VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 7.6s

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. +-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/ 99.7%

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

   4. *-rgt-identity 99.7%

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

3. Simplified 99.7%

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

4. Final simplification 99.7%

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

Alternative 2: 86.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{+64} \lor \neg \left(x \leq 1.8 \cdot 10^{+20}\right):\\ \;\;\;\;\left(B \cdot 0.16666666666666666 + \frac{1}{B}\right) - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.25e+64) (not (<= x 1.8e+20)))
   (- (+ (* B 0.16666666666666666) (/ 1.0 B)) (/ x (tan B)))
   (- (/ 1.0 (sin B)) (/ x B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.25e+64) || !(x <= 1.8e+20)) {
		tmp = ((B * 0.16666666666666666) + (1.0 / B)) - (x / tan(B));
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.25d+64)) .or. (.not. (x <= 1.8d+20))) then
        tmp = ((b * 0.16666666666666666d0) + (1.0d0 / b)) - (x / tan(b))
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -1.25e+64) || !(x <= 1.8e+20)) {
		tmp = ((B * 0.16666666666666666) + (1.0 / B)) - (x / Math.tan(B));
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -1.25e+64) or not (x <= 1.8e+20):
		tmp = ((B * 0.16666666666666666) + (1.0 / B)) - (x / math.tan(B))
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -1.25e+64) || !(x <= 1.8e+20))
		tmp = Float64(Float64(Float64(B * 0.16666666666666666) + Float64(1.0 / B)) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -1.25e+64) || ~((x <= 1.8e+20)))
		tmp = ((B * 0.16666666666666666) + (1.0 / B)) - (x / tan(B));
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -1.25e+64], N[Not[LessEqual[x, 1.8e+20]], $MachinePrecision]], N[(N[(N[(B * 0.16666666666666666), $MachinePrecision] + N[(1.0 / B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $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 < -1.25e64 or 1.8e20 < 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/ 99.7%

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

      4. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in B around 0 76.8%

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

   if -1.25e64 < x < 1.8e20

   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/ 99.7%

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

      4. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in B around 0 95.0%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{+64} \lor \neg \left(x \leq 1.8 \cdot 10^{+20}\right):\\ \;\;\;\;\left(B \cdot 0.16666666666666666 + \frac{1}{B}\right) - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]

Alternative 3: 86.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := B \cdot 0.16666666666666666 + \frac{1}{B}\\ \mathbf{if}\;x \leq -1.05 \cdot 10^{+64}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + t_0\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t_0 - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (+ (* B 0.16666666666666666) (/ 1.0 B))))
   (if (<= x -1.05e+64)
     (+ (/ -1.0 (/ (tan B) x)) t_0)
     (if (<= x 7.5e+19) (- (/ 1.0 (sin B)) (/ x B)) (- t_0 (/ x (tan B)))))))

C

double code(double B, double x) {
	double t_0 = (B * 0.16666666666666666) + (1.0 / B);
	double tmp;
	if (x <= -1.05e+64) {
		tmp = (-1.0 / (tan(B) / x)) + t_0;
	} else if (x <= 7.5e+19) {
		tmp = (1.0 / sin(B)) - (x / B);
	} else {
		tmp = t_0 - (x / tan(B));
	}
	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 = (b * 0.16666666666666666d0) + (1.0d0 / b)
    if (x <= (-1.05d+64)) then
        tmp = ((-1.0d0) / (tan(b) / x)) + t_0
    else if (x <= 7.5d+19) then
        tmp = (1.0d0 / sin(b)) - (x / b)
    else
        tmp = t_0 - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double t_0 = (B * 0.16666666666666666) + (1.0 / B);
	double tmp;
	if (x <= -1.05e+64) {
		tmp = (-1.0 / (Math.tan(B) / x)) + t_0;
	} else if (x <= 7.5e+19) {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	} else {
		tmp = t_0 - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = (B * 0.16666666666666666) + (1.0 / B)
	tmp = 0
	if x <= -1.05e+64:
		tmp = (-1.0 / (math.tan(B) / x)) + t_0
	elif x <= 7.5e+19:
		tmp = (1.0 / math.sin(B)) - (x / B)
	else:
		tmp = t_0 - (x / math.tan(B))
	return tmp

Julia

function code(B, x)
	t_0 = Float64(Float64(B * 0.16666666666666666) + Float64(1.0 / B))
	tmp = 0.0
	if (x <= -1.05e+64)
		tmp = Float64(Float64(-1.0 / Float64(tan(B) / x)) + t_0);
	elseif (x <= 7.5e+19)
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	else
		tmp = Float64(t_0 - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	t_0 = (B * 0.16666666666666666) + (1.0 / B);
	tmp = 0.0;
	if (x <= -1.05e+64)
		tmp = (-1.0 / (tan(B) / x)) + t_0;
	elseif (x <= 7.5e+19)
		tmp = (1.0 / sin(B)) - (x / B);
	else
		tmp = t_0 - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := Block[{t$95$0 = N[(N[(B * 0.16666666666666666), $MachinePrecision] + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.05e+64], N[(N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[x, 7.5e+19], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05e64

   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%

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

   3. Simplified 99.5%

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

      1. un-div-inv 99.6%

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

      2. neg-mul-1 99.6%

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

      3. associate-/l* 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in B around 0 81.9%

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

   if -1.05e64 < x < 7.5e19

   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/ 99.7%

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

      4. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in B around 0 95.0%

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

   if 7.5e19 < x

   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/ 99.9%

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

      4. *-rgt-identity 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in B around 0 73.1%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+64}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \left(B \cdot 0.16666666666666666 + \frac{1}{B}\right)\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(B \cdot 0.16666666666666666 + \frac{1}{B}\right) - \frac{x}{\tan B}\\ \end{array} \]

Alternative 4: 75.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[B_, x_] := N[(N[(N[(B - N[(B * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $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. +-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/ 99.7%

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

   4. *-rgt-identity 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in B around 0 78.9%

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

   1. frac-sub 61.8%

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

   2. associate-/r* 78.5%

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

   3. *-un-lft-identity 78.5%

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

   4. *-commutative 78.5%

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

6. Applied egg-rr 78.5%

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

7. Taylor expanded in B around 0 79.5%

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

8. Final simplification 79.5%

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

Alternative 5: 73.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 - x}{B}\\ \mathbf{if}\;x \leq -58000000:\\ \;\;\;\;B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + t_0\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{-13}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot 0.16666666666666666 + t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (let* ((t_0 (/ (- 1.0 x) B)))
   (if (<= x -58000000.0)
     (+ (* B (+ 0.16666666666666666 (* x 0.3333333333333333))) t_0)
     (if (<= x 7.8e-13) (/ 1.0 (sin B)) (+ (* B 0.16666666666666666) t_0)))))

C

double code(double B, double x) {
	double t_0 = (1.0 - x) / B;
	double tmp;
	if (x <= -58000000.0) {
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + t_0;
	} else if (x <= 7.8e-13) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = (B * 0.16666666666666666) + t_0;
	}
	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 <= (-58000000.0d0)) then
        tmp = (b * (0.16666666666666666d0 + (x * 0.3333333333333333d0))) + t_0
    else if (x <= 7.8d-13) then
        tmp = 1.0d0 / sin(b)
    else
        tmp = (b * 0.16666666666666666d0) + t_0
    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 <= -58000000.0) {
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + t_0;
	} else if (x <= 7.8e-13) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = (B * 0.16666666666666666) + t_0;
	}
	return tmp;
}

Python

def code(B, x):
	t_0 = (1.0 - x) / B
	tmp = 0
	if x <= -58000000.0:
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + t_0
	elif x <= 7.8e-13:
		tmp = 1.0 / math.sin(B)
	else:
		tmp = (B * 0.16666666666666666) + t_0
	return tmp

Julia

function code(B, x)
	t_0 = Float64(Float64(1.0 - x) / B)
	tmp = 0.0
	if (x <= -58000000.0)
		tmp = Float64(Float64(B * Float64(0.16666666666666666 + Float64(x * 0.3333333333333333))) + t_0);
	elseif (x <= 7.8e-13)
		tmp = Float64(1.0 / sin(B));
	else
		tmp = Float64(Float64(B * 0.16666666666666666) + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	t_0 = (1.0 - x) / B;
	tmp = 0.0;
	if (x <= -58000000.0)
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + t_0;
	elseif (x <= 7.8e-13)
		tmp = 1.0 / sin(B);
	else
		tmp = (B * 0.16666666666666666) + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := Block[{t$95$0 = N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]}, If[LessEqual[x, -58000000.0], N[(N[(B * N[(0.16666666666666666 + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[x, 7.8e-13], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(B * 0.16666666666666666), $MachinePrecision] + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.8e7

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. unsub-neg 99.5%

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

      3. associate-*r/ 99.7%

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

      4. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in B around 0 55.6%

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

      1. *-commutative 55.6%

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

   6. Simplified 55.6%

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

   7. Taylor expanded in B around 0 55.6%

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

      1. associate--l+ 55.6%

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

      2. *-commutative 55.6%

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

      3. sub-neg 55.6%

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

      4. *-commutative 55.6%

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

      5. distribute-rgt-neg-in 55.6%

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

      6. metadata-eval 55.6%

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

      7. div-sub 55.6%

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

   9. Simplified 55.6%

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

   if -5.8e7 < x < 7.80000000000000009e-13

   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/ 99.7%

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

      4. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in B around 0 98.4%

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

   5. Taylor expanded in B around inf 98.3%

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

   if 7.80000000000000009e-13 < x

   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/ 99.9%

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

      4. *-rgt-identity 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in B around 0 52.2%

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

   5. Taylor expanded in B around 0 52.3%

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

      1. associate--l+ 52.3%

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

      2. *-commutative 52.3%

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

      3. div-sub 52.3%

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

   7. Simplified 52.3%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -58000000:\\ \;\;\;\;B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{-13}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot 0.16666666666666666 + \frac{1 - x}{B}\\ \end{array} \]

Alternative 6: 74.5% 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. +-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/ 99.7%

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

   4. *-rgt-identity 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in B around 0 78.9%

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

5. Final simplification 78.9%

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

Alternative 7: 51.5% accurate, 16.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B} \]
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/ 99.7%

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

   4. *-rgt-identity 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in B around 0 62.4%

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

   1. *-commutative 62.4%

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

6. Simplified 62.4%

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

7. Taylor expanded in B around 0 53.9%

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

   1. associate--l+ 53.9%

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

   2. *-commutative 53.9%

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

   3. sub-neg 53.9%

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

   4. *-commutative 53.9%

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

   5. distribute-rgt-neg-in 53.9%

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

   6. metadata-eval 53.9%

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

   7. div-sub 53.9%

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

9. Simplified 53.9%

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

10. Final simplification 53.9%

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

Alternative 8: 51.5% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ B \cdot 0.16666666666666666 + \frac{1 - x}{B} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[B_, x_] := N[(N[(B * 0.16666666666666666), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / 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. +-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/ 99.7%

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

   4. *-rgt-identity 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in B around 0 78.9%

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

5. Taylor expanded in B around 0 53.8%

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

   1. associate--l+ 53.8%

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

   2. *-commutative 53.8%

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

   3. div-sub 53.8%

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

7. Simplified 53.8%

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

8. Final simplification 53.8%

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

Alternative 9: 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. +-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/ 99.7%

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

   4. *-rgt-identity 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in B around 0 78.9%

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

5. Taylor expanded in B around 0 53.6%

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

6. Final simplification 53.6%

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

Alternative 10: 26.1% accurate, 52.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[B_, x_] := N[((-x) / 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. +-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/ 99.7%

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

   4. *-rgt-identity 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in B around 0 78.9%

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

5. Taylor expanded in x around inf 23.8%

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

   1. mul-1-neg 23.8%

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

   2. distribute-frac-neg 23.8%

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

7. Simplified 23.8%

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

8. Final simplification 23.8%

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

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/ 99.7%

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

   4. *-rgt-identity 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in B around 0 78.9%

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

5. Taylor expanded in B around 0 53.8%

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

   1. associate--l+ 53.8%

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

   2. *-commutative 53.8%

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

   3. div-sub 53.8%

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

7. Simplified 53.8%

   \[\leadsto \color{blue}{B \cdot 0.16666666666666666 + \frac{1 - x}{B}} \]
8. Step-by-step derivation

   1. clear-num 53.8%

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

   2. inv-pow 53.8%

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

   3. sub-neg 53.8%

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

   4. add-sqr-sqrt 24.6%

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

   5. sqrt-unprod 42.2%

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

   6. sqr-neg 42.2%

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

   7. sqrt-unprod 18.0%

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

   8. add-sqr-sqrt 32.3%

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

9. Applied egg-rr 32.3%

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

    1. unpow-1 32.3%

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

    2. +-commutative 32.3%

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

11. Simplified 32.3%

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

12. Taylor expanded in B around inf 3.1%

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

    1. *-commutative 3.1%

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

14. Simplified 3.1%

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

15. Final simplification 3.1%

    \[\leadsto B \cdot 0.16666666666666666 \]

Reproduce

herbie shell --seed 2023200 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))