VandenBroeck and Keller, Equation (24)

Percentage Accurate: 99.7% → 99.8%

Time: 9.8s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. +-commutative 99.7%

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

   2. distribute-lft-neg-in 99.7%

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

   3. cancel-sign-sub 99.7%

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

   4. associate-*r/ 99.8%

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

   5. remove-double-neg 99.8%

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

   6. associate-/l* 99.8%

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

   7. /-rgt-identity 99.8%

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

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: 98.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -250000 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-\sin B} \cdot \left(x + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -250000.0) (not (<= x 1.0)))
   (- (/ 1.0 B) (/ x (tan B)))
   (* (/ 1.0 (- (sin B))) (+ x -1.0))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -250000.0) || !(x <= 1.0)) {
		tmp = (1.0 / B) - (x / tan(B));
	} else {
		tmp = (1.0 / -sin(B)) * (x + -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-250000.0d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = (1.0d0 / b) - (x / tan(b))
    else
        tmp = (1.0d0 / -sin(b)) * (x + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -250000.0) || !(x <= 1.0)) {
		tmp = (1.0 / B) - (x / Math.tan(B));
	} else {
		tmp = (1.0 / -Math.sin(B)) * (x + -1.0);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -250000.0) or not (x <= 1.0):
		tmp = (1.0 / B) - (x / math.tan(B))
	else:
		tmp = (1.0 / -math.sin(B)) * (x + -1.0)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -250000.0) || !(x <= 1.0))
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(1.0 / Float64(-sin(B))) * Float64(x + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -250000.0) || ~((x <= 1.0)))
		tmp = (1.0 / B) - (x / tan(B));
	else
		tmp = (1.0 / -sin(B)) * (x + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -250000.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / (-N[Sin[B], $MachinePrecision])), $MachinePrecision] * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.5e5 or 1 < x

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. distribute-lft-neg-in 99.6%

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

      3. cancel-sign-sub 99.6%

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

      4. associate-*r/ 99.9%

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

      5. remove-double-neg 99.9%

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

      6. associate-/l* 99.9%

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

      7. /-rgt-identity 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in B around 0 76.9%

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

   5. Taylor expanded in B around 0 98.9%

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

   if -2.5e5 < 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. +-commutative 99.8%

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

      2. distribute-lft-neg-in 99.8%

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

      3. cancel-sign-sub 99.8%

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

      4. associate-*r/ 99.8%

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

      5. remove-double-neg 99.8%

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

      6. associate-/l* 99.8%

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

      7. /-rgt-identity 99.8%

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

   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. Taylor expanded in B around inf 99.8%

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

      1. div-sub 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in B around 0 98.2%

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

       1. frac-2neg 98.2%

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

       2. clear-num 98.2%

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

       3. associate-/r/ 98.2%

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

       4. neg-sub0 98.2%

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

       5. associate--r- 98.2%

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

       6. metadata-eval 98.2%

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

       7. +-commutative 98.2%

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

   11. Applied egg-rr 98.2%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -250000 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-\sin B} \cdot \left(x + -1\right)\\ \end{array} \]

Alternative 3: 98.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -250000 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (or (<= x -250000.0) (not (<= x 1.0)))
   (- (/ 1.0 B) (/ x (tan B)))
   (/ (- 1.0 x) (sin B))))

C

double code(double B, double x) {
	double tmp;
	if ((x <= -250000.0) || !(x <= 1.0)) {
		tmp = (1.0 / B) - (x / tan(B));
	} else {
		tmp = (1.0 - x) / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-250000.0d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = (1.0d0 / b) - (x / tan(b))
    else
        tmp = (1.0d0 - x) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if ((x <= -250000.0) || !(x <= 1.0)) {
		tmp = (1.0 / B) - (x / Math.tan(B));
	} else {
		tmp = (1.0 - x) / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if (x <= -250000.0) or not (x <= 1.0):
		tmp = (1.0 / B) - (x / math.tan(B))
	else:
		tmp = (1.0 - x) / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if ((x <= -250000.0) || !(x <= 1.0))
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(1.0 - x) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if ((x <= -250000.0) || ~((x <= 1.0)))
		tmp = (1.0 / B) - (x / tan(B));
	else
		tmp = (1.0 - x) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[Or[LessEqual[x, -250000.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.5e5 or 1 < x

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. distribute-lft-neg-in 99.6%

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

      3. cancel-sign-sub 99.6%

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

      4. associate-*r/ 99.9%

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

      5. remove-double-neg 99.9%

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

      6. associate-/l* 99.9%

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

      7. /-rgt-identity 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in B around 0 76.9%

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

   5. Taylor expanded in B around 0 98.9%

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

   if -2.5e5 < 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. +-commutative 99.8%

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

      2. distribute-lft-neg-in 99.8%

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

      3. cancel-sign-sub 99.8%

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

      4. associate-*r/ 99.8%

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

      5. remove-double-neg 99.8%

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

      6. associate-/l* 99.8%

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

      7. /-rgt-identity 99.8%

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

   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. Taylor expanded in B around inf 99.8%

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

      1. div-sub 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in B around 0 98.2%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -250000 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \]

Alternative 4: 64.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 13.5:\\ \;\;\;\;0.16666666666666666 \cdot \left(B \cdot \left(1 - x\right)\right) + \frac{1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B 13.5)
   (+ (* 0.16666666666666666 (* B (- 1.0 x))) (/ (- 1.0 x) B))
   (/ (+ 1.0 x) (sin B))))

C

double code(double B, double x) {
	double tmp;
	if (B <= 13.5) {
		tmp = (0.16666666666666666 * (B * (1.0 - x))) + ((1.0 - x) / B);
	} else {
		tmp = (1.0 + x) / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (b <= 13.5d0) then
        tmp = (0.16666666666666666d0 * (b * (1.0d0 - x))) + ((1.0d0 - x) / b)
    else
        tmp = (1.0d0 + x) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (B <= 13.5) {
		tmp = (0.16666666666666666 * (B * (1.0 - x))) + ((1.0 - x) / B);
	} else {
		tmp = (1.0 + x) / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if B <= 13.5:
		tmp = (0.16666666666666666 * (B * (1.0 - x))) + ((1.0 - x) / B)
	else:
		tmp = (1.0 + x) / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (B <= 13.5)
		tmp = Float64(Float64(0.16666666666666666 * Float64(B * Float64(1.0 - x))) + Float64(Float64(1.0 - x) / B));
	else
		tmp = Float64(Float64(1.0 + x) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (B <= 13.5)
		tmp = (0.16666666666666666 * (B * (1.0 - x))) + ((1.0 - x) / B);
	else
		tmp = (1.0 + x) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[B, 13.5], N[(N[(0.16666666666666666 * N[(B * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 13.5

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. distribute-lft-neg-in 99.8%

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

      3. cancel-sign-sub 99.8%

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

      4. associate-*r/ 99.9%

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

      5. remove-double-neg 99.9%

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

      6. associate-/l* 99.9%

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

      7. /-rgt-identity 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{\tan B}} \]
   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. Taylor expanded in B around inf 99.8%

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

      1. div-sub 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in B around 0 86.0%

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

   10. Taylor expanded in B around 0 69.0%

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

       1. associate--l+ 69.0%

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

       2. div-sub 69.0%

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

   12. Simplified 69.0%

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

   if 13.5 < B

   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. distribute-lft-neg-in 99.6%

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

      3. cancel-sign-sub 99.6%

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

      4. associate-*r/ 99.8%

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

      5. remove-double-neg 99.8%

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

      6. associate-/l* 99.8%

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

      7. /-rgt-identity 99.8%

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

   3. Simplified 99.8%

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

      1. tan-quot 99.6%

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

      2. associate-/r/ 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in B around inf 99.6%

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

      1. div-sub 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in B around 0 54.5%

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

       1. expm1-log1p-u 29.1%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 - x}{\sin B}\right)\right)} \]

       2. expm1-udef 28.8%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1 - x}{\sin B}\right)} - 1} \]

       3. sub-neg 28.8%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{1 + \left(-x\right)}}{\sin B}\right)} - 1 \]

       4. add-sqr-sqrt 13.8%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{1 + \color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}{\sin B}\right)} - 1 \]

       5. sqrt-unprod 28.0%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{1 + \color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}}{\sin B}\right)} - 1 \]

       6. sqr-neg 28.0%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{1 + \sqrt{\color{blue}{x \cdot x}}}{\sin B}\right)} - 1 \]

       7. sqrt-unprod 15.4%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{1 + \color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sin B}\right)} - 1 \]

       8. add-sqr-sqrt 29.5%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{1 + \color{blue}{x}}{\sin B}\right)} - 1 \]

   11. Applied egg-rr 29.5%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1 + x}{\sin B}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 29.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 + x}{\sin B}\right)\right)} \]

       2. expm1-log1p 53.5%

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

   13. Simplified 53.5%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 13.5:\\ \;\;\;\;0.16666666666666666 \cdot \left(B \cdot \left(1 - x\right)\right) + \frac{1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + x}{\sin B}\\ \end{array} \]

Alternative 5: 63.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 13.5:\\ \;\;\;\;0.16666666666666666 \cdot \left(B \cdot \left(1 - x\right)\right) + \frac{1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (B x)
 :precision binary64
 (if (<= B 13.5)
   (+ (* 0.16666666666666666 (* B (- 1.0 x))) (/ (- 1.0 x) B))
   (/ 1.0 (sin B))))

C

double code(double B, double x) {
	double tmp;
	if (B <= 13.5) {
		tmp = (0.16666666666666666 * (B * (1.0 - x))) + ((1.0 - x) / B);
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(b, x)
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (b <= 13.5d0) then
        tmp = (0.16666666666666666d0 * (b * (1.0d0 - x))) + ((1.0d0 - x) / b)
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double B, double x) {
	double tmp;
	if (B <= 13.5) {
		tmp = (0.16666666666666666 * (B * (1.0 - x))) + ((1.0 - x) / B);
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(B, x):
	tmp = 0
	if B <= 13.5:
		tmp = (0.16666666666666666 * (B * (1.0 - x))) + ((1.0 - x) / B)
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(B, x)
	tmp = 0.0
	if (B <= 13.5)
		tmp = Float64(Float64(0.16666666666666666 * Float64(B * Float64(1.0 - x))) + Float64(Float64(1.0 - x) / B));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(B, x)
	tmp = 0.0;
	if (B <= 13.5)
		tmp = (0.16666666666666666 * (B * (1.0 - x))) + ((1.0 - x) / B);
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[B_, x_] := If[LessEqual[B, 13.5], N[(N[(0.16666666666666666 * N[(B * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 13.5

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. distribute-lft-neg-in 99.8%

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

      3. cancel-sign-sub 99.8%

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

      4. associate-*r/ 99.9%

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

      5. remove-double-neg 99.9%

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

      6. associate-/l* 99.9%

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

      7. /-rgt-identity 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{\tan B}} \]
   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. Taylor expanded in B around inf 99.8%

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

      1. div-sub 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in B around 0 86.0%

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

   10. Taylor expanded in B around 0 69.0%

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

       1. associate--l+ 69.0%

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

       2. div-sub 69.0%

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

   12. Simplified 69.0%

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

   if 13.5 < B

   1. Initial program 99.6%

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

      1. distribute-rgt-neg-in 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 50.4%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 13.5:\\ \;\;\;\;0.16666666666666666 \cdot \left(B \cdot \left(1 - x\right)\right) + \frac{1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

Alternative 6: 76.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. +-commutative 99.7%

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

   2. distribute-lft-neg-in 99.7%

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

   3. cancel-sign-sub 99.7%

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

   4. associate-*r/ 99.8%

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

   5. remove-double-neg 99.8%

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

   6. associate-/l* 99.8%

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

   7. /-rgt-identity 99.8%

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

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. Taylor expanded in B around inf 99.8%

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

   1. div-sub 99.8%

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

8. Simplified 99.8%

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

9. Taylor expanded in B around 0 79.4%

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

10. Final simplification 79.4%

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

Alternative 7: 51.8% accurate, 19.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. distribute-rgt-neg-in 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in B around 0 55.1%

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

   1. +-commutative 55.1%

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

   2. mul-1-neg 55.1%

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

   3. sub-neg 55.1%

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

   4. associate--l+ 55.1%

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

   5. *-commutative 55.1%

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

   6. div-sub 55.1%

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

6. Simplified 55.1%

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

7. Taylor expanded in x around inf 55.3%

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

8. Final simplification 55.3%

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

Alternative 8: 51.6% accurate, 23.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

3. Simplified 99.7%

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

4. Taylor expanded in B around 0 77.7%

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

5. Taylor expanded in B around 0 55.2%

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

6. Taylor expanded in B around -inf 55.2%

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

   1. +-commutative 55.2%

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

   2. *-commutative 55.2%

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

   3. associate-*r/ 55.2%

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

   4. sub-neg 55.2%

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

   5. metadata-eval 55.2%

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

   6. distribute-lft-in 55.2%

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

   7. metadata-eval 55.2%

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

   8. +-commutative 55.2%

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

   9. mul-1-neg 55.2%

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

   10. sub-neg 55.2%

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

8. Simplified 55.2%

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

9. Final simplification 55.2%

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

Alternative 9: 51.5% 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-rgt-neg-in 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in B around 0 55.2%

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

   1. mul-1-neg 55.2%

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

   2. sub-neg 55.2%

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

6. Simplified 55.2%

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

7. Final simplification 55.2%

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

Alternative 10: 26.8% 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-rgt-neg-in 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in B around 0 77.7%

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

5. Taylor expanded in B around 0 55.2%

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

6. Taylor expanded in x around inf 31.6%

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

   1. associate-*r/ 31.6%

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

   2. mul-1-neg 31.6%

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

8. Simplified 31.6%

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

9. Final simplification 31.6%

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

Alternative 11: 3.1% accurate, 70.0× speedup

Math

\[\begin{array}{l} \\ B \cdot 0.16666666666666666 \end{array} \]

FPCore

(FPCore (B x) :precision binary64 (* B 0.16666666666666666))

C

double code(double B, double x) {
	return B * 0.16666666666666666;
}

Fortran

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

Java

public static double code(double B, double x) {
	return B * 0.16666666666666666;
}

Python

def code(B, x):
	return B * 0.16666666666666666

Julia

function code(B, x)
	return Float64(B * 0.16666666666666666)
end

MATLAB

function tmp = code(B, x)
	tmp = B * 0.16666666666666666;
end

Wolfram

code[B_, x_] := N[(B * 0.16666666666666666), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

   1. distribute-rgt-neg-in 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in B around 0 77.7%

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

5. Taylor expanded in B around 0 55.2%

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

6. Taylor expanded in B around inf 2.9%

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

   1. *-commutative 2.9%

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

8. Simplified 2.9%

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

9. Final simplification 2.9%

   \[\leadsto B \cdot 0.16666666666666666 \]

Reproduce

herbie shell --seed 2023336 
(FPCore (B x)
  :name "VandenBroeck and Keller, Equation (24)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))