VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 9.3s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. +-commutative 99.7%

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

   2. unsub-neg 99.7%

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

   3. associate-*r/ 99.8%

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

   4. *-rgt-identity 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 98.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-9} \lor \neg \left(x \leq 1.3 \cdot 10^{-9}\right):\\ \;\;\;\;x \cdot \frac{\frac{1}{x} + -1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.45e-9) (not (<= x 1.3e-9)))
   (* x (/ (+ (/ 1.0 x) -1.0) (tan B)))
   (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.45e-9) || !(x <= 1.3e-9)) {
		tmp = x * (((1.0 / x) + -1.0) / tan(B));
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.45d-9)) .or. (.not. (x <= 1.3d-9))) then
        tmp = x * (((1.0d0 / x) + (-1.0d0)) / tan(b))
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -1.45e-9) || !(x <= 1.3e-9)) {
		tmp = x * (((1.0 / x) + -1.0) / Math.tan(B));
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -1.45e-9) or not (x <= 1.3e-9):
		tmp = x * (((1.0 / x) + -1.0) / math.tan(B))
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -1.45e-9) || !(x <= 1.3e-9))
		tmp = Float64(x * Float64(Float64(Float64(1.0 / x) + -1.0) / tan(B)));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -1.45e-9) || ~((x <= 1.3e-9)))
		tmp = x * (((1.0 / x) + -1.0) / tan(B));
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -1.45e-9], N[Not[LessEqual[x, 1.3e-9]], $MachinePrecision]], N[(x * N[(N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.44999999999999996e-9 or 1.3000000000000001e-9 < x

   1. Initial program 99.7%

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

      1. distribute-lft-neg-in 99.7%

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

   3. Simplified 99.7%

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

      1. +-commutative 99.7%

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

      2. cancel-sign-sub-inv 99.7%

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

      3. div-inv 99.8%

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

      4. clear-num 99.7%

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

      5. frac-sub 88.8%

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

      6. *-un-lft-identity 88.8%

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

      7. *-commutative 88.8%

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

      8. *-un-lft-identity 88.8%

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

   5. Applied egg-rr 88.8%

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

      1. associate-/r* 99.7%

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

      2. associate-/r/ 99.7%

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

      3. div-sub 99.7%

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

      4. *-inverses 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in B around 0 97.6%

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

   if -1.44999999999999996e-9 < x < 1.3000000000000001e-9

   1. Initial program 99.8%

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

      1. distribute-lft-neg-in 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 99.4%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-9} \lor \neg \left(x \leq 1.3 \cdot 10^{-9}\right):\\ \;\;\;\;x \cdot \frac{\frac{1}{x} + -1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 3: 98.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{-8} \lor \neg \left(x \leq 1.4 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{x \cdot \left(\frac{1}{x} + -1\right)}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -6.6e-8) (not (<= x 1.4e-9)))
   (/ (* x (+ (/ 1.0 x) -1.0)) (tan B))
   (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -6.6e-8) || !(x <= 1.4e-9)) {
		tmp = (x * ((1.0 / x) + -1.0)) / tan(B);
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-6.6d-8)) .or. (.not. (x <= 1.4d-9))) then
        tmp = (x * ((1.0d0 / x) + (-1.0d0))) / tan(b)
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -6.6e-8) || !(x <= 1.4e-9)) {
		tmp = (x * ((1.0 / x) + -1.0)) / Math.tan(B);
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -6.6e-8) or not (x <= 1.4e-9):
		tmp = (x * ((1.0 / x) + -1.0)) / math.tan(B)
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -6.6e-8) || !(x <= 1.4e-9))
		tmp = Float64(Float64(x * Float64(Float64(1.0 / x) + -1.0)) / tan(B));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -6.6e-8) || ~((x <= 1.4e-9)))
		tmp = (x * ((1.0 / x) + -1.0)) / tan(B);
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -6.6e-8], N[Not[LessEqual[x, 1.4e-9]], $MachinePrecision]], N[(N[(x * N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.59999999999999954e-8 or 1.39999999999999992e-9 < x

   1. Initial program 99.7%

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

      1. distribute-lft-neg-in 99.7%

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

   3. Simplified 99.7%

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

      1. +-commutative 99.7%

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

      2. cancel-sign-sub-inv 99.7%

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

      3. div-inv 99.8%

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

      4. clear-num 99.7%

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

      5. frac-sub 88.8%

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

      6. *-un-lft-identity 88.8%

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

      7. *-commutative 88.8%

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

      8. *-un-lft-identity 88.8%

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

   5. Applied egg-rr 88.8%

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

      1. associate-/r* 99.7%

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

      2. associate-/r/ 99.7%

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

      3. div-sub 99.7%

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

      4. *-inverses 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in B around 0 97.6%

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

      1. associate-*l/ 97.7%

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

      2. sub-neg 97.7%

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

      3. metadata-eval 97.7%

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

   10. Applied egg-rr 97.7%

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

   if -6.59999999999999954e-8 < x < 1.39999999999999992e-9

   1. Initial program 99.8%

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

      1. distribute-lft-neg-in 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 99.4%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{-8} \lor \neg \left(x \leq 1.4 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{x \cdot \left(\frac{1}{x} + -1\right)}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 4: 98.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \lor \neg \left(x \leq 4.5\right):\\ \;\;\;\;\frac{x \cdot \left(\frac{1}{x} + -1\right)}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} + x \cdot \frac{-1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -2.0) (not (<= x 4.5)))
   (/ (* x (+ (/ 1.0 x) -1.0)) (tan B))
   (+ (/ 1.0 (sin B)) (* x (/ -1.0 B)))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -2.0) || !(x <= 4.5)) {
		tmp = (x * ((1.0 / x) + -1.0)) / tan(B);
	} else {
		tmp = (1.0 / sin(B)) + (x * (-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 <= (-2.0d0)) .or. (.not. (x <= 4.5d0))) then
        tmp = (x * ((1.0d0 / x) + (-1.0d0))) / tan(b)
    else
        tmp = (1.0d0 / sin(b)) + (x * ((-1.0d0) / b))
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -2.0) || !(x <= 4.5)) {
		tmp = (x * ((1.0 / x) + -1.0)) / Math.tan(B);
	} else {
		tmp = (1.0 / Math.sin(B)) + (x * (-1.0 / B));
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -2.0) or not (x <= 4.5):
		tmp = (x * ((1.0 / x) + -1.0)) / math.tan(B)
	else:
		tmp = (1.0 / math.sin(B)) + (x * (-1.0 / B))
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -2.0) || !(x <= 4.5))
		tmp = Float64(Float64(x * Float64(Float64(1.0 / x) + -1.0)) / tan(B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) + Float64(x * Float64(-1.0 / B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -2.0) || ~((x <= 4.5)))
		tmp = (x * ((1.0 / x) + -1.0)) / tan(B);
	else
		tmp = (1.0 / sin(B)) + (x * (-1.0 / B));
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -2.0], N[Not[LessEqual[x, 4.5]], $MachinePrecision]], N[(N[(x * N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[(x * N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2 or 4.5 < x

   1. Initial program 99.7%

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

      1. distribute-lft-neg-in 99.7%

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

   3. Simplified 99.7%

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

      1. +-commutative 99.7%

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

      2. cancel-sign-sub-inv 99.7%

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

      3. div-inv 99.8%

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

      4. clear-num 99.7%

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

      5. frac-sub 89.2%

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

      6. *-un-lft-identity 89.2%

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

      7. *-commutative 89.2%

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

      8. *-un-lft-identity 89.2%

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

   5. Applied egg-rr 89.2%

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

      1. associate-/r* 99.7%

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

      2. associate-/r/ 99.7%

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

      3. div-sub 99.7%

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

      4. *-inverses 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in B around 0 98.3%

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

      1. associate-*l/ 98.4%

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

      2. sub-neg 98.4%

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

      3. metadata-eval 98.4%

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

   10. Applied egg-rr 98.4%

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

   if -2 < x < 4.5

   1. Initial program 99.8%

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

      1. distribute-lft-neg-in 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in B around 0 99.1%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \lor \neg \left(x \leq 4.5\right):\\ \;\;\;\;\frac{x \cdot \left(\frac{1}{x} + -1\right)}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} + x \cdot \frac{-1}{B}\\ \end{array} \]

Alternative 5: 97.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;x \cdot \frac{-1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -2.0) (not (<= x 1.0))) (* x (/ -1.0 (tan B))) (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -2.0) || !(x <= 1.0)) {
		tmp = x * (-1.0 / tan(B));
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-2.0d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = x * ((-1.0d0) / tan(b))
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -2.0) || !(x <= 1.0)) {
		tmp = x * (-1.0 / Math.tan(B));
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -2.0) or not (x <= 1.0):
		tmp = x * (-1.0 / math.tan(B))
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -2.0) || !(x <= 1.0))
		tmp = Float64(x * Float64(-1.0 / tan(B)));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -2.0) || ~((x <= 1.0)))
		tmp = x * (-1.0 / tan(B));
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -2.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2 or 1 < x

   1. Initial program 99.7%

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

      1. distribute-lft-neg-in 99.7%

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

   3. Simplified 99.7%

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

      1. +-commutative 99.7%

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

      2. cancel-sign-sub-inv 99.7%

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

      3. div-inv 99.8%

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

      4. clear-num 99.7%

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

      5. frac-sub 89.2%

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

      6. *-un-lft-identity 89.2%

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

      7. *-commutative 89.2%

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

      8. *-un-lft-identity 89.2%

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

   5. Applied egg-rr 89.2%

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

      1. associate-/r* 99.7%

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

      2. associate-/r/ 99.7%

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

      3. div-sub 99.7%

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

      4. *-inverses 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around inf 98.0%

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

      1. associate-*r/ 98.0%

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

      2. mul-1-neg 98.0%

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

   10. Simplified 98.0%

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

       1. distribute-frac-neg 98.0%

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

       2. neg-sub0 98.0%

          \[\leadsto \color{blue}{\left(0 - \frac{\cos B}{\sin B}\right)} \cdot x \]

       3. clear-num 98.0%

          \[\leadsto \left(0 - \color{blue}{\frac{1}{\frac{\sin B}{\cos B}}}\right) \cdot x \]

       4. quot-tan 98.0%

          \[\leadsto \left(0 - \frac{1}{\color{blue}{\tan B}}\right) \cdot x \]

   12. Applied egg-rr 98.0%

       \[\leadsto \color{blue}{\left(0 - \frac{1}{\tan B}\right)} \cdot x \]
   13. Step-by-step derivation

       1. neg-sub0 98.0%

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

       2. distribute-neg-frac 98.0%

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

       3. metadata-eval 98.0%

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

   14. Simplified 98.0%

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

   if -2 < x < 1

   1. Initial program 99.8%

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

      1. distribute-lft-neg-in 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 97.7%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;x \cdot \frac{-1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 6: 97.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35:\\ \;\;\;\;\frac{-1}{\frac{1}{\frac{x}{\tan B}}}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= x -1.35)
   (/ -1.0 (/ 1.0 (/ x (tan B))))
   (if (<= x 1.0) (/ 1.0 (sin B)) (* x (/ -1.0 (tan B))))))

C

double code(double B, double x) {
	double tmp;
	if (x <= -1.35) {
		tmp = -1.0 / (1.0 / (x / tan(B)));
	} else if (x <= 1.0) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = x * (-1.0 / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.35d0)) then
        tmp = (-1.0d0) / (1.0d0 / (x / tan(b)))
    else if (x <= 1.0d0) then
        tmp = 1.0d0 / sin(b)
    else
        tmp = x * ((-1.0d0) / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (x <= -1.35) {
		tmp = -1.0 / (1.0 / (x / Math.tan(B)));
	} else if (x <= 1.0) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = x * (-1.0 / Math.tan(B));
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if x <= -1.35:
		tmp = -1.0 / (1.0 / (x / math.tan(B)))
	elif x <= 1.0:
		tmp = 1.0 / math.sin(B)
	else:
		tmp = x * (-1.0 / math.tan(B))
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (x <= -1.35)
		tmp = Float64(-1.0 / Float64(1.0 / Float64(x / tan(B))));
	elseif (x <= 1.0)
		tmp = Float64(1.0 / sin(B));
	else
		tmp = Float64(x * Float64(-1.0 / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (x <= -1.35)
		tmp = -1.0 / (1.0 / (x / tan(B)));
	elseif (x <= 1.0)
		tmp = 1.0 / sin(B);
	else
		tmp = x * (-1.0 / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[x, -1.35], N[(-1.0 / N[(1.0 / N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.3500000000000001

   1. Initial program 99.6%

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

      1. distribute-lft-neg-in 99.6%

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

   3. Simplified 99.6%

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

      1. +-commutative 99.6%

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

      2. cancel-sign-sub-inv 99.6%

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

      3. div-inv 99.8%

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

      4. clear-num 99.7%

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

      5. frac-sub 87.7%

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

      6. *-un-lft-identity 87.7%

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

      7. *-commutative 87.7%

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

      8. *-un-lft-identity 87.7%

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

   5. Applied egg-rr 87.7%

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

      1. associate-/r* 99.7%

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

      2. associate-/r/ 99.7%

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

      3. div-sub 99.7%

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

      4. *-inverses 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around inf 96.5%

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

      1. associate-*r/ 96.5%

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

      2. mul-1-neg 96.5%

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

   10. Simplified 96.5%

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

       1. distribute-frac-neg 96.5%

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

       2. distribute-lft-neg-in 96.5%

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

       3. associate-/r/ 96.5%

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

       4. clear-num 96.4%

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

       5. distribute-neg-frac 96.4%

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

       6. metadata-eval 96.4%

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

       7. clear-num 96.5%

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

       8. add-sqr-sqrt 76.2%

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

       9. sqrt-unprod 76.4%

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

       10. sqr-neg 76.4%

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

       11. sqrt-unprod 0.1%

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

       12. add-sqr-sqrt 0.5%

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

       13. associate-/l* 0.5%

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

       14. associate-*l/ 0.5%

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

       15. clear-num 0.5%

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

       16. associate-*l/ 0.5%

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

       17. *-un-lft-identity 0.5%

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

       18. add-sqr-sqrt 0.1%

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

       19. sqrt-unprod 76.5%

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

       20. sqr-neg 76.5%

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

   12. Applied egg-rr 96.5%

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

   if -1.3500000000000001 < x < 1

   1. Initial program 99.8%

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

      1. distribute-lft-neg-in 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 97.7%

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

   if 1 < x

   1. Initial program 99.7%

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

      1. distribute-lft-neg-in 99.7%

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

   3. Simplified 99.7%

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

      1. +-commutative 99.7%

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

      2. cancel-sign-sub-inv 99.7%

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

      3. div-inv 99.8%

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

      4. clear-num 99.8%

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

      5. frac-sub 90.8%

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

      6. *-un-lft-identity 90.8%

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

      7. *-commutative 90.8%

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

      8. *-un-lft-identity 90.8%

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

   5. Applied egg-rr 90.8%

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

      1. associate-/r* 99.8%

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

      2. associate-/r/ 99.7%

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

      3. div-sub 99.7%

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

      4. *-inverses 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around inf 99.7%

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

      1. associate-*r/ 99.7%

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

      2. mul-1-neg 99.7%

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

   10. Simplified 99.7%

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

       1. distribute-frac-neg 99.7%

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

       2. neg-sub0 99.7%

          \[\leadsto \color{blue}{\left(0 - \frac{\cos B}{\sin B}\right)} \cdot x \]

       3. clear-num 99.7%

          \[\leadsto \left(0 - \color{blue}{\frac{1}{\frac{\sin B}{\cos B}}}\right) \cdot x \]

       4. quot-tan 99.7%

          \[\leadsto \left(0 - \frac{1}{\color{blue}{\tan B}}\right) \cdot x \]

   12. Applied egg-rr 99.7%

       \[\leadsto \color{blue}{\left(0 - \frac{1}{\tan B}\right)} \cdot x \]
   13. Step-by-step derivation

       1. neg-sub0 99.7%

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

       2. distribute-neg-frac 99.7%

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

       3. metadata-eval 99.7%

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

   14. Simplified 99.7%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35:\\ \;\;\;\;\frac{-1}{\frac{1}{\frac{x}{\tan B}}}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B}\\ \end{array} \]

Alternative 7: 74.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq -3.4 \cdot 10^{+21} \lor \neg \left(B \leq 0.00082\right):\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{B} + B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right)\right) - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= B -3.4e+21) (not (<= B 0.00082)))
   (/ 1.0 (sin B))
   (-
    (+ (/ 1.0 B) (* B (+ 0.16666666666666666 (* x 0.3333333333333333))))
    (/ x B))))

C

double code(double B, double x) {
	double tmp;
	if ((B <= -3.4e+21) || !(B <= 0.00082)) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = ((1.0 / B) + (B * (0.16666666666666666 + (x * 0.3333333333333333)))) - (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 ((b <= (-3.4d+21)) .or. (.not. (b <= 0.00082d0))) then
        tmp = 1.0d0 / sin(b)
    else
        tmp = ((1.0d0 / b) + (b * (0.16666666666666666d0 + (x * 0.3333333333333333d0)))) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((B <= -3.4e+21) || !(B <= 0.00082)) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = ((1.0 / B) + (B * (0.16666666666666666 + (x * 0.3333333333333333)))) - (x / B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (B <= -3.4e+21) or not (B <= 0.00082):
		tmp = 1.0 / math.sin(B)
	else:
		tmp = ((1.0 / B) + (B * (0.16666666666666666 + (x * 0.3333333333333333)))) - (x / B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((B <= -3.4e+21) || !(B <= 0.00082))
		tmp = Float64(1.0 / sin(B));
	else
		tmp = Float64(Float64(Float64(1.0 / B) + Float64(B * Float64(0.16666666666666666 + Float64(x * 0.3333333333333333)))) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((B <= -3.4e+21) || ~((B <= 0.00082)))
		tmp = 1.0 / sin(B);
	else
		tmp = ((1.0 / B) + (B * (0.16666666666666666 + (x * 0.3333333333333333)))) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[B, -3.4e+21], N[Not[LessEqual[B, 0.00082]], $MachinePrecision]], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / B), $MachinePrecision] + N[(B * N[(0.16666666666666666 + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < -3.4e21 or 8.1999999999999998e-4 < 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%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 52.7%

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

   if -3.4e21 < B < 8.1999999999999998e-4

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. unsub-neg 99.9%

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

      3. associate-*r/ 100.0%

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

      4. *-rgt-identity 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in B around 0 95.4%

      \[\leadsto \color{blue}{\left(\left(0.16666666666666666 + 0.3333333333333333 \cdot x\right) \cdot B + \frac{1}{B}\right) - \frac{x}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -3.4 \cdot 10^{+21} \lor \neg \left(B \leq 0.00082\right):\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{B} + B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right)\right) - \frac{x}{B}\\ \end{array} \]

Alternative 8: 50.8% accurate, 14.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[B_, x_] := N[(N[(N[(1.0 / B), $MachinePrecision] + N[(B * N[(0.16666666666666666 + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $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/ 99.8%

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

   4. *-rgt-identity 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in B around 0 54.0%

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

5. Final simplification 54.0%

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

Alternative 9: 50.8% 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. distribute-lft-neg-in 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in B around 0 54.0%

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

   1. +-commutative 54.0%

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

   2. mul-1-neg 54.0%

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

   3. sub-neg 54.0%

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

   4. associate--l+ 54.0%

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

   5. *-commutative 54.0%

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

   6. *-commutative 54.0%

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

   7. div-sub 54.0%

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

6. Simplified 54.0%

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

7. Final simplification 54.0%

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

Alternative 10: 50.7% accurate, 19.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[B_, x_] := N[(N[(N[(1.0 / B), $MachinePrecision] + N[(B * 0.16666666666666666), $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/ 99.8%

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

   4. *-rgt-identity 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in B around 0 67.5%

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

5. Taylor expanded in B around 0 53.9%

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

6. Final simplification 53.9%

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

Alternative 11: 48.7% accurate, 25.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.00112 \lor \neg \left(x \leq 5.2 \cdot 10^{+22}\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -0.00112) (not (<= x 5.2e+22))) (/ (- x) B) (/ 1.0 B)))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -0.00112) || !(x <= 5.2e+22)) {
		tmp = -x / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-0.00112d0)) .or. (.not. (x <= 5.2d+22))) then
        tmp = -x / b
    else
        tmp = 1.0d0 / b
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -0.00112) || !(x <= 5.2e+22)) {
		tmp = -x / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -0.00112) or not (x <= 5.2e+22):
		tmp = -x / B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -0.00112) || !(x <= 5.2e+22))
		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 <= -0.00112) || ~((x <= 5.2e+22)))
		tmp = -x / B;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -0.00112], N[Not[LessEqual[x, 5.2e+22]], $MachinePrecision]], N[((-x) / B), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0011199999999999999 or 5.2e22 < x

   1. Initial program 99.7%

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

      1. distribute-lft-neg-in 99.7%

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

   3. Simplified 99.7%

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

      1. +-commutative 99.7%

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

      2. cancel-sign-sub-inv 99.7%

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

      3. div-inv 99.8%

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

      4. clear-num 99.7%

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

      5. frac-sub 89.1%

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

      6. *-un-lft-identity 89.1%

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

      7. *-commutative 89.1%

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

      8. *-un-lft-identity 89.1%

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

   5. Applied egg-rr 89.1%

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

      1. associate-/r* 99.7%

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

      2. associate-/r/ 99.7%

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

      3. div-sub 99.7%

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

      4. *-inverses 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around inf 97.2%

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

      1. associate-*r/ 97.2%

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

      2. mul-1-neg 97.2%

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

   10. Simplified 97.2%

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

   11. Taylor expanded in B around 0 53.4%

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

       1. associate-*r/ 53.4%

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

       2. neg-mul-1 53.4%

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

   13. Simplified 53.4%

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

   if -0.0011199999999999999 < x < 5.2e22

   1. Initial program 99.8%

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

      1. distribute-lft-neg-in 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in B around 0 53.7%

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

      1. mul-1-neg 53.7%

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

      2. sub-neg 53.7%

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

   6. Simplified 53.7%

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

   7. Taylor expanded in x around 0 52.3%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00112 \lor \neg \left(x \leq 5.2 \cdot 10^{+22}\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]

Alternative 12: 50.5% accurate, 42.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. distribute-lft-neg-in 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in B around 0 53.7%

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

   1. mul-1-neg 53.7%

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

   2. sub-neg 53.7%

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

6. Simplified 53.7%

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

7. Final simplification 53.7%

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

Alternative 13: 26.3% accurate, 70.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. distribute-lft-neg-in 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in B around 0 53.7%

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

   1. mul-1-neg 53.7%

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

   2. sub-neg 53.7%

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

6. Simplified 53.7%

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

7. Taylor expanded in x around 0 27.3%

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

8. Final simplification 27.3%

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

Reproduce

herbie shell --seed 2023193 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))