.6%

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

      1. *-lft-identity 99.6%

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

   9. Simplified 99.6%

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

       1. un-div-inv 99.7%

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

       2. distribute-frac-neg 99.7%

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

       3. neg-sub0 99.7%

          \[\leadsto \color{blue}{0 - \frac{x}{\tan B}} \]

   11. Applied egg-rr 99.7%

       \[\leadsto \color{blue}{0 - \frac{x}{\tan B}} \]
   12. Step-by-step derivation

       1. neg-sub0 99.7%

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

       2. distribute-frac-neg2 99.7%

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

   13. Simplified 99.7%

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

Alternative 3: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -800:\\ \;\;\;\;x \cdot \frac{\cos B}{-\sin B}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+14}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= x -800.0)
   (* x (/ (cos B) (- (sin B))))
   (if (<= x 5e+14) (- (/ 1.0 (sin B)) (/ x B)) (/ x (- (tan B))))))

C

double code(double B, double x) {
	double tmp;
	if (x <= -800.0) {
		tmp = x * (cos(B) / -sin(B));
	} else if (x <= 5e+14) {
		tmp = (1.0 / sin(B)) - (x / 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 <= (-800.0d0)) then
        tmp = x * (cos(b) / -sin(b))
    else if (x <= 5d+14) then
        tmp = (1.0d0 / sin(b)) - (x / 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 <= -800.0) {
		tmp = x * (Math.cos(B) / -Math.sin(B));
	} else if (x <= 5e+14) {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	} else {
		tmp = x / -Math.tan(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if x <= -800.0:
		tmp = x * (math.cos(B) / -math.sin(B))
	elif x <= 5e+14:
		tmp = (1.0 / math.sin(B)) - (x / B)
	else:
		tmp = x / -math.tan(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (x <= -800.0)
		tmp = Float64(x * Float64(cos(B) / Float64(-sin(B))));
	elseif (x <= 5e+14)
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	else
		tmp = Float64(x / Float64(-tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (x <= -800.0)
		tmp = x * (cos(B) / -sin(B));
	elseif (x <= 5e+14)
		tmp = (1.0 / sin(B)) - (x / B);
	else
		tmp = x / -tan(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[x, -800.0], N[(x * N[(N[Cos[B], $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5e+14], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -800

   1. Initial program 99.4%

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

   3. Taylor expanded in x around inf 97.2%

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

      1. mul-1-neg 97.2%

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

      2. associate-/l* 97.2%

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

      3. distribute-lft-neg-in 97.2%

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

   5. Simplified 97.2%

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

   if -800 < x < 5e14

   1. Initial program 99.8%

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

      1. distribute-lft-neg-in 99.8%

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

      2. +-commutative 99.8%

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

      3. *-commutative 99.8%

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

      4. remove-double-neg 99.8%

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

      5. distribute-frac-neg2 99.8%

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

      6. tan-neg 99.8%

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

      7. cancel-sign-sub-inv 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-*r/ 99.8%

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

      10. *-rgt-identity 99.8%

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

      11. tan-neg 99.8%

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

      12. distribute-neg-frac2 99.8%

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

      13. distribute-neg-frac 99.8%

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

      14. remove-double-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in B around 0 98.3%

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

   if 5e14 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf 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* 99.6%

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

      3. distribute-lft-neg-in 99.6%

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

   5. Simplified 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. clear-num 99.5%

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

      3. tan-quot 99.6%

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

   7. Applied egg-rr 99.6%

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

      1. *-lft-identity 99.6%

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

   9. Simplified 99.6%

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

       1. un-div-inv 99.7%

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

       2. distribute-frac-neg 99.7%

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

       3. neg-sub0 99.7%

          \[\leadsto \color{blue}{0 - \frac{x}{\tan B}} \]

   11. Applied egg-rr 99.7%

       \[\leadsto \color{blue}{0 - \frac{x}{\tan B}} \]
   12. Step-by-step derivation

       1. neg-sub0 99.7%

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

       2. distribute-frac-neg2 99.7%

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

   13. Simplified 99.7%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -800:\\ \;\;\;\;x \cdot \frac{\cos B}{-\sin B}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+14}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 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. *-commutative 99.6%

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

   4. remove-double-neg 99.6%

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

   5. distribute-frac-neg2 99.6%

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

   6. tan-neg 99.6%

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

   7. cancel-sign-sub-inv 99.6%

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

   8. *-commutative 99.6%

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

   9. associate-*r/ 99.7%

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

   10. *-rgt-identity 99.7%

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

   11. tan-neg 99.7%

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

   12. distribute-neg-frac2 99.7%

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

   13. distribute-neg-frac 99.7%

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

   14. remove-double-neg 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. Add Preprocessing
5. Add Preprocessing

Alternative 5: 98.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -460 \lor \neg \left(x \leq 5 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -460.0) (not (<= x 5e+14)))
   (/ x (- (tan B)))
   (- (/ 1.0 (sin B)) (/ x B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -460.0) || !(x <= 5e+14)) {
		tmp = 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 <= (-460.0d0)) .or. (.not. (x <= 5d+14))) then
        tmp = 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 <= -460.0) || !(x <= 5e+14)) {
		tmp = x / -Math.tan(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -460.0) or not (x <= 5e+14):
		tmp = 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 <= -460.0) || !(x <= 5e+14))
		tmp = Float64(x / Float64(-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 <= -460.0) || ~((x <= 5e+14)))
		tmp = x / -tan(B);
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -460.0], N[Not[LessEqual[x, 5e+14]], $MachinePrecision]], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -460 or 5e14 < x

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf 98.2%

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

      1. mul-1-neg 98.2%

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

      2. associate-/l* 98.2%

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

      3. distribute-lft-neg-in 98.2%

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

   5. Simplified 98.2%

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

      1. *-un-lft-identity 98.2%

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

      2. clear-num 98.1%

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

      3. tan-quot 98.1%

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

   7. Applied egg-rr 98.1%

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

      1. *-lft-identity 98.1%

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

   9. Simplified 98.1%

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

       1. un-div-inv 98.2%

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

       2. distribute-frac-neg 98.2%

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

       3. neg-sub0 98.2%

          \[\leadsto \color{blue}{0 - \frac{x}{\tan B}} \]

   11. Applied egg-rr 98.2%

       \[\leadsto \color{blue}{0 - \frac{x}{\tan B}} \]
   12. Step-by-step derivation

       1. neg-sub0 98.2%

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

       2. distribute-frac-neg2 98.2%

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

   13. Simplified 98.2%

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

   if -460 < x < 5e14

   1. Initial program 99.8%

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

      1. distribute-lft-neg-in 99.8%

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

      2. +-commutative 99.8%

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

      3. *-commutative 99.8%

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

      4. remove-double-neg 99.8%

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

      5. distribute-frac-neg2 99.8%

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

      6. tan-neg 99.8%

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

      7. cancel-sign-sub-inv 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-*r/ 99.8%

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

      10. *-rgt-identity 99.8%

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

      11. tan-neg 99.8%

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

      12. distribute-neg-frac2 99.8%

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

      13. distribute-neg-frac 99.8%

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

      14. remove-double-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in B around 0 98.3%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -460 \lor \neg \left(x \leq 5 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.46 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.46) (not (<= x 1.0))) (/ x (- (tan B))) (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.46) || !(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 <= (-1.46d0)) .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 <= -1.46) || !(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 <= -1.46) 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 <= -1.46) || !(x <= 1.0))
		tmp = Float64(x / Float64(-tan(B)));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -1.46) || ~((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, -1.46], 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 < -1.46 or 1 < x

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf 98.0%

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

      1. mul-1-neg 98.0%

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

      2. associate-/l* 98.0%

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

      3. distribute-lft-neg-in 98.0%

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

   5. Simplified 98.0%

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

      1. *-un-lft-identity 98.0%

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

      2. clear-num 97.9%

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

      3. tan-quot 97.9%

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

   7. Applied egg-rr 97.9%

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

      1. *-lft-identity 97.9%

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

   9. Simplified 97.9%

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

       1. un-div-inv 98.0%

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

       2. distribute-frac-neg 98.0%

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

       3. neg-sub0 98.0%

          \[\leadsto \color{blue}{0 - \frac{x}{\tan B}} \]

   11. Applied egg-rr 98.0%

       \[\leadsto \color{blue}{0 - \frac{x}{\tan B}} \]
   12. Step-by-step derivation

       1. neg-sub0 98.0%

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

       2. distribute-frac-neg2 98.0%

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

   13. Simplified 98.0%

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

   if -1.46 < x < 1

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 97.9%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.46 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.000105:\\ \;\;\;\;\frac{1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B 0.000105) (/ (- 1.0 x) B) (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if (B <= 0.000105) {
		tmp = (1.0 - x) / B;
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (b <= 0.000105d0) then
        tmp = (1.0d0 - x) / b
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (B <= 0.000105) {
		tmp = (1.0 - x) / B;
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if B <= 0.000105:
		tmp = (1.0 - x) / B
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (B <= 0.000105)
		tmp = Float64(Float64(1.0 - x) / B);
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (B <= 0.000105)
		tmp = (1.0 - x) / B;
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[B, 0.000105], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.05e-4

   1. Initial program 99.7%

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

   3. Taylor expanded in B around 0 60.1%

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

   if 1.05e-4 < B

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 52.5%

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

Alternative 8: 50.4% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0017 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -0.0017) (not (<= x 1.0))) (/ x (- B)) (/ 1.0 B)))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -0.0017) || !(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 <= (-0.0017d0)) .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 <= -0.0017) || !(x <= 1.0)) {
		tmp = x / -B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -0.0017) 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 <= -0.0017) || !(x <= 1.0))
		tmp = Float64(x / Float64(-B));
	else
		tmp = Float64(1.0 / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -0.0017) || ~((x <= 1.0)))
		tmp = x / -B;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -0.0017], 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 < -0.00169999999999999991 or 1 < x

   1. Initial program 99.5%

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

   3. Taylor expanded in B around 0 48.9%

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

   4. Taylor expanded in x around inf 48.4%

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

      1. mul-1-neg 48.4%

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

   6. Simplified 48.4%

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

   if -0.00169999999999999991 < x < 1

   1. Initial program 99.8%

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

   3. Taylor expanded in B around 0 47.4%

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

   4. Taylor expanded in x around 0 46.9%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0017 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 51.5% accurate, 42.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in B around 0 48.1%

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

Alternative 10: 26.6% accurate, 70.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in B around 0 48.1%

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

4. Taylor expanded in x around 0 25.0%

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

Reproduce

herbie shell --seed 2024116 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))