VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 8.5s

Alternatives: 11

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. associate-*l/ 99.8%

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

   9. *-lft-identity 99.8%

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

   10. tan-neg 99.8%

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

   11. distribute-neg-frac2 99.8%

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

   12. distribute-neg-frac 99.8%

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

   13. 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. Final simplification 99.8%

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

Alternative 2: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -22000:\\ \;\;\;\;x \cdot \left(\frac{1}{B \cdot x} + \frac{-1}{\tan B}\right)\\ \mathbf{elif}\;x \leq 4200000:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot \frac{\cos B}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= x -22000.0)
   (* x (+ (/ 1.0 (* B x)) (/ -1.0 (tan B))))
   (if (<= x 4200000.0)
     (- (/ 1.0 (sin B)) (/ x B))
     (* (- x) (/ (cos B) (sin B))))))

C

double code(double B, double x) {
	double tmp;
	if (x <= -22000.0) {
		tmp = x * ((1.0 / (B * x)) + (-1.0 / tan(B)));
	} else if (x <= 4200000.0) {
		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 <= (-22000.0d0)) then
        tmp = x * ((1.0d0 / (b * x)) + ((-1.0d0) / tan(b)))
    else if (x <= 4200000.0d0) 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 <= -22000.0) {
		tmp = x * ((1.0 / (B * x)) + (-1.0 / Math.tan(B)));
	} else if (x <= 4200000.0) {
		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 <= -22000.0:
		tmp = x * ((1.0 / (B * x)) + (-1.0 / math.tan(B)))
	elif x <= 4200000.0:
		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 <= -22000.0)
		tmp = Float64(x * Float64(Float64(1.0 / Float64(B * x)) + Float64(-1.0 / tan(B))));
	elseif (x <= 4200000.0)
		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 <= -22000.0)
		tmp = x * ((1.0 / (B * x)) + (-1.0 / tan(B)));
	elseif (x <= 4200000.0)
		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, -22000.0], N[(x * N[(N[(1.0 / N[(B * x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4200000.0], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[((-x) * N[(N[Cos[B], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -22000

   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 99.5%

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

      1. clear-num 99.4%

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

      2. tan-quot 99.6%

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

      3. *-un-lft-identity 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. *-lft-identity 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in B around 0 99.0%

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

      1. *-commutative 99.0%

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

   10. Simplified 99.0%

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

   if -22000 < x < 4.2e6

   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. associate-*l/ 99.8%

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

      9. *-lft-identity 99.8%

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

      10. tan-neg 99.8%

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

      11. distribute-neg-frac2 99.8%

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

      12. distribute-neg-frac 99.8%

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

      13. 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. Step-by-step derivation

      1. div-inv 99.8%

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

      2. *-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in B around 0 98.6%

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

   if 4.2e6 < 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 98.7%

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

      1. mul-1-neg 98.7%

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

      2. associate-/l* 99.0%

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

      3. distribute-rgt-neg-in 99.0%

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

      4. distribute-neg-frac 99.0%

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

   5. Simplified 99.0%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -22000:\\ \;\;\;\;x \cdot \left(\frac{1}{B \cdot x} + \frac{-1}{\tan B}\right)\\ \mathbf{elif}\;x \leq 4200000:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot \frac{\cos B}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -22000 \lor \neg \left(x \leq 1.45\right):\\ \;\;\;\;x \cdot \left(\frac{1}{B \cdot x} + \frac{-1}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -22000.0) (not (<= x 1.45)))
   (* x (+ (/ 1.0 (* B x)) (/ -1.0 (tan B))))
   (- (/ 1.0 (sin B)) (/ x B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -22000.0) || !(x <= 1.45)) {
		tmp = x * ((1.0 / (B * x)) + (-1.0 / 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 <= (-22000.0d0)) .or. (.not. (x <= 1.45d0))) then
        tmp = x * ((1.0d0 / (b * x)) + ((-1.0d0) / 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 <= -22000.0) || !(x <= 1.45)) {
		tmp = x * ((1.0 / (B * x)) + (-1.0 / Math.tan(B)));
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -22000.0) or not (x <= 1.45):
		tmp = x * ((1.0 / (B * x)) + (-1.0 / math.tan(B)))
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -22000.0) || !(x <= 1.45))
		tmp = Float64(x * Float64(Float64(1.0 / Float64(B * x)) + Float64(-1.0 / 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 <= -22000.0) || ~((x <= 1.45)))
		tmp = x * ((1.0 / (B * x)) + (-1.0 / tan(B)));
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -22000.0], N[Not[LessEqual[x, 1.45]], $MachinePrecision]], N[(x * N[(N[(1.0 / N[(B * x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -22000 or 1.44999999999999996 < 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%

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

      1. clear-num 99.5%

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

      2. tan-quot 99.6%

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

      3. *-un-lft-identity 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. *-lft-identity 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in B around 0 98.9%

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

      1. *-commutative 98.9%

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

   10. Simplified 98.9%

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

   if -22000 < x < 1.44999999999999996

   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. associate-*l/ 99.8%

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

      9. *-lft-identity 99.8%

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

      10. tan-neg 99.8%

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

      11. distribute-neg-frac2 99.8%

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

      12. distribute-neg-frac 99.8%

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

      13. 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. Step-by-step derivation

      1. div-inv 99.8%

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

      2. *-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in B around 0 98.6%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -22000 \lor \neg \left(x \leq 1.45\right):\\ \;\;\;\;x \cdot \left(\frac{1}{B \cdot x} + \frac{-1}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.0046:\\ \;\;\;\;B \cdot 0.16666666666666666 + \left(x \cdot \left(B \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{1}{x} + -1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B 0.0046)
   (+
    (* B 0.16666666666666666)
    (+ (* x (* B 0.3333333333333333)) (/ (- 1.0 x) B)))
   (* x (/ (+ (/ 1.0 x) -1.0) (sin B)))))

C

double code(double B, double x) {
	double tmp;
	if (B <= 0.0046) {
		tmp = (B * 0.16666666666666666) + ((x * (B * 0.3333333333333333)) + ((1.0 - x) / B));
	} else {
		tmp = x * (((1.0 / x) + -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 <= 0.0046d0) then
        tmp = (b * 0.16666666666666666d0) + ((x * (b * 0.3333333333333333d0)) + ((1.0d0 - x) / b))
    else
        tmp = x * (((1.0d0 / x) + (-1.0d0)) / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (B <= 0.0046) {
		tmp = (B * 0.16666666666666666) + ((x * (B * 0.3333333333333333)) + ((1.0 - x) / B));
	} else {
		tmp = x * (((1.0 / x) + -1.0) / Math.sin(B));
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if B <= 0.0046:
		tmp = (B * 0.16666666666666666) + ((x * (B * 0.3333333333333333)) + ((1.0 - x) / B))
	else:
		tmp = x * (((1.0 / x) + -1.0) / math.sin(B))
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (B <= 0.0046)
		tmp = Float64(Float64(B * 0.16666666666666666) + Float64(Float64(x * Float64(B * 0.3333333333333333)) + Float64(Float64(1.0 - x) / B)));
	else
		tmp = Float64(x * Float64(Float64(Float64(1.0 / x) + -1.0) / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (B <= 0.0046)
		tmp = (B * 0.16666666666666666) + ((x * (B * 0.3333333333333333)) + ((1.0 - x) / B));
	else
		tmp = x * (((1.0 / x) + -1.0) / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[B, 0.0046], N[(N[(B * 0.16666666666666666), $MachinePrecision] + N[(N[(x * N[(B * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 0.0045999999999999999

   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 65.2%

      \[\leadsto \color{blue}{\frac{\left(1 + {B}^{2} \cdot \left(0.16666666666666666 + 0.3333333333333333 \cdot x\right)\right) - x}{B}} \]

   4. Taylor expanded in x around 0 65.2%

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

   5. Taylor expanded in x around inf 46.0%

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

      1. associate--l+ 46.0%

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

      2. *-commutative 46.0%

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

      3. distribute-lft-in 46.0%

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

      4. *-commutative 46.0%

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

      5. associate-/r* 46.0%

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

      6. div-sub 46.0%

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

      7. associate-/l* 65.2%

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

      8. sub-neg 65.2%

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

      9. metadata-eval 65.2%

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

      10. distribute-rgt-in 65.2%

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

      11. lft-mult-inverse 65.3%

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

      12. neg-mul-1 65.3%

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

      13. sub-neg 65.3%

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

   7. Simplified 65.3%

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

   if 0.0045999999999999999 < B

   1. Initial program 99.5%

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

      1. distribute-lft-neg-in 99.5%

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

      2. +-commutative 99.5%

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

      3. *-commutative 99.5%

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

      4. remove-double-neg 99.5%

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

      5. distribute-frac-neg2 99.5%

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

      6. tan-neg 99.5%

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

      7. cancel-sign-sub-inv 99.5%

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

      8. associate-*l/ 99.6%

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

      9. *-lft-identity 99.6%

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

      10. tan-neg 99.6%

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

      11. distribute-neg-frac2 99.6%

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

      12. distribute-neg-frac 99.6%

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

      13. remove-double-neg 99.6%

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

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. div-inv 99.5%

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

      2. *-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in x around inf 99.5%

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

      1. associate-/r* 99.5%

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

      2. div-sub 99.5%

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

   9. Simplified 99.5%

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

   10. Taylor expanded in B around 0 41.4%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 0.0046:\\ \;\;\;\;B \cdot 0.16666666666666666 + \left(x \cdot \left(B \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{1}{x} + -1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.068:\\ \;\;\;\;B \cdot 0.16666666666666666 + \left(x \cdot \left(B \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{1}{x}}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B 0.068)
   (+
    (* B 0.16666666666666666)
    (+ (* x (* B 0.3333333333333333)) (/ (- 1.0 x) B)))
   (* x (/ (/ 1.0 x) (sin B)))))

C

double code(double B, double x) {
	double tmp;
	if (B <= 0.068) {
		tmp = (B * 0.16666666666666666) + ((x * (B * 0.3333333333333333)) + ((1.0 - x) / B));
	} else {
		tmp = x * ((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 (b <= 0.068d0) then
        tmp = (b * 0.16666666666666666d0) + ((x * (b * 0.3333333333333333d0)) + ((1.0d0 - x) / b))
    else
        tmp = x * ((1.0d0 / x) / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (B <= 0.068) {
		tmp = (B * 0.16666666666666666) + ((x * (B * 0.3333333333333333)) + ((1.0 - x) / B));
	} else {
		tmp = x * ((1.0 / x) / Math.sin(B));
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if B <= 0.068:
		tmp = (B * 0.16666666666666666) + ((x * (B * 0.3333333333333333)) + ((1.0 - x) / B))
	else:
		tmp = x * ((1.0 / x) / math.sin(B))
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (B <= 0.068)
		tmp = Float64(Float64(B * 0.16666666666666666) + Float64(Float64(x * Float64(B * 0.3333333333333333)) + Float64(Float64(1.0 - x) / B)));
	else
		tmp = Float64(x * Float64(Float64(1.0 / x) / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (B <= 0.068)
		tmp = (B * 0.16666666666666666) + ((x * (B * 0.3333333333333333)) + ((1.0 - x) / B));
	else
		tmp = x * ((1.0 / x) / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[B, 0.068], N[(N[(B * 0.16666666666666666), $MachinePrecision] + N[(N[(x * N[(B * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(1.0 / x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 0.068000000000000005

   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 65.2%

      \[\leadsto \color{blue}{\frac{\left(1 + {B}^{2} \cdot \left(0.16666666666666666 + 0.3333333333333333 \cdot x\right)\right) - x}{B}} \]

   4. Taylor expanded in x around 0 65.2%

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

   5. Taylor expanded in x around inf 46.0%

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

      1. associate--l+ 46.0%

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

      2. *-commutative 46.0%

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

      3. distribute-lft-in 46.0%

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

      4. *-commutative 46.0%

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

      5. associate-/r* 46.0%

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

      6. div-sub 46.0%

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

      7. associate-/l* 65.2%

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

      8. sub-neg 65.2%

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

      9. metadata-eval 65.2%

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

      10. distribute-rgt-in 65.2%

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

      11. lft-mult-inverse 65.3%

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

      12. neg-mul-1 65.3%

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

      13. sub-neg 65.3%

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

   7. Simplified 65.3%

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

   if 0.068000000000000005 < B

   1. Initial program 99.5%

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

      1. distribute-lft-neg-in 99.5%

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

      2. +-commutative 99.5%

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

      3. *-commutative 99.5%

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

      4. remove-double-neg 99.5%

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

      5. distribute-frac-neg2 99.5%

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

      6. tan-neg 99.5%

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

      7. cancel-sign-sub-inv 99.5%

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

      8. associate-*l/ 99.6%

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

      9. *-lft-identity 99.6%

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

      10. tan-neg 99.6%

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

      11. distribute-neg-frac2 99.6%

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

      12. distribute-neg-frac 99.6%

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

      13. remove-double-neg 99.6%

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

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. div-inv 99.5%

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

      2. *-commutative 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in x around inf 99.5%

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

      1. associate-/r* 99.5%

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

      2. div-sub 99.5%

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

   9. Simplified 99.5%

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

   10. Taylor expanded in x around 0 35.8%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 0.068:\\ \;\;\;\;B \cdot 0.16666666666666666 + \left(x \cdot \left(B \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{1}{x}}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.112:\\ \;\;\;\;B \cdot 0.16666666666666666 + \left(x \cdot \left(B \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B 0.112)
   (+
    (* B 0.16666666666666666)
    (+ (* x (* B 0.3333333333333333)) (/ (- 1.0 x) B)))
   (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if (B <= 0.112) {
		tmp = (B * 0.16666666666666666) + ((x * (B * 0.3333333333333333)) + ((1.0 - x) / B));
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (b <= 0.112d0) then
        tmp = (b * 0.16666666666666666d0) + ((x * (b * 0.3333333333333333d0)) + ((1.0d0 - x) / b))
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (B <= 0.112) {
		tmp = (B * 0.16666666666666666) + ((x * (B * 0.3333333333333333)) + ((1.0 - x) / B));
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if B <= 0.112:
		tmp = (B * 0.16666666666666666) + ((x * (B * 0.3333333333333333)) + ((1.0 - x) / B))
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (B <= 0.112)
		tmp = Float64(Float64(B * 0.16666666666666666) + Float64(Float64(x * Float64(B * 0.3333333333333333)) + Float64(Float64(1.0 - x) / B)));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (B <= 0.112)
		tmp = (B * 0.16666666666666666) + ((x * (B * 0.3333333333333333)) + ((1.0 - x) / B));
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[B, 0.112], N[(N[(B * 0.16666666666666666), $MachinePrecision] + N[(N[(x * N[(B * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 0.112000000000000002

   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 65.2%

      \[\leadsto \color{blue}{\frac{\left(1 + {B}^{2} \cdot \left(0.16666666666666666 + 0.3333333333333333 \cdot x\right)\right) - x}{B}} \]

   4. Taylor expanded in x around 0 65.2%

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

   5. Taylor expanded in x around inf 46.0%

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

      1. associate--l+ 46.0%

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

      2. *-commutative 46.0%

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

      3. distribute-lft-in 46.0%

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

      4. *-commutative 46.0%

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

      5. associate-/r* 46.0%

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

      6. div-sub 46.0%

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

      7. associate-/l* 65.2%

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

      8. sub-neg 65.2%

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

      9. metadata-eval 65.2%

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

      10. distribute-rgt-in 65.2%

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

      11. lft-mult-inverse 65.3%

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

      12. neg-mul-1 65.3%

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

      13. sub-neg 65.3%

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

   7. Simplified 65.3%

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

   if 0.112000000000000002 < 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 35.8%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 0.112:\\ \;\;\;\;B \cdot 0.16666666666666666 + \left(x \cdot \left(B \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
2. 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. associate-*l/ 99.8%

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

   9. *-lft-identity 99.8%

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

   10. tan-neg 99.8%

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

   11. distribute-neg-frac2 99.8%

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

   12. distribute-neg-frac 99.8%

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

   13. 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. Step-by-step derivation

   1. div-inv 99.7%

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

   2. *-commutative 99.7%

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in B around 0 70.0%

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

8. Final simplification 70.0%

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

Alternative 8: 51.7% accurate, 14.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = (b * 0.16666666666666666d0) + ((x * (b * 0.3333333333333333d0)) + ((1.0d0 - x) / b))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in B around 0 49.7%

   \[\leadsto \color{blue}{\frac{\left(1 + {B}^{2} \cdot \left(0.16666666666666666 + 0.3333333333333333 \cdot x\right)\right) - x}{B}} \]

4. Taylor expanded in x around 0 49.8%

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

5. Taylor expanded in x around inf 35.4%

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

   1. associate--l+ 35.4%

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

   2. *-commutative 35.4%

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

   3. distribute-lft-in 35.4%

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

   4. *-commutative 35.4%

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

   5. associate-/r* 35.4%

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

   6. div-sub 35.4%

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

   7. associate-/l* 49.7%

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

   8. sub-neg 49.7%

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

   9. metadata-eval 49.7%

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

   10. distribute-rgt-in 49.7%

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

   11. lft-mult-inverse 49.8%

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

   12. neg-mul-1 49.8%

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

   13. sub-neg 49.8%

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

7. Simplified 49.8%

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

8. Final simplification 49.8%

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

Alternative 9: 50.0% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-16} \lor \neg \left(x \leq 0.00115\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -2.3e-16) (not (<= x 0.00115))) (/ (- x) B) (/ 1.0 B)))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -2.3e-16) || !(x <= 0.00115)) {
		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 <= (-2.3d-16)) .or. (.not. (x <= 0.00115d0))) 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 <= -2.3e-16) || !(x <= 0.00115)) {
		tmp = -x / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -2.3e-16) or not (x <= 0.00115):
		tmp = -x / B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -2.3e-16) || !(x <= 0.00115))
		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 <= -2.3e-16) || ~((x <= 0.00115)))
		tmp = -x / B;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -2.3e-16], N[Not[LessEqual[x, 0.00115]], $MachinePrecision]], N[((-x) / B), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.2999999999999999e-16 or 0.00115 < 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 41.2%

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

   4. Taylor expanded in x around inf 40.4%

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

      1. neg-mul-1 40.4%

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

      2. distribute-neg-frac2 40.4%

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

   6. Simplified 40.4%

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

   if -2.2999999999999999e-16 < x < 0.00115

   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 58.5%

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

   4. Taylor expanded in x around 0 58.0%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-16} \lor \neg \left(x \leq 0.00115\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 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 49.5%

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

4. Final simplification 49.5%

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

Alternative 11: 27.0% 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 49.5%

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

4. Taylor expanded in x around 0 29.4%

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

5. Final simplification 29.4%

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

Reproduce

herbie shell --seed 2024115 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))