VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 8.9s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. distribute-lft-neg-in 99.7%

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

   2. +-commutative 99.7%

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

   3. *-commutative 99.7%

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

   4. remove-double-neg 99.7%

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

   5. distribute-frac-neg2 99.7%

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

   6. tan-neg 99.7%

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

   7. cancel-sign-sub-inv 99.7%

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

   8. *-commutative 99.7%

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

   9. associate-*r/ 99.7%

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

   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 2: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -28:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;x \leq 12800000000:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(-\cos B\right)}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= x -28.0)
   (/ (- x) (tan B))
   (if (<= x 12800000000.0)
     (- (/ 1.0 (sin B)) (/ x B))
     (/ (* x (- (cos B))) (sin B)))))

C

double code(double B, double x) {
	double tmp;
	if (x <= -28.0) {
		tmp = -x / tan(B);
	} else if (x <= 12800000000.0) {
		tmp = (1.0 / sin(B)) - (x / 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 <= (-28.0d0)) then
        tmp = -x / tan(b)
    else if (x <= 12800000000.0d0) then
        tmp = (1.0d0 / sin(b)) - (x / 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 <= -28.0) {
		tmp = -x / Math.tan(B);
	} else if (x <= 12800000000.0) {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	} else {
		tmp = (x * -Math.cos(B)) / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if x <= -28.0:
		tmp = -x / math.tan(B)
	elif x <= 12800000000.0:
		tmp = (1.0 / math.sin(B)) - (x / B)
	else:
		tmp = (x * -math.cos(B)) / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (x <= -28.0)
		tmp = Float64(Float64(-x) / tan(B));
	elseif (x <= 12800000000.0)
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / 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 <= -28.0)
		tmp = -x / tan(B);
	elseif (x <= 12800000000.0)
		tmp = (1.0 / sin(B)) - (x / B);
	else
		tmp = (x * -cos(B)) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[x, -28.0], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 12800000000.0], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / 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 < -28

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 98.1%

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

      1. mul-1-neg 98.1%

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

      2. associate-/l* 98.1%

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

      3. distribute-rgt-neg-in 98.1%

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

      4. distribute-neg-frac 98.1%

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

   5. Simplified 98.1%

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

      1. frac-2neg 98.1%

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

      2. div-inv 98.0%

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

      3. remove-double-neg 98.0%

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

   7. Applied egg-rr 98.0%

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

      1. *-commutative 98.0%

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

      2. distribute-frac-neg2 98.0%

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

      3. inv-pow 98.0%

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

      4. metadata-eval 98.0%

         \[\leadsto x \cdot \left(\left(-{\sin B}^{\color{blue}{\left(-0.5 \cdot 2\right)}}\right) \cdot \cos B\right) \]

      5. pow-pow 41.4%

         \[\leadsto x \cdot \left(\left(-\color{blue}{{\left({\sin B}^{-0.5}\right)}^{2}}\right) \cdot \cos B\right) \]

      6. distribute-lft-neg-out 41.4%

         \[\leadsto x \cdot \color{blue}{\left(-{\left({\sin B}^{-0.5}\right)}^{2} \cdot \cos B\right)} \]

      7. pow-pow 98.0%

         \[\leadsto x \cdot \left(-\color{blue}{{\sin B}^{\left(-0.5 \cdot 2\right)}} \cdot \cos B\right) \]

      8. metadata-eval 98.0%

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

      9. inv-pow 98.0%

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

      10. associate-/r/ 98.0%

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

      11. tan-quot 98.2%

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

      12. distribute-rgt-neg-in 98.2%

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

      13. div-inv 98.3%

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

      14. distribute-neg-frac 98.3%

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

   9. Applied egg-rr 98.3%

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

   if -28 < x < 1.28e10

   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 97.1%

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

   if 1.28e10 < x

   1. Initial program 99.6%

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

      1. distribute-lft-neg-in 99.6%

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

      2. +-commutative 99.6%

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

      3. *-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. Step-by-step derivation

      1. div-inv 99.6%

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

      2. *-commutative 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. associate-/r/ 99.5%

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

   8. Applied egg-rr 99.5%

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

   9. Taylor expanded in x around inf 99.7%

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

       1. associate-*r/ 99.7%

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

       2. mul-1-neg 99.7%

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

       3. distribute-rgt-neg-out 99.7%

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

   11. Simplified 99.7%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -28:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;x \leq 12800000000:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(-\cos B\right)}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -62000 \lor \neg \left(x \leq 11600000000\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -62000.0) (not (<= x 11600000000.0)))
   (/ (- x) (tan B))
   (- (/ 1.0 (sin B)) (/ x B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -62000.0) || !(x <= 11600000000.0)) {
		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 <= (-62000.0d0)) .or. (.not. (x <= 11600000000.0d0))) 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 <= -62000.0) || !(x <= 11600000000.0)) {
		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 <= -62000.0) or not (x <= 11600000000.0):
		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 <= -62000.0) || !(x <= 11600000000.0))
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -62000.0) || ~((x <= 11600000000.0)))
		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, -62000.0], N[Not[LessEqual[x, 11600000000.0]], $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 < -62000 or 1.16e10 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 98.6%

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

      1. mul-1-neg 98.6%

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

      2. associate-/l* 98.6%

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

      3. distribute-rgt-neg-in 98.6%

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

      4. distribute-neg-frac 98.6%

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

   5. Simplified 98.6%

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

      1. frac-2neg 98.6%

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

      2. div-inv 98.5%

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

      3. remove-double-neg 98.5%

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

   7. Applied egg-rr 98.5%

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

      1. *-commutative 98.5%

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

      2. distribute-frac-neg2 98.5%

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

      3. inv-pow 98.5%

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

      4. metadata-eval 98.5%

         \[\leadsto x \cdot \left(\left(-{\sin B}^{\color{blue}{\left(-0.5 \cdot 2\right)}}\right) \cdot \cos B\right) \]

      5. pow-pow 47.0%

         \[\leadsto x \cdot \left(\left(-\color{blue}{{\left({\sin B}^{-0.5}\right)}^{2}}\right) \cdot \cos B\right) \]

      6. distribute-lft-neg-out 47.0%

         \[\leadsto x \cdot \color{blue}{\left(-{\left({\sin B}^{-0.5}\right)}^{2} \cdot \cos B\right)} \]

      7. pow-pow 98.5%

         \[\leadsto x \cdot \left(-\color{blue}{{\sin B}^{\left(-0.5 \cdot 2\right)}} \cdot \cos B\right) \]

      8. metadata-eval 98.5%

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

      9. inv-pow 98.5%

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

      10. associate-/r/ 98.5%

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

      11. tan-quot 98.7%

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

      12. distribute-rgt-neg-in 98.7%

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

      13. div-inv 98.8%

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

      14. distribute-neg-frac 98.8%

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

   9. Applied egg-rr 98.8%

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

   if -62000 < x < 1.16e10

   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 97.1%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -62000 \lor \neg \left(x \leq 11600000000\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15 \lor \neg \left(x \leq 1.1\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.15) (not (<= x 1.1))) (/ (- x) (tan B)) (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.15) || !(x <= 1.1)) {
		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 <= (-1.15d0)) .or. (.not. (x <= 1.1d0))) 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 <= -1.15) || !(x <= 1.1)) {
		tmp = -x / Math.tan(B);
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -1.15) or not (x <= 1.1):
		tmp = -x / math.tan(B)
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -1.15) || !(x <= 1.1))
		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 <= -1.15) || ~((x <= 1.1)))
		tmp = -x / tan(B);
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -1.15], N[Not[LessEqual[x, 1.1]], $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 < -1.1499999999999999 or 1.1000000000000001 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf 97.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   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-rgt-neg-in 97.2%

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

      4. distribute-neg-frac 97.2%

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

   5. Simplified 97.2%

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

      1. frac-2neg 97.2%

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

      2. div-inv 97.1%

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

      3. remove-double-neg 97.1%

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

   7. Applied egg-rr 97.1%

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

      1. *-commutative 97.1%

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

      2. distribute-frac-neg2 97.1%

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

      3. inv-pow 97.1%

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

      4. metadata-eval 97.1%

         \[\leadsto x \cdot \left(\left(-{\sin B}^{\color{blue}{\left(-0.5 \cdot 2\right)}}\right) \cdot \cos B\right) \]

      5. pow-pow 46.6%

         \[\leadsto x \cdot \left(\left(-\color{blue}{{\left({\sin B}^{-0.5}\right)}^{2}}\right) \cdot \cos B\right) \]

      6. distribute-lft-neg-out 46.6%

         \[\leadsto x \cdot \color{blue}{\left(-{\left({\sin B}^{-0.5}\right)}^{2} \cdot \cos B\right)} \]

      7. pow-pow 97.1%

         \[\leadsto x \cdot \left(-\color{blue}{{\sin B}^{\left(-0.5 \cdot 2\right)}} \cdot \cos B\right) \]

      8. metadata-eval 97.1%

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

      9. inv-pow 97.1%

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

      10. associate-/r/ 97.1%

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

      11. tan-quot 97.3%

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

      12. distribute-rgt-neg-in 97.3%

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

      13. div-inv 97.4%

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

      14. distribute-neg-frac 97.4%

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

   9. Applied egg-rr 97.4%

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

   if -1.1499999999999999 < x < 1.1000000000000001

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 96.9%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15 \lor \neg \left(x \leq 1.1\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.0135:\\ \;\;\;\;\frac{1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B 0.0135) (/ (- 1.0 x) B) (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if (B <= 0.0135) {
		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 <= 0.0135d0) 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 <= 0.0135) {
		tmp = (1.0 - x) / B;
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if B <= 0.0135:
		tmp = (1.0 - x) / B
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (B <= 0.0135)
		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 <= 0.0135)
		tmp = (1.0 - x) / B;
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[B, 0.0135], 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 < 0.0134999999999999998

   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 70.4%

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

   if 0.0134999999999999998 < B

   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 0 50.8%

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

Alternative 6: 50.3% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 9 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 9e-11))) (/ x (- B)) (/ 1.0 B)))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 9e-11)) {
		tmp = x / -B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 9d-11))) then
        tmp = x / -b
    else
        tmp = 1.0d0 / b
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 9e-11)) {
		tmp = x / -B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -1.0) or not (x <= 9e-11):
		tmp = x / -B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 9e-11))
		tmp = Float64(x / Float64(-B));
	else
		tmp = Float64(1.0 / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 9e-11)))
		tmp = x / -B;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 9e-11]], $MachinePrecision]], N[(x / (-B)), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 8.9999999999999999e-11 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in B around 0 47.3%

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

   4. Taylor expanded in x around inf 46.3%

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

      1. neg-mul-1 46.3%

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

      2. distribute-neg-frac2 46.3%

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

   6. Simplified 46.3%

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

   if -1 < x < 8.9999999999999999e-11

   1. Initial program 99.7%

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

   3. Taylor expanded in B around 0 51.3%

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

   4. Taylor expanded in x around 0 50.6%

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

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 9 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 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 99.7%

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

3. Taylor expanded in B around 0 49.2%

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

Alternative 8: 26.5% accurate, 70.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in B around 0 49.2%

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

4. Taylor expanded in x around 0 26.1%

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

Reproduce

herbie shell --seed 2024085 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))