VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 9.2s

Alternatives: 12

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. +-commutative 99.7%

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

   2. unsub-neg 99.7%

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

   3. associate-*r/ 99.8%

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

   4. *-rgt-identity 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -750000 \lor \neg \left(x \leq 525000000\right):\\ \;\;\;\;-\frac{\cos B}{\frac{\sin B}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -750000.0) (not (<= x 525000000.0)))
   (- (/ (cos B) (/ (sin B) x)))
   (/ (- 1.0 x) (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -750000.0) || !(x <= 525000000.0)) {
		tmp = -(cos(B) / (sin(B) / x));
	} 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 <= (-750000.0d0)) .or. (.not. (x <= 525000000.0d0))) then
        tmp = -(cos(b) / (sin(b) / x))
    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 <= -750000.0) || !(x <= 525000000.0)) {
		tmp = -(Math.cos(B) / (Math.sin(B) / x));
	} else {
		tmp = (1.0 - x) / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -750000.0) or not (x <= 525000000.0):
		tmp = -(math.cos(B) / (math.sin(B) / x))
	else:
		tmp = (1.0 - x) / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -750000.0) || !(x <= 525000000.0))
		tmp = Float64(-Float64(cos(B) / Float64(sin(B) / x)));
	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 <= -750000.0) || ~((x <= 525000000.0)))
		tmp = -(cos(B) / (sin(B) / x));
	else
		tmp = (1.0 - x) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -750000.0], N[Not[LessEqual[x, 525000000.0]], $MachinePrecision]], (-N[(N[Cos[B], $MachinePrecision] / N[(N[Sin[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.5e5 or 5.25e8 < 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%

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

   3. Simplified 99.6%

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

      1. un-div-inv 99.8%

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

      2. neg-mul-1 99.8%

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

      3. associate-/l* 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in B around inf 99.6%

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

      1. mul-1-neg 99.6%

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

      2. sub-neg 99.6%

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

      3. *-commutative 99.6%

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

      4. div-sub 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in x around inf 99.4%

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

       1. mul-1-neg 99.4%

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

       2. associate-/l* 99.4%

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

   11. Simplified 99.4%

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

   if -7.5e5 < x < 5.25e8

   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%

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

   3. Simplified 99.8%

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

      1. un-div-inv 99.8%

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

      2. neg-mul-1 99.8%

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

      3. associate-/l* 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in B around inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. sub-neg 99.8%

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

      3. *-commutative 99.8%

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

      4. div-sub 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in B around 0 98.4%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -750000 \lor \neg \left(x \leq 525000000\right):\\ \;\;\;\;-\frac{\cos B}{\frac{\sin B}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \]

Alternative 3: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -31000000 \lor \neg \left(x \leq 840000000\right):\\ \;\;\;\;\frac{\cos B \cdot \left(-x\right)}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -31000000.0) (not (<= x 840000000.0)))
   (/ (* (cos B) (- x)) (sin B))
   (/ (- 1.0 x) (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -31000000.0) || !(x <= 840000000.0)) {
		tmp = (cos(B) * -x) / sin(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 <= (-31000000.0d0)) .or. (.not. (x <= 840000000.0d0))) then
        tmp = (cos(b) * -x) / sin(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 <= -31000000.0) || !(x <= 840000000.0)) {
		tmp = (Math.cos(B) * -x) / Math.sin(B);
	} else {
		tmp = (1.0 - x) / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -31000000.0) or not (x <= 840000000.0):
		tmp = (math.cos(B) * -x) / math.sin(B)
	else:
		tmp = (1.0 - x) / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -31000000.0) || !(x <= 840000000.0))
		tmp = Float64(Float64(cos(B) * Float64(-x)) / sin(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 <= -31000000.0) || ~((x <= 840000000.0)))
		tmp = (cos(B) * -x) / sin(B);
	else
		tmp = (1.0 - x) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -31000000.0], N[Not[LessEqual[x, 840000000.0]], $MachinePrecision]], N[(N[(N[Cos[B], $MachinePrecision] * (-x)), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.1e7 or 8.4e8 < x

   1. Initial program 99.6%

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

      1. distribute-lft-neg-in 99.6%

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

   3. Simplified 99.6%

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

      1. un-div-inv 99.8%

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

      2. neg-mul-1 99.8%

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

      3. associate-/l* 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in B around inf 99.6%

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

      1. mul-1-neg 99.6%

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

      2. sub-neg 99.6%

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

      3. *-commutative 99.6%

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

      4. div-sub 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in x around inf 99.4%

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

       1. associate-*r/ 99.4%

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

       2. *-commutative 99.4%

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

       3. neg-mul-1 99.4%

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

       4. *-commutative 99.4%

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

       5. distribute-rgt-neg-out 99.4%

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

   11. Simplified 99.4%

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

   if -3.1e7 < x < 8.4e8

   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%

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

   3. Simplified 99.8%

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

      1. un-div-inv 99.8%

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

      2. neg-mul-1 99.8%

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

      3. associate-/l* 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in B around inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. sub-neg 99.8%

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

      3. *-commutative 99.8%

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

      4. div-sub 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in B around 0 98.4%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -31000000 \lor \neg \left(x \leq 840000000\right):\\ \;\;\;\;\frac{\cos B \cdot \left(-x\right)}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \]

Alternative 4: 86.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -124000000000 \lor \neg \left(x \leq 1.22 \cdot 10^{+48}\right):\\ \;\;\;\;\left(\frac{1}{B} + B \cdot 0.16666666666666666\right) - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -124000000000.0) (not (<= x 1.22e+48)))
   (- (+ (/ 1.0 B) (* B 0.16666666666666666)) (/ x (tan B)))
   (/ (- 1.0 x) (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -124000000000.0) || !(x <= 1.22e+48)) {
		tmp = ((1.0 / B) + (B * 0.16666666666666666)) - (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 <= (-124000000000.0d0)) .or. (.not. (x <= 1.22d+48))) then
        tmp = ((1.0d0 / b) + (b * 0.16666666666666666d0)) - (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 <= -124000000000.0) || !(x <= 1.22e+48)) {
		tmp = ((1.0 / B) + (B * 0.16666666666666666)) - (x / Math.tan(B));
	} else {
		tmp = (1.0 - x) / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -124000000000.0) or not (x <= 1.22e+48):
		tmp = ((1.0 / B) + (B * 0.16666666666666666)) - (x / math.tan(B))
	else:
		tmp = (1.0 - x) / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -124000000000.0) || !(x <= 1.22e+48))
		tmp = Float64(Float64(Float64(1.0 / B) + Float64(B * 0.16666666666666666)) - 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 <= -124000000000.0) || ~((x <= 1.22e+48)))
		tmp = ((1.0 / B) + (B * 0.16666666666666666)) - (x / tan(B));
	else
		tmp = (1.0 - x) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -124000000000.0], N[Not[LessEqual[x, 1.22e+48]], $MachinePrecision]], N[(N[(N[(1.0 / B), $MachinePrecision] + N[(B * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.24e11 or 1.22000000000000004e48 < x

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unsub-neg 99.6%

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

      3. associate-*r/ 99.8%

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

      4. *-rgt-identity 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in B around 0 76.9%

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

   if -1.24e11 < x < 1.22000000000000004e48

   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%

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

   3. Simplified 99.8%

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

      1. un-div-inv 99.8%

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

      2. neg-mul-1 99.8%

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

      3. associate-/l* 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in B around inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. sub-neg 99.8%

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

      3. *-commutative 99.8%

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

      4. div-sub 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in B around 0 95.2%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -124000000000 \lor \neg \left(x \leq 1.22 \cdot 10^{+48}\right):\\ \;\;\;\;\left(\frac{1}{B} + B \cdot 0.16666666666666666\right) - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \]

Alternative 5: 76.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.05)) {
		tmp = -x / sin(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 <= (-1.0d0)) .or. (.not. (x <= 1.05d0))) then
        tmp = -x / sin(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 <= -1.0) || !(x <= 1.05)) {
		tmp = -x / Math.sin(B);
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.05000000000000004 < 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%

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

   3. Simplified 99.6%

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

      1. un-div-inv 99.8%

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

      2. neg-mul-1 99.8%

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

      3. associate-/l* 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in B around inf 99.6%

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

      1. mul-1-neg 99.6%

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

      2. sub-neg 99.6%

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

      3. *-commutative 99.6%

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

      4. div-sub 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in x around inf 97.9%

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

       1. associate-*r/ 97.9%

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

       2. *-commutative 97.9%

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

       3. neg-mul-1 97.9%

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

       4. *-commutative 97.9%

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

       5. distribute-rgt-neg-out 97.9%

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

   11. Simplified 97.9%

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

   12. Taylor expanded in B around 0 47.7%

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

       1. mul-1-neg 47.7%

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

   14. Simplified 47.7%

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

   if -1 < x < 1.05000000000000004

   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%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 98.6%

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

4. Final simplification 72.2%

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

Alternative 6: 75.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 - x}{B}\\ \mathbf{if}\;x \leq -0.009:\\ \;\;\;\;B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + t_0\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t_0 + B \cdot 0.16666666666666666\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (/ (- 1.0 x) B)))
   (if (<= x -0.009)
     (+ (* B (+ 0.16666666666666666 (* x 0.3333333333333333))) t_0)
     (if (<= x 6.2e-12) (/ 1.0 (sin B)) (+ t_0 (* B 0.16666666666666666))))))

C

double code(double B, double x) {
	double t_0 = (1.0 - x) / B;
	double tmp;
	if (x <= -0.009) {
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + t_0;
	} else if (x <= 6.2e-12) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = t_0 + (B * 0.16666666666666666);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 - x) / b
    if (x <= (-0.009d0)) then
        tmp = (b * (0.16666666666666666d0 + (x * 0.3333333333333333d0))) + t_0
    else if (x <= 6.2d-12) then
        tmp = 1.0d0 / sin(b)
    else
        tmp = t_0 + (b * 0.16666666666666666d0)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double t_0 = (1.0 - x) / B;
	double tmp;
	if (x <= -0.009) {
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + t_0;
	} else if (x <= 6.2e-12) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = t_0 + (B * 0.16666666666666666);
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = (1.0 - x) / B
	tmp = 0
	if x <= -0.009:
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + t_0
	elif x <= 6.2e-12:
		tmp = 1.0 / math.sin(B)
	else:
		tmp = t_0 + (B * 0.16666666666666666)
	return tmp

Julia

function code(B, x)
	t_0 = Float64(Float64(1.0 - x) / B)
	tmp = 0.0
	if (x <= -0.009)
		tmp = Float64(Float64(B * Float64(0.16666666666666666 + Float64(x * 0.3333333333333333))) + t_0);
	elseif (x <= 6.2e-12)
		tmp = Float64(1.0 / sin(B));
	else
		tmp = Float64(t_0 + Float64(B * 0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	t_0 = (1.0 - x) / B;
	tmp = 0.0;
	if (x <= -0.009)
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + t_0;
	elseif (x <= 6.2e-12)
		tmp = 1.0 / sin(B);
	else
		tmp = t_0 + (B * 0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := Block[{t$95$0 = N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]}, If[LessEqual[x, -0.009], N[(N[(B * N[(0.16666666666666666 + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[x, 6.2e-12], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(B * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00899999999999999932

   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%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in B around 0 47.4%

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

      1. +-commutative 47.4%

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

      2. mul-1-neg 47.4%

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

      3. sub-neg 47.4%

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

      4. associate--l+ 47.4%

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

      5. *-commutative 47.4%

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

      6. *-commutative 47.4%

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

      7. div-sub 47.4%

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

   6. Simplified 47.4%

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

   if -0.00899999999999999932 < x < 6.2000000000000002e-12

   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%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 98.8%

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

   if 6.2000000000000002e-12 < x

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unsub-neg 99.6%

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

      3. associate-*r/ 99.8%

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

      4. *-rgt-identity 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in B around 0 74.7%

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

   5. Taylor expanded in B around 0 44.1%

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

      1. associate--l+ 44.1%

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

      2. *-commutative 44.1%

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

      3. sub-div 44.1%

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

   7. Applied egg-rr 44.1%

      \[\leadsto \color{blue}{B \cdot 0.16666666666666666 + \frac{1 - x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.009:\\ \;\;\;\;B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B} + B \cdot 0.16666666666666666\\ \end{array} \]

Alternative 7: 77.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

3. Simplified 99.7%

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

   1. un-div-inv 99.8%

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

   2. neg-mul-1 99.8%

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

   3. associate-/l* 99.7%

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

5. Applied egg-rr 99.7%

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

6. Taylor expanded in B around inf 99.7%

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

   1. mul-1-neg 99.7%

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

   2. sub-neg 99.7%

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

   3. *-commutative 99.7%

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

   4. div-sub 99.7%

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

8. Simplified 99.7%

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

9. Taylor expanded in B around 0 73.0%

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

10. Final simplification 73.0%

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

Alternative 8: 51.6% accurate, 19.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = ((1.0d0 - x) / b) + (0.3333333333333333d0 * (b * x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

3. Simplified 99.7%

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

   1. un-div-inv 99.8%

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

   2. neg-mul-1 99.8%

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

   3. associate-/l* 99.7%

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

5. Applied egg-rr 99.7%

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

6. Taylor expanded in B around inf 99.7%

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

   1. mul-1-neg 99.7%

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

   2. sub-neg 99.7%

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

   3. *-commutative 99.7%

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

   4. div-sub 99.7%

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

8. Simplified 99.7%

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

9. Taylor expanded in B around 0 44.5%

   \[\leadsto \color{blue}{\left(\left(0.5 \cdot x - -0.16666666666666666 \cdot \left(1 - x\right)\right) \cdot B + \frac{1}{B}\right) - \frac{x}{B}} \]
10. Step-by-step derivation

    1. associate--l+ 44.5%

       \[\leadsto \color{blue}{\left(0.5 \cdot x - -0.16666666666666666 \cdot \left(1 - x\right)\right) \cdot B + \left(\frac{1}{B} - \frac{x}{B}\right)} \]

    2. *-commutative 44.5%

       \[\leadsto \color{blue}{B \cdot \left(0.5 \cdot x - -0.16666666666666666 \cdot \left(1 - x\right)\right)} + \left(\frac{1}{B} - \frac{x}{B}\right) \]

    3. cancel-sign-sub-inv 44.5%

       \[\leadsto B \cdot \color{blue}{\left(0.5 \cdot x + \left(--0.16666666666666666\right) \cdot \left(1 - x\right)\right)} + \left(\frac{1}{B} - \frac{x}{B}\right) \]

    4. *-commutative 44.5%

       \[\leadsto B \cdot \left(\color{blue}{x \cdot 0.5} + \left(--0.16666666666666666\right) \cdot \left(1 - x\right)\right) + \left(\frac{1}{B} - \frac{x}{B}\right) \]

    5. metadata-eval 44.5%

       \[\leadsto B \cdot \left(x \cdot 0.5 + \color{blue}{0.16666666666666666} \cdot \left(1 - x\right)\right) + \left(\frac{1}{B} - \frac{x}{B}\right) \]

    6. div-sub 44.5%

       \[\leadsto B \cdot \left(x \cdot 0.5 + 0.16666666666666666 \cdot \left(1 - x\right)\right) + \color{blue}{\frac{1 - x}{B}} \]

11. Simplified 44.5%

    \[\leadsto \color{blue}{B \cdot \left(x \cdot 0.5 + 0.16666666666666666 \cdot \left(1 - x\right)\right) + \frac{1 - x}{B}} \]

12. Taylor expanded in x around inf 45.1%

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

13. Final simplification 45.1%

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

Alternative 9: 50.3% accurate, 25.8× speedup

Math

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

FPCore

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

C

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

Python

def code(B, x):
	tmp = 0
	if (x <= -1.0) 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 <= -1.0) || !(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 <= -1.0) || ~((x <= 1.0)))
		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, 1.0]], $MachinePrecision]], N[((-x) / B), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unsub-neg 99.6%

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

      3. associate-*r/ 99.8%

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

      4. *-rgt-identity 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in B around 0 74.6%

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

   5. Taylor expanded in B around 0 44.7%

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

   6. Taylor expanded in x around inf 43.4%

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

      1. mul-1-neg 43.4%

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

      2. distribute-frac-neg 43.4%

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

   8. Simplified 43.4%

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

   if -1 < x < 1

   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%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in B around 0 44.2%

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

      1. mul-1-neg 44.2%

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

      2. sub-neg 44.2%

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

   6. Simplified 44.2%

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

   7. Taylor expanded in x around 0 43.9%

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

4. Final simplification 43.7%

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

Alternative 10: 51.3% 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. Step-by-step derivation

   1. distribute-lft-neg-in 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in B around 0 44.4%

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

   1. mul-1-neg 44.4%

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

   2. sub-neg 44.4%

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

6. Simplified 44.4%

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

7. Final simplification 44.4%

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

Alternative 11: 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 99.7%

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

   1. +-commutative 99.7%

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

   2. unsub-neg 99.7%

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

   3. associate-*r/ 99.8%

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

   4. *-rgt-identity 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in B around 0 59.8%

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

5. Taylor expanded in B around inf 2.9%

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

   1. *-commutative 2.9%

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

7. Simplified 2.9%

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

8. Final simplification 2.9%

   \[\leadsto B \cdot 0.16666666666666666 \]

Alternative 12: 26.7% accurate, 70.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. distribute-lft-neg-in 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in B around 0 44.4%

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

   1. mul-1-neg 44.4%

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

   2. sub-neg 44.4%

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

6. Simplified 44.4%

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

7. Taylor expanded in x around 0 22.5%

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

8. Final simplification 22.5%

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

Reproduce

herbie shell --seed 2023213 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))