VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 9.8s

Alternatives: 14

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: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+17} \lor \neg \left(x \leq 14500000\right):\\ \;\;\;\;\frac{-\cos B}{\frac{\sin B}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} + x \cdot \frac{-1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -4e+17) (not (<= x 14500000.0)))
   (/ (- (cos B)) (/ (sin B) x))
   (+ (/ 1.0 (sin B)) (* x (/ -1.0 B)))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -4e+17) || !(x <= 14500000.0)) {
		tmp = -cos(B) / (sin(B) / x);
	} 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 <= (-4d+17)) .or. (.not. (x <= 14500000.0d0))) then
        tmp = -cos(b) / (sin(b) / x)
    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 <= -4e+17) || !(x <= 14500000.0)) {
		tmp = -Math.cos(B) / (Math.sin(B) / x);
	} else {
		tmp = (1.0 / Math.sin(B)) + (x * (-1.0 / B));
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -4e+17) or not (x <= 14500000.0):
		tmp = -math.cos(B) / (math.sin(B) / x)
	else:
		tmp = (1.0 / math.sin(B)) + (x * (-1.0 / B))
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -4e+17) || !(x <= 14500000.0))
		tmp = Float64(Float64(-cos(B)) / Float64(sin(B) / x));
	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 <= -4e+17) || ~((x <= 14500000.0)))
		tmp = -cos(B) / (sin(B) / x);
	else
		tmp = (1.0 / sin(B)) + (x * (-1.0 / B));
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -4e+17], N[Not[LessEqual[x, 14500000.0]], $MachinePrecision]], N[((-N[Cos[B], $MachinePrecision]) / N[(N[Sin[B], $MachinePrecision] / x), $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 < -4e17 or 1.45e7 < x

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unsub-neg 99.6%

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

      3. associate-*r/ 99.8%

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

      4. *-rgt-identity 99.8%

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

   3. Simplified 99.8%

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

      1. tan-quot 99.7%

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

      2. associate-/r/ 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. associate-/r/ 99.7%

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

      2. tan-quot 99.8%

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

      3. clear-num 99.7%

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

      4. div-inv 99.7%

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

      5. associate-/r* 99.6%

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

   7. Applied egg-rr 99.6%

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

   8. Taylor expanded in x around inf 99.7%

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

      1. mul-1-neg 99.7%

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

      2. associate-/l* 99.6%

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

   10. Simplified 99.6%

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

   if -4e17 < x < 1.45e7

   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 98.0%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+17} \lor \neg \left(x \leq 14500000\right):\\ \;\;\;\;\frac{-\cos B}{\frac{\sin B}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} + x \cdot \frac{-1}{B}\\ \end{array} \]

Alternative 3: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+17} \lor \neg \left(x \leq 34000\right):\\ \;\;\;\;\frac{x \cdot \left(-\cos B\right)}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} + x \cdot \frac{-1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -4e+17) (not (<= x 34000.0)))
   (/ (* x (- (cos B))) (sin B))
   (+ (/ 1.0 (sin B)) (* x (/ -1.0 B)))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -4e+17) || !(x <= 34000.0)) {
		tmp = (x * -cos(B)) / sin(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 <= (-4d+17)) .or. (.not. (x <= 34000.0d0))) then
        tmp = (x * -cos(b)) / sin(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 <= -4e+17) || !(x <= 34000.0)) {
		tmp = (x * -Math.cos(B)) / Math.sin(B);
	} else {
		tmp = (1.0 / Math.sin(B)) + (x * (-1.0 / B));
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -4e+17) or not (x <= 34000.0):
		tmp = (x * -math.cos(B)) / math.sin(B)
	else:
		tmp = (1.0 / math.sin(B)) + (x * (-1.0 / B))
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -4e+17) || !(x <= 34000.0))
		tmp = Float64(Float64(x * Float64(-cos(B))) / sin(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 <= -4e+17) || ~((x <= 34000.0)))
		tmp = (x * -cos(B)) / sin(B);
	else
		tmp = (1.0 / sin(B)) + (x * (-1.0 / B));
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -4e+17], N[Not[LessEqual[x, 34000.0]], $MachinePrecision]], N[(N[(x * (-N[Cos[B], $MachinePrecision])), $MachinePrecision] / N[Sin[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 < -4e17 or 34000 < x

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unsub-neg 99.6%

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

      3. associate-*r/ 99.8%

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

      4. *-rgt-identity 99.8%

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

   3. Simplified 99.8%

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

      1. tan-quot 99.7%

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

      2. associate-/r/ 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. associate-/r/ 99.7%

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

      2. tan-quot 99.8%

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

      3. clear-num 99.7%

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

      4. div-inv 99.7%

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

      5. associate-/r* 99.6%

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

   7. Applied egg-rr 99.6%

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

   8. Taylor expanded in x around inf 99.7%

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

      1. associate-*r/ 99.7%

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

      2. *-commutative 99.7%

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

      3. associate-*r* 99.7%

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

      4. neg-mul-1 99.7%

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

   10. Simplified 99.7%

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

   if -4e17 < x < 34000

   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 98.0%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+17} \lor \neg \left(x \leq 34000\right):\\ \;\;\;\;\frac{x \cdot \left(-\cos B\right)}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} + x \cdot \frac{-1}{B}\\ \end{array} \]

Alternative 4: 86.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+18} \lor \neg \left(x \leq 1.3 \cdot 10^{+43}\right):\\ \;\;\;\;\left(\frac{1}{B} + B \cdot 0.16666666666666666\right) - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} + x \cdot \frac{-1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -1.6e+18) (not (<= x 1.3e+43)))
   (- (+ (/ 1.0 B) (* B 0.16666666666666666)) (/ x (tan B)))
   (+ (/ 1.0 (sin B)) (* x (/ -1.0 B)))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -1.6e+18) || !(x <= 1.3e+43)) {
		tmp = ((1.0 / B) + (B * 0.16666666666666666)) - (x / 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 <= (-1.6d+18)) .or. (.not. (x <= 1.3d+43))) then
        tmp = ((1.0d0 / b) + (b * 0.16666666666666666d0)) - (x / 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 <= -1.6e+18) || !(x <= 1.3e+43)) {
		tmp = ((1.0 / B) + (B * 0.16666666666666666)) - (x / 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 <= -1.6e+18) or not (x <= 1.3e+43):
		tmp = ((1.0 / B) + (B * 0.16666666666666666)) - (x / 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 <= -1.6e+18) || !(x <= 1.3e+43))
		tmp = Float64(Float64(Float64(1.0 / B) + Float64(B * 0.16666666666666666)) - Float64(x / 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 <= -1.6e+18) || ~((x <= 1.3e+43)))
		tmp = ((1.0 / B) + (B * 0.16666666666666666)) - (x / 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, -1.6e+18], N[Not[LessEqual[x, 1.3e+43]], $MachinePrecision]], N[(N[(N[(1.0 / B), $MachinePrecision] + N[(B * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[(x * N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.6e18 or 1.3000000000000001e43 < x

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unsub-neg 99.6%

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

      3. associate-*r/ 99.8%

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

      4. *-rgt-identity 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in B around 0 80.8%

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

   if -1.6e18 < x < 1.3000000000000001e43

   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 95.5%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{+18} \lor \neg \left(x \leq 1.3 \cdot 10^{+43}\right):\\ \;\;\;\;\left(\frac{1}{B} + B \cdot 0.16666666666666666\right) - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} + x \cdot \frac{-1}{B}\\ \end{array} \]

Alternative 5: 77.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq -0.068 \lor \neg \left(B \leq 0.05\right):\\ \;\;\;\;\frac{1 + x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= B -0.068) (not (<= B 0.05)))
   (/ (+ 1.0 x) (sin B))
   (+ (* B (+ 0.16666666666666666 (* x 0.3333333333333333))) (/ (- 1.0 x) B))))

C

double code(double B, double x) {
	double tmp;
	if ((B <= -0.068) || !(B <= 0.05)) {
		tmp = (1.0 + x) / sin(B);
	} else {
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - 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 <= (-0.068d0)) .or. (.not. (b <= 0.05d0))) then
        tmp = (1.0d0 + x) / sin(b)
    else
        tmp = (b * (0.16666666666666666d0 + (x * 0.3333333333333333d0))) + ((1.0d0 - x) / b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((B <= -0.068) || !(B <= 0.05)) {
		tmp = (1.0 + x) / Math.sin(B);
	} else {
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (B <= -0.068) or not (B <= 0.05):
		tmp = (1.0 + x) / math.sin(B)
	else:
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((B <= -0.068) || !(B <= 0.05))
		tmp = Float64(Float64(1.0 + x) / sin(B));
	else
		tmp = Float64(Float64(B * Float64(0.16666666666666666 + Float64(x * 0.3333333333333333))) + Float64(Float64(1.0 - x) / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((B <= -0.068) || ~((B <= 0.05)))
		tmp = (1.0 + x) / sin(B);
	else
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[B, -0.068], N[Not[LessEqual[B, 0.05]], $MachinePrecision]], N[(N[(1.0 + x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(B * N[(0.16666666666666666 + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < -0.068000000000000005 or 0.050000000000000003 < 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 B around 0 54.0%

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

      1. +-commutative 54.0%

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

      2. un-div-inv 54.0%

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

      3. frac-add 54.0%

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

      4. *-un-lft-identity 54.0%

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

      5. add-sqr-sqrt 25.2%

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

      6. sqrt-unprod 54.4%

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

      7. sqr-neg 54.4%

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

      8. sqrt-unprod 29.2%

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

      9. add-sqr-sqrt 54.5%

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

   6. Applied egg-rr 54.5%

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

   7. Taylor expanded in B around 0 56.2%

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

   8. Taylor expanded in B around inf 57.7%

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

   if -0.068000000000000005 < B < 0.050000000000000003

   1. Initial program 99.9%

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

      1. distribute-lft-neg-in 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in B around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. associate--l+ 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. div-sub 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -0.068 \lor \neg \left(B \leq 0.05\right):\\ \;\;\;\;\frac{1 + x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \end{array} \]

Alternative 6: 75.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq -0.059:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{elif}\;B \leq 0.175:\\ \;\;\;\;B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{B}{B \cdot \sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B -0.059)
   (/ 1.0 (sin B))
   (if (<= B 0.175)
     (+ (* B (+ 0.16666666666666666 (* x 0.3333333333333333))) (/ (- 1.0 x) B))
     (/ B (* B (sin B))))))

C

double code(double B, double x) {
	double tmp;
	if (B <= -0.059) {
		tmp = 1.0 / sin(B);
	} else if (B <= 0.175) {
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B);
	} else {
		tmp = B / (B * sin(B));
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (b <= (-0.059d0)) then
        tmp = 1.0d0 / sin(b)
    else if (b <= 0.175d0) then
        tmp = (b * (0.16666666666666666d0 + (x * 0.3333333333333333d0))) + ((1.0d0 - x) / b)
    else
        tmp = b / (b * sin(b))
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (B <= -0.059) {
		tmp = 1.0 / Math.sin(B);
	} else if (B <= 0.175) {
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B);
	} else {
		tmp = B / (B * Math.sin(B));
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if B <= -0.059:
		tmp = 1.0 / math.sin(B)
	elif B <= 0.175:
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B)
	else:
		tmp = B / (B * math.sin(B))
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (B <= -0.059)
		tmp = Float64(1.0 / sin(B));
	elseif (B <= 0.175)
		tmp = Float64(Float64(B * Float64(0.16666666666666666 + Float64(x * 0.3333333333333333))) + Float64(Float64(1.0 - x) / B));
	else
		tmp = Float64(B / Float64(B * sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (B <= -0.059)
		tmp = 1.0 / sin(B);
	elseif (B <= 0.175)
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B);
	else
		tmp = B / (B * sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[B, -0.059], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 0.175], N[(N[(B * N[(0.16666666666666666 + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], N[(B / N[(B * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < -0.058999999999999997

   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. Taylor expanded in x around 0 56.7%

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

   if -0.058999999999999997 < B < 0.17499999999999999

   1. Initial program 99.9%

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

      1. distribute-lft-neg-in 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in B around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. associate--l+ 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. div-sub 100.0%

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

   6. Simplified 100.0%

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

   if 0.17499999999999999 < 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 B around 0 49.7%

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

      1. +-commutative 49.7%

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

      2. un-div-inv 49.7%

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

      3. frac-add 49.7%

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

      4. *-un-lft-identity 49.7%

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

      5. add-sqr-sqrt 24.0%

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

      6. sqrt-unprod 50.7%

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

      7. sqr-neg 50.7%

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

      8. sqrt-unprod 26.5%

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

      9. add-sqr-sqrt 50.4%

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

   6. Applied egg-rr 50.4%

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

   7. Taylor expanded in B around inf 49.7%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -0.059:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{elif}\;B \leq 0.175:\\ \;\;\;\;B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{B}{B \cdot \sin B}\\ \end{array} \]

Alternative 7: 76.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[B_, x_] := N[(N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision] / N[(N[Sin[B], $MachinePrecision] / x), $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. Step-by-step derivation

   1. tan-quot 99.8%

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

   2. associate-/r/ 99.8%

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

5. Applied egg-rr 99.8%

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

   1. associate-/r/ 99.8%

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

   2. tan-quot 99.8%

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

   3. clear-num 99.8%

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

   4. div-inv 99.8%

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

   5. associate-/r* 99.7%

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

7. Applied egg-rr 99.7%

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

   1. frac-sub 99.6%

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

   2. *-un-lft-identity 99.6%

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

9. Applied egg-rr 99.6%

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

    1. associate-*r/ 99.6%

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

    2. *-rgt-identity 99.6%

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

    3. associate-*r/ 99.5%

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

    4. *-rgt-identity 99.5%

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

11. Simplified 99.5%

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

12. Taylor expanded in B around 0 77.6%

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

13. Final simplification 77.6%

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

Alternative 8: 75.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq -0.034 \lor \neg \left(B \leq 0.0255\right):\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= B -0.034) (not (<= B 0.0255)))
   (/ 1.0 (sin B))
   (+ (* B (+ 0.16666666666666666 (* x 0.3333333333333333))) (/ (- 1.0 x) B))))

C

double code(double B, double x) {
	double tmp;
	if ((B <= -0.034) || !(B <= 0.0255)) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - 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 <= (-0.034d0)) .or. (.not. (b <= 0.0255d0))) then
        tmp = 1.0d0 / sin(b)
    else
        tmp = (b * (0.16666666666666666d0 + (x * 0.3333333333333333d0))) + ((1.0d0 - x) / b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((B <= -0.034) || !(B <= 0.0255)) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (B <= -0.034) or not (B <= 0.0255):
		tmp = 1.0 / math.sin(B)
	else:
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((B <= -0.034) || !(B <= 0.0255))
		tmp = Float64(1.0 / sin(B));
	else
		tmp = Float64(Float64(B * Float64(0.16666666666666666 + Float64(x * 0.3333333333333333))) + Float64(Float64(1.0 - x) / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((B <= -0.034) || ~((B <= 0.0255)))
		tmp = 1.0 / sin(B);
	else
		tmp = (B * (0.16666666666666666 + (x * 0.3333333333333333))) + ((1.0 - x) / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[B, -0.034], N[Not[LessEqual[B, 0.0255]], $MachinePrecision]], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(B * N[(0.16666666666666666 + N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < -0.034000000000000002 or 0.0254999999999999984 < 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 53.9%

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

   if -0.034000000000000002 < B < 0.0254999999999999984

   1. Initial program 99.9%

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

      1. distribute-lft-neg-in 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in B around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. mul-1-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. associate--l+ 100.0%

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

      5. *-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. div-sub 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -0.034 \lor \neg \left(B \leq 0.0255\right):\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;B \cdot \left(0.16666666666666666 + x \cdot 0.3333333333333333\right) + \frac{1 - x}{B}\\ \end{array} \]

Alternative 9: 51.1% 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 51.3%

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

   1. +-commutative 51.3%

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

   2. mul-1-neg 51.3%

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

   3. sub-neg 51.3%

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

   4. associate--l+ 51.3%

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

   5. *-commutative 51.3%

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

   6. *-commutative 51.3%

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

   7. div-sub 51.3%

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

6. Simplified 51.3%

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

7. Final simplification 51.3%

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

Alternative 10: 49.8% accurate, 23.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.5 \lor \neg \left(x \leq 15.5\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -0.5) (not (<= x 15.5))) (/ (- x) B) (/ (+ 1.0 x) B)))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -0.5) || !(x <= 15.5)) {
		tmp = -x / B;
	} else {
		tmp = (1.0 + 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 <= (-0.5d0)) .or. (.not. (x <= 15.5d0))) then
        tmp = -x / b
    else
        tmp = (1.0d0 + x) / b
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -0.5) || !(x <= 15.5)) {
		tmp = -x / B;
	} else {
		tmp = (1.0 + x) / B;
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -0.5) or not (x <= 15.5):
		tmp = -x / B
	else:
		tmp = (1.0 + x) / B
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -0.5) || !(x <= 15.5))
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(1.0 + x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -0.5) || ~((x <= 15.5)))
		tmp = -x / B;
	else
		tmp = (1.0 + x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -0.5], N[Not[LessEqual[x, 15.5]], $MachinePrecision]], N[((-x) / B), $MachinePrecision], N[(N[(1.0 + x), $MachinePrecision] / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.5 or 15.5 < x

   1. Initial program 99.6%

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

      1. distribute-lft-neg-in 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in B around 0 51.1%

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

   5. Taylor expanded in B around 0 51.5%

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

   6. Taylor expanded in x around inf 50.2%

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

      1. associate-*r/ 50.2%

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

      2. neg-mul-1 50.2%

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

   8. Simplified 50.2%

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

   if -0.5 < x < 15.5

   1. Initial program 99.9%

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

      1. distribute-lft-neg-in 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in B around 0 98.6%

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

      1. +-commutative 98.6%

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

      2. un-div-inv 98.6%

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

      3. frac-add 75.4%

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

      4. *-un-lft-identity 75.4%

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

      5. add-sqr-sqrt 40.2%

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

      6. sqrt-unprod 75.4%

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

      7. sqr-neg 75.4%

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

      8. sqrt-unprod 35.2%

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

      9. add-sqr-sqrt 74.4%

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

   6. Applied egg-rr 74.4%

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

   7. Taylor expanded in B around 0 48.6%

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

      1. +-commutative 48.6%

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

   9. Simplified 48.6%

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

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.5 \lor \neg \left(x \leq 15.5\right):\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x}{B}\\ \end{array} \]

Alternative 11: 51.0% 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. 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 76.3%

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

5. Taylor expanded in B around 0 51.1%

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

   1. *-commutative 51.1%

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

   2. fma-def 51.1%

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

   3. +-commutative 51.1%

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

   4. neg-mul-1 51.1%

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

   5. unsub-neg 51.1%

      \[\leadsto \mathsf{fma}\left(B, 0.16666666666666666, \color{blue}{\frac{1}{B} - \frac{x}{B}}\right) \]

7. Simplified 51.1%

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

   1. fma-udef 51.1%

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

   2. *-commutative 51.1%

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

   3. sub-neg 51.1%

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

   4. mul-1-neg 51.1%

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

   5. +-commutative 51.1%

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

   6. +-commutative 51.1%

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

   7. +-commutative 51.1%

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

   8. mul-1-neg 51.1%

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

   9. sub-neg 51.1%

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

   10. sub-div 51.1%

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

   11. *-commutative 51.1%

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

9. Applied egg-rr 51.1%

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

10. Final simplification 51.1%

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

Alternative 12: 50.8% 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 50.7%

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

   1. mul-1-neg 50.7%

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

   2. sub-neg 50.7%

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

6. Simplified 50.7%

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

7. Final simplification 50.7%

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

Alternative 13: 26.0% 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. 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 76.3%

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

5. Taylor expanded in B around 0 51.1%

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

6. Taylor expanded in x around inf 25.4%

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

   1. associate-*r/ 25.4%

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

   2. neg-mul-1 25.4%

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

8. Simplified 25.4%

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

9. Final simplification 25.4%

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

Alternative 14: 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. 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 76.3%

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

5. Taylor expanded in B around 0 51.1%

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

6. Taylor expanded in B around inf 3.5%

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

   1. *-commutative 3.5%

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

8. Simplified 3.5%

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

9. Final simplification 3.5%

   \[\leadsto B \cdot 0.16666666666666666 \]

Reproduce

herbie shell --seed 2023214 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))